Subgroups — C₂×C₇×#2976 #13237440

Order 1792.

2306 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#525	C₂	1	1	yes	maximal, characteristic
2	C₇×#2976	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₇×#436	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#435	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#435	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×C₈×D₄	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#433	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#408	C₂	1	2	yes	maximal, complemented
9	C₂×C₇×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₇×M₁₆	C₂²	1	2	yes	complemented
16	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₂³×C₇×C₈	C₂²	1	2	yes	complemented
19	C₇×#525	C₄	1	8	yes	—
20	C₇×#525	C₂²	1	8	yes	—
21	C₂×C₇×#123	C₄	1	2	yes	—
22	C₂×C₇×C₂²⋊Q₈	C₄	1	1	yes	characteristic
23	C₂×C₇×C₂²⋊Q₈	C₄	1	1	yes	characteristic
24	C₂×C₇×#121	C₄	1	1	yes	characteristic
25	C₂×C₇×#121	C₄	1	1	yes	characteristic
26	C₂×C₇×#120	C₄	1	2	yes	—
27	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
29	C₂×C₇×#117	C₂²	1	1	yes	characteristic
30	C₇×#436	C₂²	1	16	yes	complemented
31	C₇×#435	C₂²	1	16	yes	complemented
32	C₇×#435	C₂²	1	16	yes	complemented
33	C₇×C₈×D₄	C₂²	1	16	yes	complemented
34	C₇×#433	C₂²	1	16	yes	complemented
35	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₇×#408	C₂²	1	16	yes	complemented
38	C₇×#407	C₂²	1	8	yes	complemented
39	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₇×C₂²⋊C₈	C₂²	1	8	yes	complemented
44	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
45	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
46	C₂×C₇×#97	C₂²	1	2	yes	complemented
47	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
48	C₂×#2976	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₇×#41	—	4	2	no	—
51	C₂×C₇×#41	—	4	2	no	—
52	C₂×C₇×#41	C₂×C₄	1	4	yes	—
53	C₂×C₇×#41	C₂³	1	4	yes	—
54	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
56	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
57	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
59	C₂³×C₄×C₇	—	4	2	no	—
60	C₂³×C₄×C₇	—	4	2	no	—
61	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
65	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₇	C₂³	1	4	yes	—
70	C₂×C₇×M₁₆	—	4	2	no	—
71	C₂×C₇×M₁₆	—	4	2	no	—
72	C₂×C₇×M₁₆	—	4	4	no	—
73	C₂×C₇×M₁₆	C₂³	1	4	yes	—
74	C₂×C₇×M₁₆	C₂³	1	4	yes	—
75	C₂×C₇×M₁₆	C₂³	1	8	yes	complemented
76	C₂²×C₇×C₈	—	4	2	no	—
77	C₂²×C₇×C₈	—	4	2	no	—
78	C₂²×C₇×C₈	—	4	4	no	—
79	C₂²×C₇×C₈	C₂³	1	2	yes	—
80	C₂²×C₇×C₈	C₂³	1	2	yes	—
81	C₂²×C₇×C₈	C₂³	1	2	yes	—
82	C₂²×C₇×C₈	C₂³	1	2	yes	—
83	C₂²×C₇×C₈	C₂³	1	4	yes	—
84	C₂²×C₇×C₈	C₂³	1	4	yes	—
85	C₂²×C₇×C₈	C₂³	1	8	yes	complemented
86	C₇×#123	C₂×C₄	1	16	yes	—
87	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
88	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
89	C₇×#121	C₂×C₄	1	8	yes	—
90	C₇×#121	C₂×C₄	1	8	yes	—
91	C₇×#120	C₂×C₄	1	16	yes	—
92	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
93	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
94	C₄×C₇×D₄	C₂³	1	8	yes	—
95	C₄×C₇×D₄	C₂³	1	8	yes	—
96	C₇×#117	C₂×C₄	1	4	yes	—
97	C₇×#117	C₂³	1	4	yes	—
98	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
99	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
100	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
101	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
102	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
103	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
104	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
105	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
106	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
107	C₂×C₄²×C₇	C₂³	1	2	yes	—
