Subgroups — C₂×C₇×#2974 #13237438

Order 1792.

2654 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#2974	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₇×#436	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#435	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×C₈×D₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#407	C₂	1	2	yes	maximal, complemented
9	C₂²×C₇×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
10	C₂³×C₇×D₄	C₄	1	1	yes	characteristic
11	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
12	C₂²×C₇×M₁₆	C₂²	1	4	yes	complemented
13	C₂³×C₇×C₈	C₂²	1	4	yes	complemented
14	C₂×C₄×C₇×D₄	C₄	1	2	yes	—
15	C₂×C₄×C₇×D₄	C₄	1	8	yes	—
16	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
17	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
18	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
20	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×D₄	C₂²	1	8	yes	—
22	C₂²×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
23	C₂²×C₇×C₂²⋊C₄	C₄	1	2	yes	—
24	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₇×#436	C₂²	1	16	yes	complemented
26	C₇×#435	C₂²	1	32	yes	complemented
27	C₇×C₈×D₄	C₂²	1	16	yes	complemented
28	C₇×#432	C₂²	1	8	yes	complemented
29	C₇×#431	C₂²	1	8	yes	complemented
30	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
31	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
32	C₇×#407	C₂²	1	16	yes	complemented
33	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₂×C₇×C₂²⋊C₈	C₂²	1	16	yes	complemented
37	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	complemented
38	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	complemented
39	C₂×C₇×#97	C₂²	1	2	yes	complemented
40	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
41	C₂×#2974	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂⁵×C₇	C₂×C₄	1	2	yes	—
43	C₂²×C₇×D₄	—	2	8	no	—
44	C₂²×C₇×D₄	C₂×C₄	1	2	yes	—
45	C₂²×C₇×D₄	C₂×C₄	1	8	yes	—
46	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₇	—	4	8	no	—
49	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
52	C₂³×C₄×C₇	C₂×C₄	1	8	yes	—
53	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
54	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
55	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₇	C₂³	1	2	yes	—
58	C₂³×C₄×C₇	C₂³	1	8	yes	—
59	C₂×C₇×M₁₆	—	4	8	no	—
60	C₂×C₇×M₁₆	C₂³	1	16	yes	complemented
61	C₂²×C₇×C₈	—	4	8	no	—
62	C₂²×C₇×C₈	C₂³	1	2	yes	—
63	C₂²×C₇×C₈	C₂³	1	2	yes	—
64	C₂²×C₇×C₈	C₂³	1	4	yes	—
65	C₂²×C₇×C₈	C₂³	1	16	yes	complemented
66	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
67	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
68	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
69	C₄×C₇×D₄	C₂×C₄	1	16	yes	—
70	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
71	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
72	C₄×C₇×D₄	C₂³	1	4	yes	—
73	C₄×C₇×D₄	C₂³	1	8	yes	—
74	C₄×C₇×D₄	C₂³	1	4	yes	—
75	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
76	C₂×C₇×C₄⋊C₄	C₂×C₄	1	8	yes	—
77	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
80	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
81	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
82	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
83	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	16	yes	—
84	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
