Subgroups — C₇×#22503 #13199719

Order 1792.

1246 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#555	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1895	C₂	1	2	yes	maximal, complemented
7	C₇×#1874	C₂	1	2	yes	maximal, complemented
8	C₇×#1825	C₂	1	2	yes	maximal, complemented
9	C₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1736	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
13	C₇×#555	—	2	4	no	—
14	C₇×#555	C₄	1	8	yes	complemented
15	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×#124	—	2	2	no	—
17	C₂×C₇×#121	—	2	1	no	—
18	C₂×C₇×#121	C₂²	1	2	yes	—
19	C₂×C₄×C₇×D₄	—	2	1	no	—
20	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×#117	—	2	1	no	—
23	C₂×C₇×#117	C₂²	1	2	yes	—
24	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
25	C₇×#436	—	2	4	no	—
26	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₇×#102	C₂²	1	1	yes	characteristic
28	C₂×C₇×#102	C₂²	1	1	yes	characteristic
29	C₂×C₇×#102	C₂²	1	2	yes	—
30	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
32	C₇×#389	—	2	1	no	—
33	C₇×#385	—	2	2	no	—
34	C₇×#384	C₂²	1	1	yes	characteristic
35	C₇×#380	C₂²	1	2	yes	—
36	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₇×#340	C₂²	1	2	yes	—
38	C₇×#340	C₂²	1	2	yes	—
39	C₇×#340	C₂²	1	2	yes	—
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	1	yes	characteristic
42	C₇×#340	C₂²	1	2	yes	—
43	C₇×#340	C₂²	1	1	yes	characteristic
44	#22503	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×Q₈	—	2	1	no	—
46	C₂²×C₇×D₄	—	2	2	no	—
47	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
48	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₇	—	2	1	no	—
50	C₂³×C₄×C₇	—	2	2	no	—
51	C₂³×C₄×C₇	C₂³	1	2	yes	—
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	C₂³	1	1	yes	characteristic
56	C₂²×C₇×C₈	C₂³	1	2	yes	—
57	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₇×C₄⋊Q₈	—	2	2	no	—
59	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
60	C₇×#124	—	2	16	no	—
61	C₇×#121	—	2	8	no	—
62	C₇×#121	—	2	4	no	—
63	C₇×#121	C₂×C₄	1	8	yes	complemented
64	C₄×C₇×D₄	—	2	2	no	—
65	C₄×C₇×D₄	—	4	4	no	—
66	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
67	C₄×C₇×D₄	C₂×C₄	1	4	yes	complemented
68	C₄×C₇×D₄	D₄	1	4	yes	—
69	C₇×#117	—	2	4	no	—
70	C₇×#117	—	2	8	no	—
71	C₇×#117	C₂×C₄	1	8	yes	complemented
72	C₂×C₇×C₄⋊C₄	—	2	1	no	—
73	C₂×C₇×C₄⋊C₄	—	2	4	no	—
74	C₂×C₇×C₄⋊C₄	—	2	2	no	—
75	C₂×C₇×C₄⋊C₄	—	2	1	no	—
76	C₂×C₇×C₄⋊C₄	—	2	1	no	—
77	C₂×C₇×C₄⋊C₄	—	4	2	no	—
78	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
79	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
80	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
83	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
86	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
87	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
88	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
89	C₂×C₄²×C₇	—	2	2	no	—
90	C₂×C₄²×C₇	—	2	1	no	—
91	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
92	C₇×C₄⋊C₈	—	2	2	no	—
93	C₇×#102	—	2	2	no	—
94	C₇×#102	—	2	2	no	—
95	C₇×#102	—	4	2	no	—
96	C₇×C₂²⋊C₈	—	2	4	no	—
97	C₄×C₇×C₈	—	2	2	no	—
98	C₇×#95	—	2	1	no	—
99	C₇×#95	—	2	2	no	—