108	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
109	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
110	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
111	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
112	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
113	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
114	C₇×C₂²⋊C₈	C₂³	1	8	yes	complemented
115	C₇×C₂²⋊C₈	C₂³	1	8	yes	complemented
116	C₇×#97	C₂³	1	8	yes	complemented
117	C₄×C₇×C₈	C₂³	1	8	yes	complemented
118	C₂×#525	C₂×C₇	1	1	yes	characteristic
119	#2976	C₂×C₇	1	16	yes	direct product factor
120	C₂×#436	C₂×C₇	1	2	yes	complemented
121	C₂×#435	C₂×C₇	1	2	yes	complemented
122	C₂×#435	C₂×C₇	1	2	yes	complemented
123	C₂×C₈×D₄	C₂×C₇	1	2	yes	complemented
124	C₂×#433	C₂×C₇	1	2	yes	complemented
125	C₂×#408	C₂×C₇	1	2	yes	complemented
126	C₂×#407	C₂×C₇	1	1	yes	characteristic, complemented
127	C₂²×C₂²⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
128	C₂⁴×C₇	—	2	4	no	—
129	C₂⁴×C₇	—	4	2	no	—
130	C₂⁴×C₇	—	4	2	no	—
131	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
132	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
133	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
134	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
135	C₂⁴×C₇	C₂²×C₄	1	4	yes	—
136	C₇×#41	—	4	8	no	—
137	C₇×#41	—	4	4	no	—
138	C₇×#41	—	4	4	no	—
139	C₂×C₇×Q₈	—	4	2	no	—
140	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
141	C₂×C₇×D₄	—	2	8	no	—
142	C₂×C₇×D₄	—	4	4	no	—
143	C₂×C₇×D₄	—	4	2	no	—
144	C₂×C₇×D₄	—	4	4	no	—
145	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
146	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
147	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
148	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₇	—	2	2	no	—
150	C₂²×C₄×C₇	—	2	2	no	—
151	C₂²×C₄×C₇	—	2	4	no	—
152	C₂²×C₄×C₇	—	2	2	no	—
153	C₂²×C₄×C₇	—	2	2	no	—
154	C₂²×C₄×C₇	—	2	4	no	—
155	C₂²×C₄×C₇	—	2	2	no	—
156	C₂²×C₄×C₇	—	2	2	no	—
157	C₂²×C₄×C₇	—	4	8	no	—
158	C₂²×C₄×C₇	—	4	4	no	—
159	C₂²×C₄×C₇	—	4	4	no	—
160	C₂²×C₄×C₇	—	4	2	no	—
161	C₂²×C₄×C₇	—	4	2	no	—
162	C₂²×C₄×C₇	—	4	2	no	—
163	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
164	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
165	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
168	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
170	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
171	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
172	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
173	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
174	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
175	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
176	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
181	C₇×M₁₆	—	4	4	no	—
182	C₇×M₁₆	—	4	4	no	—
183	C₇×M₁₆	—	4	8	no	—
184	C₂×C₇×C₈	—	2	4	no	—
185	C₂×C₇×C₈	—	2	4	no	—
186	C₂×C₇×C₈	—	2	4	no	—
187	C₂×C₇×C₈	—	2	4	no	—
188	C₂×C₇×C₈	—	4	4	no	—
189	C₂×C₇×C₈	—	4	4	no	—
190	C₂×C₇×C₈	—	4	8	no	—
191	C₂×C₇×C₈	C₂⁴	1	4	yes	—
192	C₂×C₇×C₈	C₂⁴	1	4	yes	—
193	C₂×C₇×C₈	C₂⁴	1	4	yes	—
194	C₂×C₇×C₈	C₂⁴	1	4	yes	—
195	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
196	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
197	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
198	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
199	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
200	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
201	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
202	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
203	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
204	C₄²×C₇	C₂²×C₄	1	4	yes	—
205	C₄²×C₇	C₂⁴	1	4	yes	—
206	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