86	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₇	C₂³	1	4	yes	—
88	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
90	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
91	C₇×C₄⋊C₈	C₂³	1	8	yes	complemented
92	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
93	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
94	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
95	C₇×C₂²⋊C₈	C₂³	1	16	yes	complemented
96	C₇×C₂²⋊C₈	C₂³	1	16	yes	complemented
97	C₇×#97	C₂³	1	8	yes	complemented
98	C₄×C₇×C₈	C₂³	1	8	yes	complemented
99	C₂²×C₄×D₄	C₂×C₇	1	1	yes	characteristic
100	#2974	C₂×C₇	1	16	yes	direct product factor
101	C₂×#436	C₂×C₇	1	2	yes	complemented
102	C₂×#435	C₂×C₇	1	4	yes	complemented
103	C₂×C₈×D₄	C₂×C₇	1	2	yes	complemented
104	C₂×#432	C₂×C₇	1	1	yes	characteristic, complemented
105	C₂×#431	C₂×C₇	1	1	yes	characteristic, complemented
106	C₂×#407	C₂×C₇	1	2	yes	complemented
107	C₂²×C₂²⋊C₈	C₂×C₇	1	2	yes	complemented
108	C₂⁴×C₇	—	2	8	no	—
109	C₂⁴×C₇	—	2	8	no	—
110	C₂⁴×C₇	—	4	4	no	—
111	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
112	C₂⁴×C₇	C₂²×C₄	1	8	yes	—
113	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
114	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₇×D₄	—	2	8	no	—
116	C₂×C₇×D₄	—	2	32	no	—
117	C₂×C₇×D₄	—	2	4	no	—
118	C₂×C₇×D₄	M₁₆	1	8	yes	complemented
119	C₂×C₇×D₄	C₂²×C₄	1	8	yes	—
120	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
121	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
122	C₂²×C₄×C₇	—	2	4	no	—
123	C₂²×C₄×C₇	—	2	4	no	—
124	C₂²×C₄×C₇	—	2	8	no	—
125	C₂²×C₄×C₇	—	2	8	no	—
126	C₂²×C₄×C₇	—	2	4	no	—
127	C₂²×C₄×C₇	—	4	16	no	—
128	C₂²×C₄×C₇	—	4	16	no	—
129	C₂²×C₄×C₇	—	4	4	no	—
130	C₂²×C₄×C₇	—	4	4	no	—
131	C₂²×C₄×C₇	—	4	4	no	—
132	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
135	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
139	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
140	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
142	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
143	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
144	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
145	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
146	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
151	C₇×M₁₆	—	4	16	no	—
152	C₂×C₇×C₈	—	2	4	no	—
153	C₂×C₇×C₈	—	2	8	no	—
154	C₂×C₇×C₈	—	2	4	no	—
155	C₂×C₇×C₈	—	4	16	no	—
156	C₂×C₇×C₈	C₂⁴	1	4	yes	—
157	C₂×C₇×C₈	C₂⁴	1	4	yes	—
158	C₂×C₇×C₈	C₂⁴	1	8	yes	—
159	C₇×C₄⋊C₄	C₂²×C₄	1	8	yes	—
160	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
161	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
162	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
163	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
164	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
165	C₇×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
166	C₄²×C₇	C₂²×C₄	1	4	yes	—
167	C₄²×C₇	C₂²×C₄	1	4	yes	—
168	C₄²×C₇	C₂²×C₄	1	2	yes	—
169	C₄²×C₇	C₂²×C₄	1	2	yes	—
170	C₄²×C₇	C₂⁴	1	2	yes	—
171	C₄²×C₇	C₂⁴	1	2	yes	—
172	C₂³×D₄	C₄×C₇	1	1	yes	characteristic
173	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
174	C₂²×M₁₆	C₂²×C₇	1	4	yes	complemented
175	C₂³×C₈	C₂²×C₇	1	4	yes	complemented
176	C₂×C₄×D₄	C₄×C₇	1	2	yes	—
177	C₂×C₄×D₄	C₄×C₇	1	8	yes	—
178	C₂×C₄×D₄	C₄×C₇	1	1	yes	characteristic
179	C₂×C₄×D₄	C₄×C₇	1	1	yes	characteristic
