Subgroups — C₇×#22546 #13199762

Order 1792.

1162 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#559	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1983	C₂	1	2	yes	maximal, complemented
5	C₇×#1982	C₂	1	1	yes	maximal, characteristic
6	C₇×#1959	C₂	1	2	yes	maximal, complemented
7	C₇×#1957	C₂	1	1	yes	maximal, characteristic
8	C₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1802	C₂	1	2	yes	maximal, complemented
10	C₇×#1762	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#559	—	2	4	no	—
13	C₇×#559	C₄	1	8	yes	complemented
14	C₂×C₇×#127	C₂²	1	1	yes	characteristic
15	C₂×C₇×#124	—	2	1	no	—
16	C₂×C₇×#124	C₂²	1	2	yes	—
17	C₂×C₇×#123	—	2	2	no	—
18	C₂×C₄×C₇×Q₈	—	2	1	no	—
19	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×D₄	—	2	1	no	—
22	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
23	C₇×#446	—	2	4	no	—
24	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂×C₇×#103	C₂²	1	1	yes	characteristic
27	C₂×C₇×#103	C₂²	1	1	yes	characteristic
28	C₂×C₇×#103	C₂²	1	2	yes	—
29	C₂×C₇×#102	C₂²	1	2	yes	—
30	C₂×C₇×#102	C₂²	1	2	yes	—
31	C₂×C₇×#102	C₂²	1	2	yes	—
32	C₂×C₇×#102	C₂²	1	2	yes	—
33	C₂×C₇×#97	C₂²	1	2	yes	—
34	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
35	C₇×#389	—	2	1	no	—
36	C₇×#389	—	2	2	no	—
37	C₇×#384	C₂²	1	2	yes	—
38	C₇×#384	C₂²	1	1	yes	characteristic
39	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	1	yes	characteristic
42	C₇×#340	C₂²	1	1	yes	characteristic
43	C₇×#340	C₂²	1	2	yes	—
44	#22546	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
46	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇×D₄	—	2	1	no	—
48	C₂²×C₇×D₄	—	2	1	no	—
49	C₂²×C₇×D₄	C₂³	1	2	yes	—
50	C₂³×C₄×C₇	—	2	2	no	—
51	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₇×C₈	C₂³	1	2	yes	—
53	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₇×#127	—	2	2	no	—
55	C₇×#127	C₂×C₄	1	4	yes	—
56	C₇×#124	—	2	8	no	—
57	C₇×#124	—	2	4	no	—
58	C₇×#124	C₂×C₄	1	8	yes	—
59	C₇×#123	—	2	16	no	—
60	C₄×C₇×Q₈	—	2	2	no	—
61	C₄×C₇×Q₈	—	4	4	no	—
62	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
63	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
64	C₄×C₇×Q₈	D₄	1	4	yes	—
65	C₄×C₇×D₄	—	2	8	no	—
66	C₄×C₇×D₄	—	2	4	no	—
67	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
68	C₂×C₇×C₄⋊C₄	—	2	1	no	—
69	C₂×C₇×C₄⋊C₄	—	2	2	no	—
70	C₂×C₇×C₄⋊C₄	—	2	1	no	—
71	C₂×C₇×C₄⋊C₄	—	2	2	no	—
72	C₂×C₇×C₄⋊C₄	—	2	1	no	—
73	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
75	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
76	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
80	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
81	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
82	C₂×C₄²×C₇	—	2	1	no	—
83	C₂×C₄²×C₇	—	2	2	no	—
84	C₂×C₄²×C₇	—	2	1	no	—
85	C₂×C₄²×C₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
87	C₇×C₄⋊C₈	—	2	2	no	—
88	C₇×C₄⋊C₈	—	2	4	no	—
89	C₇×#103	—	2	2	no	—
90	C₇×#103	—	2	2	no	—
91	C₇×#103	—	4	2	no	—
92	C₇×#102	—	2	4	no	—
93	C₇×#102	—	2	4	no	—