207	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
208	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
209	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
210	C₂²×M₁₆	C₂²×C₇	1	2	yes	complemented
211	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
212	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
213	C₂³×C₈	C₂²×C₇	1	2	yes	complemented
214	#525	C₄×C₇	1	8	yes	—
215	#525	C₂²×C₇	1	8	yes	—
216	C₂×#123	C₄×C₇	1	2	yes	—
217	C₂×C₂²⋊Q₈	C₄×C₇	1	1	yes	characteristic
218	C₂×C₂²⋊Q₈	C₄×C₇	1	1	yes	characteristic
219	C₂×#121	C₄×C₇	1	1	yes	characteristic
220	C₂×#121	C₄×C₇	1	1	yes	characteristic
221	C₂×#120	C₄×C₇	1	2	yes	—
222	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
223	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
224	C₂×#117	C₂²×C₇	1	1	yes	characteristic
225	#436	C₂²×C₇	1	16	yes	complemented
226	#435	C₂²×C₇	1	16	yes	complemented
227	#435	C₂²×C₇	1	16	yes	complemented
228	C₈×D₄	C₂²×C₇	1	16	yes	complemented
229	#433	C₂²×C₇	1	16	yes	complemented
230	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
231	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
232	#408	C₂²×C₇	1	16	yes	complemented
233	#407	C₂²×C₇	1	8	yes	complemented
234	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
235	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
236	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
237	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
238	C₂×C₂²⋊C₈	C₂²×C₇	1	8	yes	complemented
239	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	complemented
240	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	complemented
241	C₂×#97	C₂²×C₇	1	2	yes	complemented
242	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
243	C₂³×C₇	—	2	2	no	—
244	C₂³×C₇	—	2	2	no	—
245	C₂³×C₇	—	2	2	no	—
246	C₂³×C₇	—	2	2	no	—
247	C₂³×C₇	—	2	4	no	—
248	C₂³×C₇	—	2	4	no	—
249	C₂³×C₇	—	2	8	no	—
250	C₂³×C₇	—	2	2	no	—
251	C₂³×C₇	—	2	2	no	—
252	C₂³×C₇	—	2	2	no	—
253	C₂³×C₇	—	2	4	no	—
254	C₂³×C₇	—	2	2	no	—
255	C₂³×C₇	—	4	4	no	—
256	C₂³×C₇	—	4	4	no	—
257	C₂³×C₇	—	4	16	no	—
258	C₂³×C₇	#131	1	4	yes	—
259	C₂³×C₇	C₂³×C₄	1	2	yes	—
260	C₂³×C₇	C₂³×C₄	1	2	yes	—
261	C₂³×C₇	C₂³×C₄	1	4	yes	—
262	C₂³×C₇	C₂³×C₄	1	2	yes	—
263	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
264	C₇×Q₈	—	4	4	no	—
265	C₇×D₄	—	2	16	no	—
266	C₇×D₄	—	4	8	no	—
267	C₇×D₄	—	4	4	no	—
268	C₇×D₄	—	4	8	no	—
269	C₂×C₄×C₇	—	2	2	no	—
270	C₂×C₄×C₇	—	2	2	no	—
271	C₂×C₄×C₇	—	2	2	no	—
272	C₂×C₄×C₇	—	2	4	no	—
273	C₂×C₄×C₇	—	2	4	no	—
274	C₂×C₄×C₇	—	2	4	no	—
275	C₂×C₄×C₇	—	2	4	no	—
276	C₂×C₄×C₇	—	2	4	no	—
277	C₂×C₄×C₇	—	2	4	no	—
278	C₂×C₄×C₇	—	2	8	no	—
279	C₂×C₄×C₇	—	2	2	no	—
280	C₂×C₄×C₇	—	2	2	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	2	no	—
283	C₂×C₄×C₇	—	2	2	no	—
284	C₂×C₄×C₇	—	2	4	no	—
285	C₂×C₄×C₇	—	2	2	no	—
286	C₂×C₄×C₇	—	2	2	no	—
287	C₂×C₄×C₇	—	4	4	no	—
288	C₂×C₄×C₇	—	4	4	no	—
289	C₂×C₄×C₇	—	4	4	no	—
290	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
291	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
292	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
293	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
294	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
295	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
296	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
297	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
298	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
299	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
300	C₇×C₈	—	2	4	no	—
301	C₇×C₈	—	2	4	no	—
302	C₇×C₈	—	2	4	no	—
303	C₇×C₈	—	2	4	no	—
304	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
305	C₂×#41	—	4	2	no	—
306	C₂×#41	—	4	2	no	—
307	C₂×#41	C₂×C₄×C₇	1	4	yes	—