100	C₂×#555	C₂×C₇	1	1	yes	characteristic
101	C₂×#436	C₂×C₇	1	1	yes	characteristic, complemented
102	D₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
103	#1990	C₂×C₇	1	1	yes	characteristic, complemented
104	#1987	C₂×C₇	1	1	yes	characteristic, complemented
105	#1895	C₂×C₇	1	2	yes	complemented
106	#1874	C₂×C₇	1	2	yes	complemented
107	#1825	C₂×C₇	1	2	yes	complemented
108	#1802	C₂×C₇	1	1	yes	characteristic, complemented
109	C₄×#102	C₂×C₇	1	1	yes	characteristic, complemented
110	#1736	C₂×C₇	1	2	yes	complemented
111	C₂⁴×C₇	—	2	1	no	—
112	C₂⁴×C₇	C₂×D₄	1	2	yes	—
113	C₂×C₇×Q₈	—	2	4	no	—
114	C₂×C₇×Q₈	—	8	1	no	—
115	C₂×C₇×Q₈	—	8	1	no	—
116	C₂×C₇×D₄	—	2	8	no	—
117	C₂×C₇×D₄	—	2	2	no	—
118	C₂×C₇×D₄	—	2	2	no	—
119	C₂×C₇×D₄	—	2	1	no	—
120	C₂×C₇×D₄	—	2	1	no	—
121	C₂×C₇×D₄	—	2	1	no	—
122	C₂×C₇×D₄	—	2	1	no	—
123	C₂×C₇×D₄	—	2	1	no	—
124	C₂×C₇×D₄	—	8	2	no	—
125	C₂×C₇×D₄	—	8	2	no	—
126	C₂×C₇×D₄	C₂²⋊C₄	1	4	yes	—
127	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₇	—	2	2	no	—
129	C₂²×C₄×C₇	—	2	2	no	—
130	C₂²×C₄×C₇	—	2	2	no	—
131	C₂²×C₄×C₇	—	2	2	no	—
132	C₂²×C₄×C₇	—	2	1	no	—
133	C₂²×C₄×C₇	—	2	1	no	—
134	C₂²×C₄×C₇	—	2	2	no	—
135	C₂²×C₄×C₇	—	2	4	no	—
136	C₂²×C₄×C₇	—	2	1	no	—
137	C₂²×C₄×C₇	—	2	2	no	—
138	C₂²×C₄×C₇	—	4	4	no	—
139	C₂²×C₄×C₇	—	4	2	no	—
140	C₂²×C₄×C₇	—	4	2	no	—
141	C₂²×C₄×C₇	—	4	4	no	—
142	C₂²×C₄×C₇	—	8	2	no	—
143	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
149	C₇×M₁₆	—	4	8	no	—
150	C₂×C₇×C₈	—	2	1	no	—
151	C₂×C₇×C₈	—	2	1	no	—
152	C₂×C₇×C₈	—	2	1	no	—
153	C₂×C₇×C₈	—	2	2	no	—
154	C₂×C₇×C₈	—	2	1	no	—
155	C₂×C₇×C₈	—	2	2	no	—
156	C₂×C₇×C₈	—	2	2	no	—
157	C₂×C₇×C₈	—	2	1	no	—
158	C₂×C₇×C₈	#41	1	2	yes	—
159	C₇×C₄⋊C₄	—	2	4	no	—
160	C₇×C₄⋊C₄	—	2	1	no	—
161	C₇×C₄⋊C₄	—	2	2	no	—
162	C₇×C₄⋊C₄	—	2	1	no	—
163	C₇×C₄⋊C₄	—	2	1	no	—
164	C₇×C₄⋊C₄	—	2	2	no	—
165	C₇×C₄⋊C₄	—	2	2	no	—
166	C₇×C₄⋊C₄	—	2	1	no	—
167	C₇×C₄⋊C₄	—	4	4	no	—
168	C₇×C₄⋊C₄	—	4	2	no	—
169	C₇×C₄⋊C₄	—	8	1	no	—
170	C₇×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
171	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
172	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
173	C₇×C₄⋊C₄	#41	1	2	yes	—
174	C₇×C₂²⋊C₄	—	2	8	no	—
175	C₇×C₂²⋊C₄	—	2	8	no	—
176	C₇×C₂²⋊C₄	—	2	4	no	—
177	C₇×C₂²⋊C₄	—	4	4	no	—
178	C₇×C₂²⋊C₄	C₂²⋊C₄	1	8	yes	—
179	C₄²×C₇	—	2	8	no	—
180	C₄²×C₇	—	4	2	no	—
181	C₄²×C₇	C₂²×C₄	1	2	yes	—
182	C₄²×C₇	C₂×D₄	1	2	yes	—
183	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
184	#555	—	2	4	no	—
185	#555	C₄×C₇	1	8	yes	complemented
186	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
187	C₂×#124	—	2	2	no	—
188	C₂×#121	—	2	1	no	—
189	C₂×#121	C₂²×C₇	1	2	yes	—
190	C₂×C₄×D₄	—	2	1	no	—
191	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
192	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
193	C₂×#117	—	2	1	no	—
194	C₂×#117	C₂²×C₇	1	2	yes	—
195	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
196	#436	—	2	4	no	—
197	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
198	C₂×#102	C₂²×C₇	1	1	yes	characteristic
199	C₂×#102	C₂²×C₇	1	1	yes	characteristic
200	C₂×#102	C₂²×C₇	1	2	yes	—
201	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