94	C₇×#102	—	4	2	no	—
95	C₇×#102	—	4	2	no	—
96	C₇×#97	—	2	4	no	—
97	C₄×C₇×C₈	—	2	2	no	—
98	C₇×#95	—	2	2	no	—
99	C₇×#95	—	2	1	no	—
100	C₂×#559	C₂×C₇	1	1	yes	characteristic
101	C₂×#446	C₂×C₇	1	1	yes	characteristic, complemented
102	Q₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
103	#1983	C₂×C₇	1	2	yes	complemented
104	#1982	C₂×C₇	1	1	yes	characteristic
105	#1959	C₂×C₇	1	2	yes	complemented
106	#1957	C₂×C₇	1	1	yes	characteristic
107	#1803	C₂×C₇	1	1	yes	characteristic, complemented
108	#1802	C₂×C₇	1	2	yes	complemented
109	#1762	C₂×C₇	1	2	yes	complemented
110	C₄×#103	C₂×C₇	1	1	yes	characteristic, complemented
111	C₂⁴×C₇	—	2	2	no	—
112	C₂×C₇×Q₈	—	2	2	no	—
113	C₂×C₇×Q₈	—	2	2	no	—
114	C₂×C₇×Q₈	—	2	1	no	—
115	C₂×C₇×Q₈	—	2	1	no	—
116	C₂×C₇×Q₈	—	2	1	no	—
117	C₂×C₇×Q₈	—	2	1	no	—
118	C₂×C₇×Q₈	—	2	1	no	—
119	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
120	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
121	C₂×C₇×D₄	—	2	4	no	—
122	C₂×C₇×D₄	—	2	4	no	—
123	C₂×C₇×D₄	—	2	2	no	—
124	C₂×C₇×D₄	—	2	2	no	—
125	C₂×C₇×D₄	—	2	2	no	—
126	C₂×C₇×D₄	—	2	2	no	—
127	C₂×C₇×D₄	—	2	2	no	—
128	C₂×C₇×D₄	—	4	2	no	—
129	C₂×C₇×D₄	—	4	2	no	—
130	C₂×C₇×D₄	—	8	1	no	—
131	C₂×C₇×D₄	—	8	1	no	—
132	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₇	—	2	1	no	—
134	C₂²×C₄×C₇	—	2	1	no	—
135	C₂²×C₄×C₇	—	2	1	no	—
136	C₂²×C₄×C₇	—	2	8	no	—
137	C₂²×C₄×C₇	—	2	2	no	—
138	C₂²×C₄×C₇	—	2	1	no	—
139	C₂²×C₄×C₇	—	8	2	no	—
140	C₂²×C₄×C₇	—	8	2	no	—
141	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
146	C₂×C₇×C₈	—	2	1	no	—
147	C₂×C₇×C₈	—	2	1	no	—
148	C₂×C₇×C₈	—	2	1	no	—
149	C₂×C₇×C₈	—	2	2	no	—
150	C₂×C₇×C₈	—	2	2	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	#41	1	2	yes	—
155	C₇×C₄⋊C₄	—	2	4	no	—
156	C₇×C₄⋊C₄	—	2	1	no	—
157	C₇×C₄⋊C₄	—	2	1	no	—
158	C₇×C₄⋊C₄	—	2	2	no	—
159	C₇×C₄⋊C₄	—	2	1	no	—
160	C₇×C₄⋊C₄	—	2	1	no	—
161	C₇×C₄⋊C₄	—	2	2	no	—
162	C₇×C₄⋊C₄	—	4	4	no	—
163	C₇×C₄⋊C₄	—	4	2	no	—
164	C₇×C₄⋊C₄	—	8	2	no	—
165	C₇×C₄⋊C₄	—	8	1	no	—
166	C₇×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
167	C₇×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
168	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
169	C₇×C₄⋊C₄	#41	1	2	yes	—
170	C₇×C₂²⋊C₄	—	2	8	no	—
171	C₇×C₂²⋊C₄	—	2	8	no	—
172	C₇×C₂²⋊C₄	—	2	8	no	—
173	C₄²×C₇	—	2	4	no	—
174	C₄²×C₇	—	4	4	no	—
175	C₄²×C₇	—	4	2	no	—
176	C₄²×C₇	C₂²×C₄	1	4	yes	—
177	C₄²×C₇	C₂²×C₄	1	2	yes	—
178	C₄²×C₇	C₂×D₄	1	2	yes	—
179	C₄²×C₇	C₂×D₄	1	4	yes	—
180	#559	—	2	4	no	—
181	#559	C₄×C₇	1	8	yes	complemented
182	C₂×#127	C₂²×C₇	1	1	yes	characteristic
183	C₂×#124	—	2	1	no	—
184	C₂×#124	C₂²×C₇	1	2	yes	—
185	C₂×#123	—	2	2	no	—
186	C₂×C₄×Q₈	—	2	1	no	—
187	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
188	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
189	C₂×C₄×D₄	—	2	1	no	—
190	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
191	#446	—	2	4	no	—
192	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
194	C₂×#103	C₂²×C₇	1	1	yes	characteristic