308	C₂×#41	C₂³×C₇	1	4	yes	—
309	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
310	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
311	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
312	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
313	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
314	C₂³×C₄	—	4	2	no	—
315	C₂³×C₄	—	4	2	no	—
316	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
317	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
318	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
319	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
320	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₇	1	2	yes	—
323	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
324	C₂³×C₄	C₂³×C₇	1	4	yes	—
325	C₂×M₁₆	—	4	2	no	—
326	C₂×M₁₆	—	4	2	no	—
327	C₂×M₁₆	—	4	4	no	—
328	C₂×M₁₆	C₂³×C₇	1	4	yes	—
329	C₂×M₁₆	C₂³×C₇	1	4	yes	—
330	C₂×M₁₆	C₂³×C₇	1	8	yes	complemented
331	C₂²×C₈	—	4	2	no	—
332	C₂²×C₈	—	4	2	no	—
333	C₂²×C₈	—	4	4	no	—
334	C₂²×C₈	C₂³×C₇	1	2	yes	—
335	C₂²×C₈	C₂³×C₇	1	2	yes	—
336	C₂²×C₈	C₂³×C₇	1	2	yes	—
337	C₂²×C₈	C₂³×C₇	1	2	yes	—
338	C₂²×C₈	C₂³×C₇	1	4	yes	—
339	C₂²×C₈	C₂³×C₇	1	4	yes	—
340	C₂²×C₈	C₂³×C₇	1	8	yes	complemented
341	#123	C₂×C₄×C₇	1	16	yes	—
342	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	—
343	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	—
344	#121	C₂×C₄×C₇	1	8	yes	—
345	#121	C₂×C₄×C₇	1	8	yes	—
346	#120	C₂×C₄×C₇	1	16	yes	—
347	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
348	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
349	C₄×D₄	C₂³×C₇	1	8	yes	—
350	C₄×D₄	C₂³×C₇	1	8	yes	—
351	#117	C₂×C₄×C₇	1	4	yes	—
352	#117	C₂³×C₇	1	4	yes	—
353	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
356	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
357	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
358	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
359	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
360	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
361	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
362	C₂×C₄²	C₂³×C₇	1	2	yes	—
363	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
364	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
365	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
366	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
367	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
368	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
369	C₂²⋊C₈	C₂³×C₇	1	8	yes	complemented
370	C₂²⋊C₈	C₂³×C₇	1	8	yes	complemented
371	#97	C₂³×C₇	1	8	yes	complemented
372	C₄×C₈	C₂³×C₇	1	8	yes	complemented
373	C₂²×C₇	—	2	2	no	—
374	C₂²×C₇	—	2	8	no	—
375	C₂²×C₇	—	2	2	no	—
376	C₂²×C₇	—	2	2	no	—
377	C₂²×C₇	—	2	2	no	—
378	C₂²×C₇	—	2	4	no	—
379	C₂²×C₇	—	2	4	no	—
380	C₂²×C₇	—	2	2	no	—
381	C₂²×C₇	—	2	2	no	—
382	C₂²×C₇	—	2	2	no	—
383	C₂²×C₇	—	2	4	no	—
384	C₂²×C₇	—	2	2	no	—
385	C₂²×C₇	—	4	16	no	—
386	C₂²×C₇	—	4	8	no	—
387	C₂²×C₇	—	4	4	no	—
388	C₂²×C₇	—	4	4	no	—
389	C₂²×C₇	#518	1	2	yes	—
390	C₂²×C₇	C₂×#131	1	4	yes	—
391	C₂²×C₇	C₂×#131	1	2	yes	—
392	C₂²×C₇	#568	1	2	yes	—
393	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
394	C₄×C₇	—	2	2	no	—
395	C₄×C₇	—	2	2	no	—
396	C₄×C₇	—	2	2	no	—
397	C₄×C₇	—	2	4	no	—
398	C₄×C₇	—	2	4	no	—
399	C₄×C₇	—	2	4	no	—
400	C₄×C₇	#518	1	2	yes	—
401	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
402	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	4	2	no	—
405	C₂⁴	—	4	2	no	—
406	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
407	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
408	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
409	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
410	C₂⁴	C₂²×C₄×C₇	1	4	yes	—