202	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
203	#389	—	2	1	no	—
204	#385	—	2	2	no	—
205	#384	C₂²×C₇	1	1	yes	characteristic
206	#380	C₂²×C₇	1	2	yes	—
207	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
208	#340	C₂²×C₇	1	2	yes	—
209	#340	C₂²×C₇	1	1	yes	characteristic
210	#340	C₂²×C₇	1	2	yes	—
211	#340	C₂²×C₇	1	2	yes	—
212	#340	C₂²×C₇	1	2	yes	—
213	#340	C₂²×C₇	1	2	yes	—
214	#340	C₂²×C₇	1	1	yes	characteristic
215	C₂³×C₇	—	2	2	no	—
216	C₂³×C₇	—	2	2	no	—
217	C₂³×C₇	—	2	2	no	—
218	C₂³×C₇	—	2	2	no	—
219	C₂³×C₇	—	2	2	no	—
220	C₂³×C₇	—	2	2	no	—
221	C₂³×C₇	—	2	2	no	—
222	C₂³×C₇	—	4	1	no	—
223	C₂³×C₇	—	4	1	no	—
224	C₂³×C₇	—	4	1	no	—
225	C₂³×C₇	—	4	1	no	—
226	C₂³×C₇	—	4	1	no	—
227	C₂³×C₇	—	4	1	no	—
228	C₂³×C₇	C₂×C₂²⋊C₄	1	4	yes	—
229	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
230	C₇×Q₈	—	8	4	no	—
231	C₇×D₄	—	2	2	no	—
232	C₇×D₄	—	2	2	no	—
233	C₇×D₄	—	2	8	no	—
234	C₇×D₄	—	4	1	no	—
235	C₇×D₄	—	4	1	no	—
236	C₇×D₄	—	8	8	no	—
237	C₂×C₄×C₇	—	2	2	no	—
238	C₂×C₄×C₇	—	2	2	no	—
239	C₂×C₄×C₇	—	2	1	no	—
240	C₂×C₄×C₇	—	2	1	no	—
241	C₂×C₄×C₇	—	2	4	no	—
242	C₂×C₄×C₇	—	2	4	no	—
243	C₂×C₄×C₇	—	2	4	no	—
244	C₂×C₄×C₇	—	2	4	no	—
245	C₂×C₄×C₇	—	2	2	no	—
246	C₂×C₄×C₇	—	2	4	no	—
247	C₂×C₄×C₇	—	2	2	no	—
248	C₂×C₄×C₇	—	4	1	no	—
249	C₂×C₄×C₇	—	4	1	no	—
250	C₂×C₄×C₇	—	4	1	no	—
251	C₂×C₄×C₇	—	4	1	no	—
252	C₂×C₄×C₇	—	4	2	no	—
253	C₂×C₄×C₇	—	4	2	no	—
254	C₂×C₄×C₇	—	4	8	no	—
255	C₂×C₄×C₇	—	4	2	no	—
256	C₂×C₄×C₇	—	4	1	no	—
257	C₂×C₄×C₇	—	4	1	no	—
258	C₂×C₄×C₇	—	4	2	no	—
259	C₂×C₄×C₇	—	4	2	no	—
260	C₂×C₄×C₇	—	4	1	no	—
261	C₂×C₄×C₇	—	8	8	no	—
262	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
263	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
264	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
265	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
266	C₂×C₄×C₇	#137	1	2	yes	—
267	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
268	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
272	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
273	C₇×C₈	—	2	2	no	—
274	C₇×C₈	—	4	2	no	—
275	C₇×C₈	—	4	1	no	—
276	C₂²×Q₈	—	2	1	no	—
277	C₂²×D₄	—	2	2	no	—
278	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
279	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
280	C₂³×C₄	—	2	1	no	—
281	C₂³×C₄	—	2	2	no	—
282	C₂³×C₄	C₂³×C₇	1	2	yes	—
283	C₂×M₁₆	—	2	4	no	—
284	C₂×M₁₆	—	2	4	no	—
285	C₂×M₁₆	—	4	2	no	—
286	C₂²×C₈	C₂³×C₇	1	2	yes	—
287	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
289	C₄⋊Q₈	—	2	2	no	—
290	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	complemented
291	#124	—	2	16	no	—
292	#121	—	2	8	no	—
293	#121	—	2	4	no	—
294	#121	C₂×C₄×C₇	1	8	yes	complemented
295	C₄×D₄	—	2	2	no	—
296	C₄×D₄	—	4	4	no	—
297	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
298	C₄×D₄	C₂×C₄×C₇	1	4	yes	complemented
299	C₄×D₄	C₇×D₄	1	4	yes	—
300	#117	—	2	4	no	—
301	#117	—	2	8	no	—
302	#117	C₂×C₄×C₇	1	8	yes	complemented
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	4	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	4	2	no	—
309	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
314	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