180	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
181	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
182	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
183	C₂×C₄×D₄	C₂²×C₇	1	8	yes	—
184	C₂²×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
185	C₂²×C₂²⋊C₄	C₄×C₇	1	2	yes	—
186	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
187	#436	C₂²×C₇	1	16	yes	complemented
188	#435	C₂²×C₇	1	32	yes	complemented
189	C₈×D₄	C₂²×C₇	1	16	yes	complemented
190	#432	C₂²×C₇	1	8	yes	complemented
191	#431	C₂²×C₇	1	8	yes	complemented
192	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
193	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	complemented
194	#407	C₂²×C₇	1	16	yes	complemented
195	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
196	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
198	C₂×C₂²⋊C₈	C₂²×C₇	1	16	yes	complemented
199	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	complemented
200	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	complemented
201	C₂×#97	C₂²×C₇	1	2	yes	complemented
202	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
203	C₂³×C₇	—	2	2	no	—
204	C₂³×C₇	—	2	4	no	—
205	C₂³×C₇	—	2	2	no	—
206	C₂³×C₇	—	2	4	no	—
207	C₂³×C₇	—	2	4	no	—
208	C₂³×C₇	—	2	16	no	—
209	C₂³×C₇	—	2	4	no	—
210	C₂³×C₇	—	2	16	no	—
211	C₂³×C₇	—	2	4	no	—
212	C₂³×C₇	—	4	8	no	—
213	C₂³×C₇	—	4	32	no	—
214	C₂³×C₇	C₂×M₁₆	1	8	yes	—
215	C₂³×C₇	#131	1	4	yes	—
216	C₂³×C₇	C₂³×C₄	1	4	yes	—
217	C₂³×C₇	C₂³×C₄	1	4	yes	—
218	C₂³×C₇	C₂³×C₄	1	2	yes	—
219	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
220	C₇×D₄	—	2	16	no	—
221	C₇×D₄	—	2	8	no	—
222	C₇×D₄	C₂×M₁₆	1	16	yes	complemented
223	C₂×C₄×C₇	—	2	2	no	—
224	C₂×C₄×C₇	—	2	4	no	—
225	C₂×C₄×C₇	—	2	4	no	—
226	C₂×C₄×C₇	—	2	2	no	—
227	C₂×C₄×C₇	—	2	8	no	—
228	C₂×C₄×C₇	—	2	2	no	—
229	C₂×C₄×C₇	—	2	2	no	—
230	C₂×C₄×C₇	—	2	4	no	—
231	C₂×C₄×C₇	—	2	8	no	—
232	C₂×C₄×C₇	—	2	8	no	—
233	C₂×C₄×C₇	—	2	8	no	—
234	C₂×C₄×C₇	—	2	8	no	—
235	C₂×C₄×C₇	—	2	8	no	—
236	C₂×C₄×C₇	—	2	4	no	—
237	C₂×C₄×C₇	—	4	8	no	—
238	C₂×C₄×C₇	—	4	8	no	—
239	C₂×C₄×C₇	—	4	8	no	—
240	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
241	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
243	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
244	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
245	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
246	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
247	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
249	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
250	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
251	C₇×C₈	—	2	4	no	—
252	C₇×C₈	—	2	8	no	—
253	C₇×C₈	—	2	4	no	—
254	C₂⁵	C₂×C₄×C₇	1	2	yes	—
255	C₂²×D₄	—	2	8	no	—
256	C₂²×D₄	C₂×C₄×C₇	1	2	yes	—
257	C₂²×D₄	C₂×C₄×C₇	1	8	yes	—
258	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
259	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
260	C₂³×C₄	—	4	8	no	—
261	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
262	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
263	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
264	C₂³×C₄	C₂×C₄×C₇	1	8	yes	—
265	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
266	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
267	C₂³×C₄	C₂³×C₇	1	2	yes	—
268	C₂³×C₄	C₂³×C₇	1	2	yes	—
269	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
