Subgroups — C₇×#22698 #13199914

Order 1792.

1032 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#435	C₂	1	2	yes	maximal
2	C₄²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#1940	C₂	1	2	yes	maximal, complemented
4	C₇×#1851	C₂	1	2	yes	maximal, complemented
5	C₇×#1850	C₂	1	2	yes	maximal, complemented
6	C₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1728	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
13	C₂³×C₇×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
15	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
16	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
19	C₇×#435	—	2	8	no	—
20	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₇×#97	C₂²	1	2	yes	—
27	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
28	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
29	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₄×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
31	C₄×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
32	C₄×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
33	C₄×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
34	C₄×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
35	C₄×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
36	C₄³×C₇	C₂²	1	1	yes	characteristic
37	C₇×#336	C₂²	1	1	yes	characteristic
38	C₇×#336	C₂²	1	1	yes	characteristic
39	C₇×#336	C₂²	1	2	yes	—
40	C₇×#336	C₂²	1	1	yes	characteristic
41	C₇×#336	C₂²	1	2	yes	—
42	C₇×#336	C₂²	1	1	yes	characteristic
43	C₇×#336	C₂²	1	2	yes	—
44	#22698	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×D₄	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₇×M₁₆	—	2	4	no	—
53	C₂×C₇×M₁₆	—	2	4	no	—
54	C₂×C₇×M₁₆	—	4	2	no	—
55	C₂²×C₇×C₈	—	2	4	no	—
56	C₂²×C₇×C₈	—	2	4	no	—
57	C₂²×C₇×C₈	—	4	2	no	—
58	C₂²×C₇×C₈	C₂³	1	2	yes	—
59	C₂²×C₇×C₈	C₂³	1	2	yes	—
60	C₂²×C₇×C₈	C₂³	1	2	yes	—
61	C₂²×C₇×C₈	C₂³	1	2	yes	—
62	C₄×C₇×D₄	—	2	4	no	—
63	C₄×C₇×D₄	—	2	4	no	—
64	C₄×C₇×D₄	D₄	1	4	yes	—
65	C₄×C₇×D₄	Q₈	1	4	yes	—
66	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
69	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
70	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
72	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₇	—	4	2	no	—
76	C₂×C₄²×C₇	—	4	2	no	—
77	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
90	C₇×C₄⋊C₈	—	2	4	no	—
91	C₇×C₄⋊C₈	—	2	4	no	—
92	C₇×C₄⋊C₈	—	4	2	no	—
93	C₇×C₄⋊C₈	—	4	2	no	—
94	C₇×C₂²⋊C₈	—	2	4	no	—
95	C₇×C₂²⋊C₈	—	2	4	no	—
96	C₇×#97	—	2	4	no	—
97	C₇×#95	C₂×C₄	1	1	yes	characteristic
98	C₇×#95	C₂×C₄	1	1	yes	characteristic
99	C₇×#95	C₂×C₄	1	1	yes	characteristic
100	C₇×#95	C₂×C₄	1	1	yes	characteristic
101	C₇×#95	C₂×C₄	1	1	yes	characteristic
102	C₇×#95	C₂×C₄	1	1	yes	characteristic
103	C₂×#435	C₂×C₇	1	2	yes	—
104	C₄²×D₄	C₂×C₇	1	1	yes	characteristic
105	#1940	C₂×C₇	1	2	yes	complemented
106	#1851	C₂×C₇	1	2	yes	complemented
107	#1850	C₂×C₇	1	2	yes	complemented
108	#1831	C₂×C₇	1	1	yes	characteristic, complemented
109	#1797	C₂×C₇	1	1	yes	characteristic, complemented
110	#1797	C₂×C₇	1	1	yes	characteristic, complemented
111	#1767	C₂×C₇	1	1	yes	characteristic, complemented
112	#1728	C₂×C₇	1	1	yes	characteristic, complemented
113	C₂×#336	C₂×C₇	1	1	yes	characteristic, complemented
114	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
115	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₇×D₄	—	2	2	no	—
117	C₂×C₇×D₄	—	2	2	no	—
118	C₂×C₇×D₄	C₄⋊C₄	1	4	yes	—
119	C₂²×C₄×C₇	—	2	2	no	—