195	C₂×#103	C₂²×C₇	1	1	yes	characteristic
196	C₂×#103	C₂²×C₇	1	2	yes	—
197	C₂×#102	C₂²×C₇	1	2	yes	—
198	C₂×#102	C₂²×C₇	1	2	yes	—
199	C₂×#102	C₂²×C₇	1	2	yes	—
200	C₂×#102	C₂²×C₇	1	2	yes	—
201	C₂×#97	C₂²×C₇	1	2	yes	—
202	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
203	#389	—	2	1	no	—
204	#389	—	2	2	no	—
205	#384	C₂²×C₇	1	2	yes	—
206	#384	C₂²×C₇	1	1	yes	characteristic
207	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
208	C₄×C₄⋊C₄	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	1	yes	characteristic
212	C₂³×C₇	—	2	4	no	—
213	C₂³×C₇	—	4	2	no	—
214	C₂³×C₇	—	4	2	no	—
215	C₂³×C₇	—	4	2	no	—
216	C₂³×C₇	—	4	2	no	—
217	C₂³×C₇	—	4	2	no	—
218	C₂³×C₇	—	4	2	no	—
219	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
220	C₇×Q₈	—	2	8	no	—
221	C₇×Q₈	—	2	2	no	—
222	C₇×Q₈	—	2	2	no	—
223	C₇×Q₈	—	4	1	no	—
224	C₇×Q₈	—	4	1	no	—
225	C₇×D₄	—	2	4	no	—
226	C₇×D₄	—	2	4	no	—
227	C₇×D₄	—	4	2	no	—
228	C₇×D₄	—	4	2	no	—
229	C₇×D₄	—	4	8	no	—
230	C₇×D₄	—	8	4	no	—
231	C₂×C₄×C₇	—	2	2	no	—
232	C₂×C₄×C₇	—	2	2	no	—
233	C₂×C₄×C₇	—	2	2	no	—
234	C₂×C₄×C₇	—	2	1	no	—
235	C₂×C₄×C₇	—	2	2	no	—
236	C₂×C₄×C₇	—	2	1	no	—
237	C₂×C₄×C₇	—	2	2	no	—
238	C₂×C₄×C₇	—	2	2	no	—
239	C₂×C₄×C₇	—	2	2	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	1	no	—
242	C₂×C₄×C₇	—	4	1	no	—
243	C₂×C₄×C₇	—	4	1	no	—
244	C₂×C₄×C₇	—	4	1	no	—
245	C₂×C₄×C₇	—	4	1	no	—
246	C₂×C₄×C₇	—	4	2	no	—
247	C₂×C₄×C₇	—	4	1	no	—
248	C₂×C₄×C₇	—	4	2	no	—
249	C₂×C₄×C₇	—	4	1	no	—
250	C₂×C₄×C₇	—	4	1	no	—
251	C₂×C₄×C₇	—	4	2	no	—
252	C₂×C₄×C₇	—	8	8	no	—
253	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
254	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
255	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
256	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
257	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
258	C₂×C₄×C₇	#137	1	2	yes	—
259	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
260	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
263	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
264	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
265	C₇×C₈	—	2	2	no	—
266	C₇×C₈	—	4	1	no	—
267	C₇×C₈	—	4	2	no	—
268	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
269	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
270	C₂²×D₄	—	2	1	no	—
271	C₂²×D₄	—	2	1	no	—
272	C₂²×D₄	C₂³×C₇	1	2	yes	—
273	C₂³×C₄	—	2	2	no	—
274	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₇	1	2	yes	—
276	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
277	#127	—	2	2	no	—
278	#127	C₂×C₄×C₇	1	4	yes	—
279	#124	—	2	8	no	—
280	#124	—	2	4	no	—
281	#124	C₂×C₄×C₇	1	8	yes	—
282	#123	—	2	16	no	—
283	C₄×Q₈	—	2	2	no	—
284	C₄×Q₈	—	4	4	no	—
285	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
286	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
287	C₄×Q₈	C₇×D₄	1	4	yes	—
288	C₄×D₄	—	2	8	no	—
289	C₄×D₄	—	2	4	no	—
290	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	—	2	2	no	—
307	C₂×C₄²	—	2	1	no	—