411	#41	—	4	8	no	—
412	#41	—	4	4	no	—
413	#41	—	4	4	no	—
414	C₂×Q₈	—	4	2	no	—
415	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	4	2	no	—
418	C₂×D₄	—	4	4	no	—
419	C₂×D₄	—	4	4	no	—
420	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
421	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
422	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
423	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	4	8	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	4	2	no	—
436	C₂²×C₄	—	4	2	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
439	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
440	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
441	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
442	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
443	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
444	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
445	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
446	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
447	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
448	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
449	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
450	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
451	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
452	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
455	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
456	M₁₆	—	4	4	no	—
457	M₁₆	—	4	4	no	—
458	M₁₆	—	4	8	no	—
459	C₂×C₈	—	2	4	no	—
460	C₂×C₈	—	2	4	no	—
461	C₂×C₈	—	2	4	no	—
462	C₂×C₈	—	2	4	no	—
463	C₂×C₈	—	4	4	no	—
464	C₂×C₈	—	4	4	no	—
465	C₂×C₈	—	4	8	no	—
466	C₂×C₈	C₂⁴×C₇	1	4	yes	—
467	C₂×C₈	C₂⁴×C₇	1	4	yes	—
468	C₂×C₈	C₂⁴×C₇	1	4	yes	—
469	C₂×C₈	C₂⁴×C₇	1	4	yes	—
470	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
471	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
472	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
473	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
474	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
475	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
476	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
477	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
478	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
479	C₄²	C₂²×C₄×C₇	1	4	yes	—
480	C₄²	C₂⁴×C₇	1	4	yes	—
481	C₂×C₇	—	2	4	no	—
482	C₂×C₇	—	4	4	no	—
483	C₂×C₇	—	4	2	no	—
484	C₂×C₇	—	4	2	no	—
485	C₂×C₇	#2976	1	4	yes	direct product factor
486	C₂×C₇	C₂×#518	1	1	yes	characteristic
487	C₂×C₇	C₂²×#131	1	1	yes	characteristic
488	C₂×C₇	C₂×#568	1	1	yes	characteristic
489	C₂³	—	2	2	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	4	no	—
494	C₂³	—	2	4	no	—
495	C₂³	—	2	8	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	4	4	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	4	16	no	—
504	C₂³	C₇×#131	1	4	yes	—
505	C₂³	C₂³×C₄×C₇	1	2	yes	—
506	C₂³	C₂³×C₄×C₇	1	2	yes	—
507	C₂³	C₂³×C₄×C₇	1	4	yes	—
508	C₂³	C₂³×C₄×C₇	1	2	yes	—
509	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
510	Q₈	—	4	4	no	—
511	D₄	—	2	16	no	—
512	D₄	—	4	8	no	—
513	D₄	—	4	4	no	—
514	D₄	—	4	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	8	no	—
533	C₂×C₄	—	4	4	no	—
534	C₂×C₄	—	4	4	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
537	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
538	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
539	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
540	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
541	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
542	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
543	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