120	C₂²×C₄×C₇	—	2	2	no	—
121	C₂²×C₄×C₇	—	2	2	no	—
122	C₂²×C₄×C₇	—	2	2	no	—
123	C₂²×C₄×C₇	—	2	2	no	—
124	C₂²×C₄×C₇	—	2	2	no	—
125	C₂²×C₄×C₇	—	2	2	no	—
126	C₂²×C₄×C₇	—	2	2	no	—
127	C₂²×C₄×C₇	—	2	2	no	—
128	C₂²×C₄×C₇	—	4	1	no	—
129	C₂²×C₄×C₇	—	4	1	no	—
130	C₂²×C₄×C₇	—	4	1	no	—
131	C₂²×C₄×C₇	—	4	1	no	—
132	C₂²×C₄×C₇	—	4	1	no	—
133	C₂²×C₄×C₇	—	4	1	no	—
134	C₂²×C₄×C₇	M₁₆	1	4	yes	complemented
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	1	yes	characteristic
139	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
150	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
151	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
152	C₇×M₁₆	—	4	8	no	—
153	C₂×C₇×C₈	—	2	2	no	—
154	C₂×C₇×C₈	—	2	2	no	—
155	C₂×C₇×C₈	—	2	2	no	—
156	C₂×C₇×C₈	—	2	2	no	—
157	C₂×C₇×C₈	—	2	2	no	—
158	C₂×C₇×C₈	—	2	2	no	—
159	C₂×C₇×C₈	—	2	2	no	—
160	C₂×C₇×C₈	—	2	2	no	—
161	C₂×C₇×C₈	—	2	2	no	—
162	C₂×C₇×C₈	—	2	2	no	—
163	C₂×C₇×C₈	—	2	2	no	—
164	C₂×C₇×C₈	—	4	8	no	—
165	C₂×C₇×C₈	#41	1	4	yes	—
166	C₇×C₄⋊C₄	—	2	2	no	—
167	C₇×C₄⋊C₄	—	2	2	no	—
168	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
169	C₇×C₂²⋊C₄	—	2	2	no	—
170	C₇×C₂²⋊C₄	—	2	2	no	—
171	C₇×C₂²⋊C₄	—	2	2	no	—
172	C₇×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
173	C₇×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
174	C₇×C₂²⋊C₄	M₁₆	1	4	yes	complemented
175	C₄²×C₇	—	2	2	no	—
176	C₄²×C₇	—	2	2	no	—
177	C₄²×C₇	—	2	2	no	—
178	C₄²×C₇	—	2	2	no	—
179	C₄²×C₇	—	2	2	no	—
180	C₄²×C₇	—	2	2	no	—
181	C₄²×C₇	—	2	2	no	—
182	C₄²×C₇	—	4	1	no	—
183	C₄²×C₇	—	4	1	no	—
184	C₄²×C₇	—	4	1	no	—
185	C₄²×C₇	—	4	1	no	—
186	C₄²×C₇	—	4	1	no	—
187	C₄²×C₇	C₂×D₄	1	2	yes	—
188	C₄²×C₇	C₂×Q₈	1	2	yes	—
189	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
190	C₂³×C₈	C₂²×C₇	1	2	yes	—
191	C₂×C₄×D₄	C₄×C₇	1	1	yes	characteristic
192	C₂×C₄×D₄	C₄×C₇	1	1	yes	characteristic
193	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
194	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
195	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
196	#435	—	2	8	no	—
197	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
198	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
199	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
200	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
201	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
202	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
203	C₂×#97	C₂²×C₇	1	2	yes	—
204	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
205	C₄×C₄⋊C₄	C₄×C₇	1	1	yes	characteristic
206	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
207	C₄×C₂²⋊C₄	C₄×C₇	1	1	yes	characteristic
208	C₄×C₂²⋊C₄	C₄×C₇	1	1	yes	characteristic
209	C₄×C₂²⋊C₄	C₄×C₇	1	1	yes	characteristic
210	C₄×C₂²⋊C₄	C₄×C₇	1	1	yes	characteristic
211	C₄×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
212	C₄×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
213	C₄³	C₂²×C₇	1	1	yes	characteristic
214	#336	C₂²×C₇	1	1	yes	characteristic
215	#336	C₂²×C₇	1	1	yes	characteristic
216	#336	C₂²×C₇	1	1	yes	characteristic
217	#336	C₂²×C₇	1	2	yes	—
218	#336	C₂²×C₇	1	2	yes	—
219	#336	C₂²×C₇	1	2	yes	—
220	#336	C₂²×C₇	1	1	yes	characteristic
221	C₂³×C₇	—	2	1	no	—
222	C₂³×C₇	—	2	1	no	—
223	C₂³×C₇	—	2	1	no	—
224	C₂³×C₇	—	2	1	no	—
225	C₂³×C₇	—	2	1	no	—
226	C₂³×C₇	—	2	1	no	—
227	C₂³×C₇	—	2	1	no	—
228	C₂³×C₇	—	2	1	no	—
229	C₂³×C₇	—	2	1	no	—
230	C₂³×C₇	—	2	1	no	—
231	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