308	C₂×C₄²	C₂³×C₇	1	2	yes	—
309	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
310	C₄⋊C₈	—	2	2	no	—
311	C₄⋊C₈	—	2	4	no	—
312	#103	—	2	2	no	—
313	#103	—	2	2	no	—
314	#103	—	4	2	no	—
315	#102	—	2	4	no	—
316	#102	—	2	4	no	—
317	#102	—	4	2	no	—
318	#102	—	4	2	no	—
319	#97	—	2	4	no	—
320	C₄×C₈	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	—	2	1	no	—
323	C₂²×C₇	—	4	4	no	—
324	C₂²×C₇	—	4	4	no	—
325	C₂²×C₇	—	4	4	no	—
326	C₂²×C₇	—	8	2	no	—
327	C₂²×C₇	—	8	2	no	—
328	C₂²×C₇	—	8	2	no	—
329	C₂²×C₇	—	8	2	no	—
330	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
331	C₂²×C₇	#548	1	1	yes	characteristic
332	C₂²×C₇	#550	1	1	yes	characteristic
333	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂²×C₇	C₂×#137	1	1	yes	characteristic
335	C₂²×C₇	#576	1	1	yes	characteristic
336	C₂²×C₇	#577	1	1	yes	characteristic
337	C₄×C₇	—	2	4	no	—
338	C₄×C₇	—	2	4	no	—
339	C₄×C₇	—	2	2	no	—
340	C₄×C₇	—	4	2	no	—
341	C₄×C₇	—	4	2	no	—
342	C₄×C₇	—	4	2	no	—
343	C₄×C₇	—	8	2	no	—
344	C₄×C₇	—	8	1	no	—
345	C₄×C₇	#417	1	4	yes	—
346	C₄×C₇	C₄×#41	1	1	yes	characteristic
347	C₄×C₇	#520	1	1	yes	characteristic
348	C₄×C₇	#548	1	1	yes	characteristic
349	C₄×C₇	#550	1	1	yes	characteristic
350	C₂⁴	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
359	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
360	C₂×D₄	—	2	4	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	4	2	no	—
368	C₂×D₄	—	4	2	no	—
369	C₂×D₄	—	8	1	no	—
370	C₂×D₄	—	8	1	no	—
371	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	8	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	8	2	no	—
379	C₂²×C₄	—	8	2	no	—
380	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
381	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
382	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
383	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
384	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	C₇×#41	1	2	yes	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	4	4	no	—
402	C₄⋊C₄	—	4	2	no	—
403	C₄⋊C₄	—	8	2	no	—
404	C₄⋊C₄	—	8	1	no	—
405	C₄⋊C₄	C₇×C₂²⋊C₄	1	4	yes	—
406	C₄⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
407	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
408	C₄⋊C₄	C₇×#41	1	2	yes	—
409	C₂²⋊C₄	—	2	8	no	—
410	C₂²⋊C₄	—	2	8	no	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₄²	—	2	4	no	—
413	C₄²	—	4	4	no	—
414	C₄²	—	4	2	no	—
415	C₄²	C₂²×C₄×C₇	1	4	yes	—
416	C₄²	C₂²×C₄×C₇	1	2	yes	—
417	C₄²	C₂×C₇×D₄	1	2	yes	—
418	C₄²	C₂×C₇×D₄	1	4	yes	—
419	C₂×C₇	—	8	4	no	—
420	C₂×C₇	#2341	1	1	yes	characteristic
421	C₂×C₇	C₂×#417	1	1	yes	characteristic
422	C₂×C₇	#2897	1	1	yes	characteristic
423	C₂×C₇	#3373	1	1	yes	characteristic
424	C₂×C₇	#3375	1	1	yes	characteristic
425	C₂×C₇	#3399	1	1	yes	characteristic
426	C₂×C₇	#3400	1	1	yes	characteristic
427	C₂³	—	2	4	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	2	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
435	Q₈	—	2	8	no	—
436	Q₈	—	2	2	no	—
437	Q₈	—	2	2	no	—