270	C₂³×C₄	C₂³×C₇	1	8	yes	—
271	C₂×M₁₆	—	4	8	no	—
272	C₂×M₁₆	C₂³×C₇	1	16	yes	complemented
273	C₂²×C₈	—	4	8	no	—
274	C₂²×C₈	C₂³×C₇	1	2	yes	—
275	C₂²×C₈	C₂³×C₇	1	4	yes	—
276	C₂²×C₈	C₂³×C₇	1	2	yes	—
277	C₂²×C₈	C₂³×C₇	1	16	yes	complemented
278	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
279	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
280	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
281	C₄×D₄	C₂×C₄×C₇	1	16	yes	—
282	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
283	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
284	C₄×D₄	C₂³×C₇	1	4	yes	—
285	C₄×D₄	C₂³×C₇	1	8	yes	—
286	C₄×D₄	C₂³×C₇	1	4	yes	—
287	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
288	C₂×C₄⋊C₄	C₂×C₄×C₇	1	8	yes	—
289	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
292	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
293	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
294	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	2	yes	—
295	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	16	yes	—
296	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
297	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
298	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₇	1	4	yes	—
302	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
303	C₄⋊C₈	C₂³×C₇	1	8	yes	complemented
304	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
305	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
306	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
307	C₂²⋊C₈	C₂³×C₇	1	16	yes	complemented
308	C₂²⋊C₈	C₂³×C₇	1	16	yes	complemented
309	#97	C₂³×C₇	1	8	yes	complemented
310	C₄×C₈	C₂³×C₇	1	8	yes	complemented
311	C₂²×C₇	—	2	2	no	—
312	C₂²×C₇	—	2	4	no	—
313	C₂²×C₇	—	2	8	no	—
314	C₂²×C₇	—	2	16	no	—
315	C₂²×C₇	—	2	2	no	—
316	C₂²×C₇	—	2	4	no	—
317	C₂²×C₇	—	2	4	no	—
318	C₂²×C₇	—	2	4	no	—
319	C₂²×C₇	—	2	4	no	—
320	C₂²×C₇	—	4	32	no	—
321	C₂²×C₇	—	4	8	no	—
322	C₂²×C₇	#518	1	2	yes	—
323	C₂²×C₇	C₂²×M₁₆	1	8	yes	—
324	C₂²×C₇	C₂²×M₁₆	1	2	yes	—
325	C₂²×C₇	C₂×#131	1	4	yes	—
326	C₂²×C₇	C₂×#131	1	2	yes	—
327	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
328	C₄×C₇	—	2	2	no	—
329	C₄×C₇	—	2	4	no	—
330	C₄×C₇	—	2	4	no	—
331	C₄×C₇	—	2	2	no	—
332	C₄×C₇	—	2	2	no	—
333	C₄×C₇	—	2	2	no	—
334	C₄×C₇	#518	1	2	yes	—
335	C₄×C₇	C₂²×M₁₆	1	4	yes	—
336	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
337	C₂⁴	—	2	8	no	—
338	C₂⁴	—	2	8	no	—
339	C₂⁴	—	4	4	no	—
340	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
341	C₂⁴	C₂²×C₄×C₇	1	8	yes	—
342	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
343	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
344	C₂×D₄	—	2	8	no	—
345	C₂×D₄	—	2	32	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	C₇×M₁₆	1	8	yes	complemented
348	C₂×D₄	C₂²×C₄×C₇	1	8	yes	—
349	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
350	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	8	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	4	16	no	—
357	C₂²×C₄	—	4	16	no	—
358	C₂²×C₄	—	4	4	no	—
359	C₂²×C₄	—	4	4	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
362	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
363	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
364	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
365	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