544	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
545	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
546	C₈	—	2	4	no	—
547	C₈	—	2	4	no	—
548	C₈	—	2	4	no	—
549	C₈	—	2	4	no	—
550	C₇	C₂×#2976	1	1	yes	characteristic, direct product factor, Sylow
551	C₂²	—	2	2	no	—
552	C₂²	—	2	8	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	4	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	4	16	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	8	no	—
567	C₂²	C₇×#518	1	2	yes	—
568	C₂²	C₂×C₇×#131	1	4	yes	—
569	C₂²	C₂×C₇×#131	1	2	yes	—
570	C₂²	C₇×#568	1	2	yes	—
571	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
572	C₄	—	2	4	no	—
573	C₄	—	2	4	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	4	no	—
578	C₄	C₇×#518	1	2	yes	—
579	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
580	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
581	C₂	—	2	4	no	—
582	C₂	—	4	2	no	—
583	C₂	—	4	2	no	—
584	C₂	—	4	4	no	—
585	C₂	C₇×#2976	1	4	yes	direct product factor
586	C₂	C₂×C₇×#518	1	1	yes	characteristic
587	C₂	C₂²×C₇×#131	1	1	yes	characteristic
588	C₂	C₂×C₇×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂×C₇×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₇×#2976	C₂	direct (N × H)
3. C₂×C₇×#436	C₂	semidirect
4. C₂×C₇×#435	C₂	semidirect
5. C₂×C₇×#435	C₂	semidirect
6. C₂×C₇×C₈×D₄	C₂	semidirect
7. C₂×C₇×#433	C₂	semidirect
8. C₂×C₇×#408	C₂	semidirect
9. C₂×C₇×#407	C₂	semidirect
10. C₂²×C₇×C₂²⋊C₈	C₂	semidirect
15. C₂²×C₇×M₁₆	C₂²	semidirect
18. C₂³×C₇×C₈	C₂²	semidirect
30. C₇×#436	C₂²	semidirect
31. C₇×#435	C₂²	semidirect
32. C₇×#435	C₂²	semidirect
33. C₇×C₈×D₄	C₂²	semidirect
34. C₇×#433	C₂²	semidirect
35. C₂×C₇×C₄⋊C₈	C₂²	semidirect
36. C₂×C₇×C₄⋊C₈	C₂²	semidirect
37. C₇×#408	C₂²	semidirect
38. C₇×#407	C₂²	semidirect
43. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
44. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
45. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
46. C₂×C₇×#97	C₂²	semidirect
47. C₂×C₄×C₇×C₈	C₂²	semidirect
48. C₂×#2976	C₇	direct (N × H)
75. C₂×C₇×M₁₆	C₂³	semidirect
85. C₂²×C₇×C₈	C₂³	semidirect
108. C₇×C₄⋊C₈	C₂³	semidirect
109. C₇×C₄⋊C₈	C₂³	semidirect
114. C₇×C₂²⋊C₈	C₂³	semidirect
115. C₇×C₂²⋊C₈	C₂³	semidirect
116. C₇×#97	C₂³	semidirect
117. C₄×C₇×C₈	C₂³	semidirect
119. #2976	C₂×C₇	direct (N × H)
120. C₂×#436	C₂×C₇	semidirect
121. C₂×#435	C₂×C₇	semidirect
122. C₂×#435	C₂×C₇	semidirect
123. C₂×C₈×D₄	C₂×C₇	semidirect
124. C₂×#433	C₂×C₇	semidirect
125. C₂×#408	C₂×C₇	semidirect
126. C₂×#407	C₂×C₇	semidirect
127. C₂²×C₂²⋊C₈	C₂×C₇	semidirect
210. C₂²×M₁₆	C₂²×C₇	semidirect
213. C₂³×C₈	C₂²×C₇	semidirect
225. #436	C₂²×C₇	semidirect
226. #435	C₂²×C₇	semidirect
227. #435	C₂²×C₇	semidirect
228. C₈×D₄	C₂²×C₇	semidirect
229. #433	C₂²×C₇	semidirect
230. C₂×C₄⋊C₈	C₂²×C₇	semidirect
231. C₂×C₄⋊C₈	C₂²×C₇	semidirect
232. #408	C₂²×C₇	semidirect
233. #407	C₂²×C₇	semidirect
238. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
239. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
240. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
241. C₂×#97	C₂²×C₇	semidirect
242. C₂×C₄×C₈	C₂²×C₇	semidirect
330. C₂×M₁₆	C₂³×C₇	semidirect
340. C₂²×C₈	C₂³×C₇	semidirect
363. C₄⋊C₈	C₂³×C₇	semidirect
364. C₄⋊C₈	C₂³×C₇	semidirect
369. C₂²⋊C₈	C₂³×C₇	semidirect
370. C₂²⋊C₈	C₂³×C₇	semidirect
371. #97	C₂³×C₇	semidirect
372. C₄×C₈	C₂³×C₇	semidirect
485. C₂×C₇	#2976	direct (N × H)
550. C₇	C₂×#2976	direct (N × H)
585. C₂	C₇×#2976	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#525	2
C₇×#2976	2
C₂×C₇×#436	2
C₂×C₇×#435	2
C₂×C₇×C₈×D₄	2
C₂×C₇×#433	2
C₂×C₇×#408	2
C₂×C₇×#407	2
C₂²×C₇×C₂²⋊C₈	2
C₂×#2976	7

Subgroups — C2×C7×#2976 #13237440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2976 #13237440