438	Q₈	—	4	1	no	—
439	Q₈	—	4	1	no	—
440	D₄	—	2	4	no	—
441	D₄	—	2	4	no	—
442	D₄	—	4	2	no	—
443	D₄	—	4	2	no	—
444	D₄	—	4	8	no	—
445	D₄	—	8	4	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	8	8	no	—
468	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
469	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
470	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
471	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
472	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
473	C₂×C₄	C₇×#137	1	2	yes	—
474	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
475	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
476	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
477	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
478	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
479	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
480	C₈	—	2	2	no	—
481	C₈	—	4	1	no	—
482	C₈	—	4	2	no	—
483	C₇	#22546	1	1	yes	characteristic, direct product factor, Sylow
484	C₂²	—	4	4	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	8	2	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	2	no	—
490	C₂²	—	8	2	no	—
491	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
492	C₂²	C₇×#548	1	1	yes	characteristic
493	C₂²	C₇×#550	1	1	yes	characteristic
494	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
495	C₂²	C₂×C₇×#137	1	1	yes	characteristic
496	C₂²	C₇×#576	1	1	yes	characteristic
497	C₂²	C₇×#577	1	1	yes	characteristic
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	8	2	no	—
505	C₄	—	8	1	no	—
506	C₄	C₇×#417	1	4	yes	—
507	C₄	C₄×C₇×#41	1	1	yes	characteristic
508	C₄	C₇×#520	1	1	yes	characteristic
509	C₄	C₇×#548	1	1	yes	characteristic
510	C₄	C₇×#550	1	1	yes	characteristic
511	C₂	—	8	4	no	—
512	C₂	C₇×#2341	1	1	yes	characteristic
513	C₂	C₂×C₇×#417	1	1	yes	characteristic
514	C₂	C₇×#2897	1	1	yes	characteristic
515	C₂	C₇×#3373	1	1	yes	characteristic
516	C₂	C₇×#3375	1	1	yes	characteristic
517	C₂	C₇×#3399	1	1	yes	characteristic
518	C₂	C₇×#3400	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×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#446	C₂	semidirect
3. C₇×Q₈×C₄⋊C₄	C₂	semidirect
4. C₇×#1983	C₂	semidirect
6. C₇×#1959	C₂	semidirect
8. C₇×#1803	C₂	semidirect
9. C₇×#1802	C₂	semidirect
10. C₇×#1762	C₂	semidirect
11. C₄×C₇×#103	C₂	semidirect
13. C₇×#559	C₄	semidirect
44. #22546	C₇	direct (N × H)
101. C₂×#446	C₂×C₇	semidirect
102. Q₈×C₄⋊C₄	C₂×C₇	semidirect
103. #1983	C₂×C₇	semidirect
105. #1959	C₂×C₇	semidirect
107. #1803	C₂×C₇	semidirect
108. #1802	C₂×C₇	semidirect
109. #1762	C₂×C₇	semidirect
110. C₄×#103	C₂×C₇	semidirect
181. #559	C₄×C₇	semidirect
483. C₇	#22546	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×#559	2
C₂×C₇×#446	2
C₇×Q₈×C₄⋊C₄	2
C₇×#1983	2
C₇×#1982	2
C₇×#1959	2
C₇×#1957	2
C₇×#1803	2
C₇×#1802	2
C₇×#1762	2
C₄×C₇×#103	2
#22546	7

Subgroups — C7×#22546 #13199762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22546 #13199762