232	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
233	C₂³×C₇	C₂×M₁₆	1	2	yes	—
234	C₂³×C₇	C₂×M₁₆	1	2	yes	—
235	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
236	C₇×D₄	—	2	8	no	—
237	C₂×C₄×C₇	—	2	1	no	—
238	C₂×C₄×C₇	—	2	1	no	—
239	C₂×C₄×C₇	—	2	1	no	—
240	C₂×C₄×C₇	—	2	1	no	—
241	C₂×C₄×C₇	—	2	1	no	—
242	C₂×C₄×C₇	—	2	1	no	—
243	C₂×C₄×C₇	—	2	1	no	—
244	C₂×C₄×C₇	—	2	1	no	—
245	C₂×C₄×C₇	—	2	1	no	—
246	C₂×C₄×C₇	—	2	1	no	—
247	C₂×C₄×C₇	—	2	1	no	—
248	C₂×C₄×C₇	—	2	1	no	—
249	C₂×C₄×C₇	—	2	1	no	—
250	C₂×C₄×C₇	—	2	1	no	—
251	C₂×C₄×C₇	—	2	1	no	—
252	C₂×C₄×C₇	—	2	1	no	—
253	C₂×C₄×C₇	—	2	1	no	—
254	C₂×C₄×C₇	—	2	1	no	—
255	C₂×C₄×C₇	—	2	4	no	—
256	C₂×C₄×C₇	—	2	4	no	—
257	C₂×C₄×C₇	—	2	4	no	—
258	C₂×C₄×C₇	—	2	4	no	—
259	C₂×C₄×C₇	—	2	1	no	—
260	C₂×C₄×C₇	—	2	1	no	—
261	C₂×C₄×C₇	—	2	1	no	—
262	C₂×C₄×C₇	—	2	1	no	—
263	C₂×C₄×C₇	—	2	1	no	—
264	C₂×C₄×C₇	—	2	1	no	—
265	C₂×C₄×C₇	—	2	1	no	—
266	C₂×C₄×C₇	—	2	1	no	—
267	C₂×C₄×C₇	—	2	1	no	—
268	C₂×C₄×C₇	—	4	4	no	—
269	C₂×C₄×C₇	—	4	4	no	—
270	C₂×C₄×C₇	—	4	4	no	—
271	C₂×C₄×C₇	—	4	4	no	—
272	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
273	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
274	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
275	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
276	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
277	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
278	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
279	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
280	C₂×C₄×C₇	#131	1	2	yes	—
281	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
283	C₂×C₄×C₇	C₂×#41	1	2	yes	—
284	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
285	C₇×C₈	—	2	4	no	—
286	C₇×C₈	—	4	2	no	—
287	C₇×C₈	—	4	2	no	—
288	C₇×C₈	—	4	2	no	—
289	C₂²×D₄	C₂×C₄×C₇	1	1	yes	characteristic
290	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
291	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
292	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
293	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
294	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
295	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
296	C₂×M₁₆	—	2	4	no	—
297	C₂×M₁₆	—	2	4	no	—
298	C₂×M₁₆	—	4	2	no	—
299	C₂²×C₈	—	2	4	no	—
300	C₂²×C₈	—	2	4	no	—
301	C₂²×C₈	—	4	2	no	—
302	C₂²×C₈	C₂³×C₇	1	2	yes	—
303	C₂²×C₈	C₂³×C₇	1	2	yes	—
304	C₂²×C₈	C₂³×C₇	1	2	yes	—
305	C₂²×C₈	C₂³×C₇	1	2	yes	—
306	C₄×D₄	—	2	4	no	—
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	C₇×D₄	1	4	yes	—
309	C₄×D₄	C₇×Q₈	1	4	yes	—
310	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
314	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
317	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
318	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	1	yes	characteristic
319	C₂×C₄²	—	4	2	no	—
320	C₂×C₄²	—	4	2	no	—
321	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
322	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
323	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
324	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
325	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
326	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
327	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
328	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
334	C₄⋊C₈	—	2	4	no	—
335	C₄⋊C₈	—	2	4	no	—
336	C₄⋊C₈	—	4	2	no	—
337	C₄⋊C₈	—	4	2	no	—