320	C₂×C₄²	—	2	2	no	—
321	C₂×C₄²	—	2	1	no	—
322	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
323	C₄⋊C₈	—	2	2	no	—
324	#102	—	2	2	no	—
325	#102	—	2	2	no	—
326	#102	—	4	2	no	—
327	C₂²⋊C₈	—	2	4	no	—
328	C₄×C₈	—	2	2	no	—
329	#95	—	2	1	no	—
330	#95	—	2	2	no	—
331	C₂²×C₇	—	2	4	no	—
332	C₂²×C₇	—	2	4	no	—
333	C₂²×C₇	—	2	4	no	—
334	C₂²×C₇	—	4	2	no	—
335	C₂²×C₇	—	4	2	no	—
336	C₂²×C₇	—	4	2	no	—
337	C₂²×C₇	—	4	2	no	—
338	C₂²×C₇	—	4	2	no	—
339	C₂²×C₇	—	4	2	no	—
340	C₂²×C₇	—	4	2	no	—
341	C₂²×C₇	—	8	1	no	—
342	C₂²×C₇	—	8	1	no	—
343	C₂²×C₇	—	8	1	no	—
344	C₂²×C₇	—	8	1	no	—
345	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
346	C₂²×C₇	#546	1	1	yes	characteristic
347	C₂²×C₇	#547	1	1	yes	characteristic
348	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
349	C₂²×C₇	C₂×#137	1	1	yes	characteristic
350	C₂²×C₇	#576	1	1	yes	characteristic
351	C₂²×C₇	#577	1	1	yes	characteristic
352	C₄×C₇	—	2	4	no	—
353	C₄×C₇	—	2	2	no	—
354	C₄×C₇	—	4	2	no	—
355	C₄×C₇	—	4	4	no	—
356	C₄×C₇	—	4	2	no	—
357	C₄×C₇	—	8	1	no	—
358	C₄×C₇	—	8	2	no	—
359	C₄×C₇	#417	1	4	yes	—
360	C₄×C₇	C₄×#41	1	1	yes	characteristic
361	C₄×C₇	#520	1	1	yes	characteristic
362	C₄×C₇	#546	1	1	yes	characteristic
363	C₄×C₇	#547	1	1	yes	characteristic
364	C₂⁴	—	2	1	no	—
365	C₂⁴	C₂×C₇×D₄	1	2	yes	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	8	1	no	—
368	C₂×Q₈	—	8	1	no	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	8	2	no	—
378	C₂×D₄	—	8	2	no	—
379	C₂×D₄	C₇×C₂²⋊C₄	1	4	yes	—
380	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	2	no	—
394	C₂²×C₄	—	4	4	no	—
395	C₂²×C₄	—	8	2	no	—
396	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
397	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
398	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
399	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
400	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
402	M₁₆	—	4	8	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	C₇×#41	1	2	yes	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	4	4	no	—
421	C₄⋊C₄	—	4	2	no	—
422	C₄⋊C₄	—	8	1	no	—
423	C₄⋊C₄	C₇×C₂²⋊C₄	1	4	yes	—
424	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
425	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
426	C₄⋊C₄	C₇×#41	1	2	yes	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
432	C₄²	—	2	8	no	—
433	C₄²	—	4	2	no	—
434	C₄²	C₂²×C₄×C₇	1	2	yes	—
435	C₄²	C₂×C₇×D₄	1	2	yes	—
436	C₂×C₇	—	4	4	no	—
437	C₂×C₇	—	8	2	no	—
438	C₂×C₇	#2339	1	1	yes	characteristic
439	C₂×C₇	C₂×#417	1	1	yes	characteristic
440	C₂×C₇	#2897	1	1	yes	characteristic
441	C₂×C₇	#3325	1	1	yes	characteristic
442	C₂×C₇	#3329	1	1	yes	characteristic
443	C₂×C₇	#3354	1	1	yes	characteristic
444	C₂×C₇	#3357	1	1	yes	characteristic
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	1	no	—
453	C₂³	—	4	1	no	—
454	C₂³	—	4	1	no	—
455	C₂³	—	4	1	no	—
456	C₂³	—	4	1	no	—
457	C₂³	—	4	1	no	—
458	C₂³	C₂×C₇×C₂²⋊C₄	1	4	yes	—
459	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
460	Q₈	—	8	4	no	—
461	D₄	—	2	2	no	—
462	D₄	—	2	2	no	—
463	D₄	—	2	8	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	1	no	—