366	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
367	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
368	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
369	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
370	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
371	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
372	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
373	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
374	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
375	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
376	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
377	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
378	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
379	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
380	M₁₆	—	4	16	no	—
381	C₂×C₈	—	2	4	no	—
382	C₂×C₈	—	2	8	no	—
383	C₂×C₈	—	2	4	no	—
384	C₂×C₈	—	4	16	no	—
385	C₂×C₈	C₂⁴×C₇	1	4	yes	—
386	C₂×C₈	C₂⁴×C₇	1	4	yes	—
387	C₂×C₈	C₂⁴×C₇	1	8	yes	—
388	C₄⋊C₄	C₂²×C₄×C₇	1	8	yes	—
389	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
390	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
391	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
392	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
393	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
394	C₂²⋊C₄	C₂²×C₄×C₇	1	8	yes	—
395	C₄²	C₂²×C₄×C₇	1	4	yes	—
396	C₄²	C₂²×C₄×C₇	1	4	yes	—
397	C₄²	C₂²×C₄×C₇	1	2	yes	—
398	C₄²	C₂²×C₄×C₇	1	2	yes	—
399	C₄²	C₂⁴×C₇	1	2	yes	—
400	C₄²	C₂⁴×C₇	1	2	yes	—
401	C₂×C₇	—	2	4	no	—
402	C₂×C₇	—	2	8	no	—
403	C₂×C₇	—	4	4	no	—
404	C₂×C₇	#2974	1	4	yes	direct product factor
405	C₂×C₇	C₂×#518	1	1	yes	characteristic
406	C₂×C₇	C₂³×M₁₆	1	1	yes	characteristic
407	C₂×C₇	C₂²×#131	1	1	yes	characteristic
408	C₂³	—	2	2	no	—
409	C₂³	—	2	4	no	—
410	C₂³	—	2	2	no	—
411	C₂³	—	2	4	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	16	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	16	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	4	8	no	—
418	C₂³	—	4	32	no	—
419	C₂³	C₂×C₇×M₁₆	1	8	yes	—
420	C₂³	C₇×#131	1	4	yes	—
421	C₂³	C₂³×C₄×C₇	1	4	yes	—
422	C₂³	C₂³×C₄×C₇	1	4	yes	—
423	C₂³	C₂³×C₄×C₇	1	2	yes	—
424	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
425	D₄	—	2	16	no	—
426	D₄	—	2	8	no	—
427	D₄	C₂×C₇×M₁₆	1	16	yes	complemented
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	8	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	8	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	4	8	no	—
443	C₂×C₄	—	4	8	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
446	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
447	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
448	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
449	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
450	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
451	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
452	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
453	C₂×C₄	C₂³×C₄×C₇	1	2	yes	—
454	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
455	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
456	C₈	—	2	4	no	—
457	C₈	—	2	8	no	—
458	C₈	—	2	4	no	—
459	C₇	C₂×#2974	1	1	yes	characteristic, direct product factor, Sylow
460	C₂²	—	2	2	no	—
461	C₂²	—	2	4	no	—
462	C₂²	—	2	8	no	—
463	C₂²	—	2	16	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	4	no	—