338	C₂²⋊C₈	—	2	4	no	—
339	C₂²⋊C₈	—	2	4	no	—
340	#97	—	2	4	no	—
341	#95	C₂×C₄×C₇	1	1	yes	characteristic
342	#95	C₂×C₄×C₇	1	1	yes	characteristic
343	#95	C₂×C₄×C₇	1	1	yes	characteristic
344	#95	C₂×C₄×C₇	1	1	yes	characteristic
345	#95	C₂×C₄×C₇	1	1	yes	characteristic
346	#95	C₂×C₄×C₇	1	1	yes	characteristic
347	C₂²×C₇	—	2	2	no	—
348	C₂²×C₇	—	2	2	no	—
349	C₂²×C₇	—	2	2	no	—
350	C₂²×C₇	—	2	2	no	—
351	C₂²×C₇	—	2	2	no	—
352	C₂²×C₇	—	2	2	no	—
353	C₂²×C₇	—	2	2	no	—
354	C₂²×C₇	—	2	2	no	—
355	C₂²×C₇	—	2	2	no	—
356	C₂²×C₇	—	4	1	no	—
357	C₂²×C₇	—	4	1	no	—
358	C₂²×C₇	—	4	1	no	—
359	C₂²×C₇	—	4	1	no	—
360	C₂²×C₇	#423	1	4	yes	—
361	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
362	C₂²×C₇	C₄×#41	1	1	yes	characteristic
363	C₂²×C₇	#518	1	1	yes	characteristic
364	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
365	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
366	C₂²×C₇	C₂×#131	1	1	yes	characteristic
367	C₂²×C₇	#568	1	1	yes	characteristic
368	C₄×C₇	—	2	2	no	—
369	C₄×C₇	—	2	2	no	—
370	C₄×C₇	—	2	2	no	—
371	C₄×C₇	—	2	2	no	—
372	C₄×C₇	—	2	2	no	—
373	C₄×C₇	—	2	2	no	—
374	C₄×C₇	—	2	2	no	—
375	C₄×C₇	—	2	2	no	—
376	C₄×C₇	—	2	2	no	—
377	C₄×C₇	—	4	1	no	—
378	C₄×C₇	—	4	1	no	—
379	C₄×C₇	—	4	1	no	—
380	C₄×C₇	#546	1	2	yes	—
381	C₄×C₇	#554	1	2	yes	—
382	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
383	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	C₇×C₄⋊C₄	1	4	yes	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	4	1	no	—
397	C₂²×C₄	—	4	1	no	—
398	C₂²×C₄	—	4	1	no	—
399	C₂²×C₄	—	4	1	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	—	4	1	no	—
402	C₂²×C₄	C₇×M₁₆	1	4	yes	complemented
403	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
404	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
405	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
406	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
407	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
408	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
409	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
410	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
411	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
412	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
413	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
414	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
415	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
416	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
417	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
418	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
419	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
420	M₁₆	—	4	8	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	4	8	no	—
433	C₂×C₈	C₇×#41	1	4	yes	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	2	2	no	—
440	C₂²⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
441	C₂²⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
442	C₂²⋊C₄	C₇×M₁₆	1	4	yes	complemented
443	C₄²	—	2	2	no	—
444	C₄²	—	2	2	no	—
445	C₄²	—	2	2	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	2	2	no	—
450	C₄²	—	4	1	no	—
451	C₄²	—	4	1	no	—
452	C₄²	—	4	1	no	—
453	C₄²	—	4	1	no	—
454	C₄²	—	4	1	no	—
455	C₄²	C₂×C₇×D₄	1	2	yes	—
456	C₄²	C₂×C₇×Q₈	1	2	yes	—
457	C₂×C₇	—	2	4	no	—
458	C₂×C₇	—	4	2	no	—
459	C₂×C₇	#2349	1	1	yes	characteristic
460	C₂×C₇	C₂×#423	1	1	yes	characteristic
461	C₂×C₇	#2906	1	1	yes	characteristic