466	D₄	—	8	8	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	8	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	8	8	no	—
492	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
493	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
494	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
495	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
496	C₂×C₄	C₇×#137	1	2	yes	—
497	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
498	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
499	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
500	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
501	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
502	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
503	C₈	—	2	2	no	—
504	C₈	—	4	2	no	—
505	C₈	—	4	1	no	—
506	C₇	#22503	1	1	yes	characteristic, direct product factor, Sylow
507	C₂²	—	2	4	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	8	1	no	—
518	C₂²	—	8	1	no	—
519	C₂²	—	8	1	no	—
520	C₂²	—	8	1	no	—
521	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
522	C₂²	C₇×#546	1	1	yes	characteristic
523	C₂²	C₇×#547	1	1	yes	characteristic
524	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
525	C₂²	C₂×C₇×#137	1	1	yes	characteristic
526	C₂²	C₇×#576	1	1	yes	characteristic
527	C₂²	C₇×#577	1	1	yes	characteristic
528	C₄	—	2	4	no	—
529	C₄	—	2	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	4	no	—
533	C₄	—	8	2	no	—
534	C₄	—	8	1	no	—
535	C₄	C₇×#417	1	4	yes	—
536	C₄	C₄×C₇×#41	1	1	yes	characteristic
537	C₄	C₇×#520	1	1	yes	characteristic
538	C₄	C₇×#546	1	1	yes	characteristic
539	C₄	C₇×#547	1	1	yes	characteristic
540	C₂	—	4	4	no	—
541	C₂	—	8	2	no	—
542	C₂	C₇×#2339	1	1	yes	characteristic
543	C₂	C₂×C₇×#417	1	1	yes	characteristic
544	C₂	C₇×#2897	1	1	yes	characteristic
545	C₂	C₇×#3325	1	1	yes	characteristic
546	C₂	C₇×#3329	1	1	yes	characteristic
547	C₂	C₇×#3354	1	1	yes	characteristic
548	C₂	C₇×#3357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#436	C₂	semidirect
3. C₇×D₄×C₄⋊C₄	C₂	semidirect
4. C₇×#1990	C₂	semidirect
5. C₇×#1987	C₂	semidirect
6. C₇×#1895	C₂	semidirect
7. C₇×#1874	C₂	semidirect
8. C₇×#1825	C₂	semidirect
9. C₇×#1802	C₂	semidirect
10. C₄×C₇×#102	C₂	semidirect
11. C₇×#1736	C₂	semidirect
14. C₇×#555	C₄	semidirect
44. #22503	C₇	direct (N × H)
59. C₇×C₄⋊Q₈	C₂×C₄	semidirect
63. C₇×#121	C₂×C₄	semidirect
67. C₄×C₇×D₄	C₂×C₄	semidirect
71. C₇×#117	C₂×C₄	semidirect
101. C₂×#436	C₂×C₇	semidirect
102. D₄×C₄⋊C₄	C₂×C₇	semidirect
103. #1990	C₂×C₇	semidirect
104. #1987	C₂×C₇	semidirect
105. #1895	C₂×C₇	semidirect
106. #1874	C₂×C₇	semidirect
107. #1825	C₂×C₇	semidirect
108. #1802	C₂×C₇	semidirect
109. C₄×#102	C₂×C₇	semidirect
110. #1736	C₂×C₇	semidirect
185. #555	C₄×C₇	semidirect
290. C₄⋊Q₈	C₂×C₄×C₇	semidirect
294. #121	C₂×C₄×C₇	semidirect
298. C₄×D₄	C₂×C₄×C₇	semidirect
302. #117	C₂×C₄×C₇	semidirect
506. C₇	#22503	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×#555	2
C₂×C₇×#436	2
C₇×D₄×C₄⋊C₄	2
C₇×#1990	2
C₇×#1987	2
C₇×#1895	2
C₇×#1874	2
C₇×#1825	2
C₇×#1802	2
C₄×C₇×#102	2
C₇×#1736	2
#22503	7

Subgroups — C7×#22503 #13199719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22503 #13199719