466	C₂²	—	2	4	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	4	32	no	—
470	C₂²	—	4	8	no	—
471	C₂²	C₇×#518	1	2	yes	—
472	C₂²	C₂²×C₇×M₁₆	1	8	yes	—
473	C₂²	C₂²×C₇×M₁₆	1	2	yes	—
474	C₂²	C₂×C₇×#131	1	4	yes	—
475	C₂²	C₂×C₇×#131	1	2	yes	—
476	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
477	C₄	—	2	2	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	4	no	—
483	C₄	C₇×#518	1	2	yes	—
484	C₄	C₂²×C₇×M₁₆	1	4	yes	—
485	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
486	C₂	—	2	4	no	—
487	C₂	—	2	8	no	—
488	C₂	—	4	4	no	—
489	C₂	C₇×#2974	1	4	yes	direct product factor
490	C₂	C₂×C₇×#518	1	1	yes	characteristic
491	C₂	C₂³×C₇×M₁₆	1	1	yes	characteristic
492	C₂	C₂²×C₇×#131	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	C₂³×C₇×D₄	C₄
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×#2974	C₂	direct (N × H)
3. C₂×C₇×#436	C₂	semidirect
4. C₂×C₇×#435	C₂	semidirect
5. C₂×C₇×C₈×D₄	C₂	semidirect
6. C₂×C₇×#432	C₂	semidirect
7. C₂×C₇×#431	C₂	semidirect
8. C₂×C₇×#407	C₂	semidirect
9. C₂²×C₇×C₂²⋊C₈	C₂	semidirect
12. C₂²×C₇×M₁₆	C₂²	semidirect
13. C₂³×C₇×C₈	C₂²	semidirect
25. C₇×#436	C₂²	semidirect
26. C₇×#435	C₂²	semidirect
27. C₇×C₈×D₄	C₂²	semidirect
28. C₇×#432	C₂²	semidirect
29. C₇×#431	C₂²	semidirect
30. C₂×C₇×C₄⋊C₈	C₂²	semidirect
31. C₂×C₇×C₄⋊C₈	C₂²	semidirect
32. C₇×#407	C₂²	semidirect
36. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
37. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
38. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
39. C₂×C₇×#97	C₂²	semidirect
40. C₂×C₄×C₇×C₈	C₂²	semidirect
41. C₂×#2974	C₇	direct (N × H)
60. C₂×C₇×M₁₆	C₂³	semidirect
65. C₂²×C₇×C₈	C₂³	semidirect
90. C₇×C₄⋊C₈	C₂³	semidirect
91. C₇×C₄⋊C₈	C₂³	semidirect
95. C₇×C₂²⋊C₈	C₂³	semidirect
96. C₇×C₂²⋊C₈	C₂³	semidirect
97. C₇×#97	C₂³	semidirect
98. C₄×C₇×C₈	C₂³	semidirect
100. #2974	C₂×C₇	direct (N × H)
101. C₂×#436	C₂×C₇	semidirect
102. C₂×#435	C₂×C₇	semidirect
103. C₂×C₈×D₄	C₂×C₇	semidirect
104. C₂×#432	C₂×C₇	semidirect
105. C₂×#431	C₂×C₇	semidirect
106. C₂×#407	C₂×C₇	semidirect
107. C₂²×C₂²⋊C₈	C₂×C₇	semidirect
118. C₂×C₇×D₄	M₁₆	semidirect
174. C₂²×M₁₆	C₂²×C₇	semidirect
175. C₂³×C₈	C₂²×C₇	semidirect
187. #436	C₂²×C₇	semidirect
188. #435	C₂²×C₇	semidirect
189. C₈×D₄	C₂²×C₇	semidirect
190. #432	C₂²×C₇	semidirect
191. #431	C₂²×C₇	semidirect
192. C₂×C₄⋊C₈	C₂²×C₇	semidirect
193. C₂×C₄⋊C₈	C₂²×C₇	semidirect
194. #407	C₂²×C₇	semidirect
198. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
199. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
200. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
201. C₂×#97	C₂²×C₇	semidirect
202. C₂×C₄×C₈	C₂²×C₇	semidirect
222. C₇×D₄	C₂×M₁₆	semidirect
272. C₂×M₁₆	C₂³×C₇	semidirect
277. C₂²×C₈	C₂³×C₇	semidirect
302. C₄⋊C₈	C₂³×C₇	semidirect
303. C₄⋊C₈	C₂³×C₇	semidirect
307. C₂²⋊C₈	C₂³×C₇	semidirect
308. C₂²⋊C₈	C₂³×C₇	semidirect
309. #97	C₂³×C₇	semidirect
310. C₄×C₈	C₂³×C₇	semidirect
347. C₂×D₄	C₇×M₁₆	semidirect
404. C₂×C₇	#2974	direct (N × H)
427. D₄	C₂×C₇×M₁₆	semidirect
459. C₇	C₂×#2974	direct (N × H)
489. C₂	C₇×#2974	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×D₄	2
C₇×#2974	2
C₂×C₇×#436	2
C₂×C₇×#435	2
C₂×C₇×C₈×D₄	2
C₂×C₇×#432	2
C₂×C₇×#431	2
C₂×C₇×#407	2
C₂²×C₇×C₂²⋊C₈	2
C₂×#2974	7

Subgroups — C2×C7×#2974 #13237438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2974 #13237438