462	C₂×C₇	#2965	1	1	yes	characteristic
463	C₂×C₇	#2970	1	1	yes	characteristic
464	C₂×C₇	#2973	1	1	yes	characteristic
465	C₂×C₇	#2974	1	1	yes	characteristic
466	C₂³	—	2	1	no	—
467	C₂³	—	2	1	no	—
468	C₂³	—	2	1	no	—
469	C₂³	—	2	1	no	—
470	C₂³	—	2	1	no	—
471	C₂³	—	2	1	no	—
472	C₂³	—	2	1	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	1	no	—
475	C₂³	—	2	1	no	—
476	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
477	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
478	C₂³	C₂×C₇×M₁₆	1	2	yes	—
479	C₂³	C₂×C₇×M₁₆	1	2	yes	—
480	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
481	D₄	—	2	8	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
518	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
519	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
520	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
521	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
522	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
523	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
524	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
525	C₂×C₄	C₇×#131	1	2	yes	—
526	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
527	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
528	C₂×C₄	C₂×C₇×#41	1	2	yes	—
529	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
530	C₈	—	2	4	no	—
531	C₈	—	4	2	no	—
532	C₈	—	4	2	no	—
533	C₈	—	4	2	no	—
534	C₇	#22698	1	1	yes	characteristic, direct product factor, Sylow
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	1	no	—
548	C₂²	C₇×#423	1	4	yes	—
549	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
550	C₂²	C₄×C₇×#41	1	1	yes	characteristic
551	C₂²	C₇×#518	1	1	yes	characteristic
552	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
553	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
554	C₂²	C₂×C₇×#131	1	1	yes	characteristic
555	C₂²	C₇×#568	1	1	yes	characteristic
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	2	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	2	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	C₇×#546	1	2	yes	—
569	C₄	C₇×#554	1	2	yes	—
570	C₂	—	2	4	no	—
571	C₂	—	4	2	no	—
572	C₂	C₇×#2349	1	1	yes	characteristic
573	C₂	C₂×C₇×#423	1	1	yes	characteristic
574	C₂	C₇×#2906	1	1	yes	characteristic
575	C₂	C₇×#2965	1	1	yes	characteristic
576	C₂	C₇×#2970	1	1	yes	characteristic
577	C₂	C₇×#2973	1	1	yes	characteristic
578	C₂	C₇×#2974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂²×C₇×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1940	C₂	semidirect
4. C₇×#1851	C₂	semidirect
5. C₇×#1850	C₂	semidirect
6. C₇×#1831	C₂	semidirect
7. C₇×#1797	C₂	semidirect
8. C₇×#1797	C₂	semidirect
9. C₇×#1767	C₂	semidirect
10. C₇×#1728	C₂	semidirect
11. C₂×C₇×#336	C₂	semidirect
44. #22698	C₇	direct (N × H)
105. #1940	C₂×C₇	semidirect
106. #1851	C₂×C₇	semidirect
107. #1850	C₂×C₇	semidirect
108. #1831	C₂×C₇	semidirect
109. #1797	C₂×C₇	semidirect
110. #1797	C₂×C₇	semidirect
111. #1767	C₂×C₇	semidirect
112. #1728	C₂×C₇	semidirect
113. C₂×#336	C₂×C₇	semidirect
134. C₂²×C₄×C₇	M₁₆	semidirect
174. C₇×C₂²⋊C₄	M₁₆	semidirect
402. C₂²×C₄	C₇×M₁₆	semidirect
442. C₂²⋊C₄	C₇×M₁₆	semidirect
534. C₇	#22698	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×#435	2
C₄²×C₇×D₄	2
C₇×#1940	2
C₇×#1851	2
C₇×#1850	2
C₇×#1831	2
C₇×#1797	2
C₇×#1767	2
C₇×#1728	2
C₂×C₇×#336	2
#22698	7

Subgroups — C7×#22698 #13199914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22698 #13199914