Subgroups — C₇×#22759 #13199975

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#2354	C₂	1	1	yes	maximal, characteristic
3	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1936	C₂	1	2	yes	maximal
5	C₇×#1919	C₂	1	2	yes	maximal, complemented
6	C₇×#1880	C₂	1	2	yes	maximal
7	C₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1802	C₂	1	1	yes	maximal, characteristic
10	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1763	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
14	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₇×#117	—	2	1	no	—
17	C₄×C₇×SD₁₆	C₄	1	8	yes	complemented
18	C₂×C₇×#106	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×#103	C₂²	1	2	yes	—
21	C₂×C₇×#103	C₂²	1	1	yes	characteristic
22	C₂×C₇×#102	C₂²	1	2	yes	—
23	C₂×C₇×#102	C₂²	1	1	yes	characteristic
24	C₂×C₇×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₇×#390	C₂²	1	2	yes	—
27	C₇×#389	—	2	2	no	—
28	C₇×#389	C₂²	1	2	yes	—
29	C₇×#388	—	2	2	no	—
30	C₇×#384	—	2	1	no	—
31	C₇×#384	C₂²	1	1	yes	characteristic
32	C₇×#384	C₂²	1	1	yes	characteristic
33	C₇×#380	—	2	1	no	—
34	C₄×C₇×C₄⋊C₄	—	2	1	no	—
35	C₄×C₇×C₄⋊C₄	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×#336	C₂²	1	2	yes	—
42	#22759	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
44	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₇	—	2	1	no	—
46	C₂×C₇×SD₁₆	—	2	2	no	—
47	C₂×C₇×SD₁₆	—	2	2	no	—
48	C₂×C₇×SD₁₆	C₂×C₄	1	4	yes	—
49	C₂²×C₇×C₈	C₂³	1	2	yes	—
50	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₄×C₇×Q₈	—	2	2	no	—
53	C₄×C₇×Q₈	—	2	4	no	—
54	C₄×C₇×Q₈	—	4	2	no	—
55	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
56	C₄×C₇×D₄	—	2	2	no	—
57	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
58	C₇×#117	—	4	4	no	—
59	C₂×C₇×C₄⋊C₄	—	2	1	no	—
60	C₂×C₇×C₄⋊C₄	—	2	1	no	—
61	C₂×C₇×C₄⋊C₄	—	2	1	no	—
62	C₂×C₇×C₄⋊C₄	—	2	2	no	—
63	C₂×C₇×C₄⋊C₄	—	2	2	no	—
64	C₂×C₇×C₄⋊C₄	—	2	1	no	—
65	C₂×C₇×C₄⋊C₄	—	2	1	no	—
66	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
67	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
70	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
73	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
74	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
75	C₂×C₄²×C₇	—	2	1	no	—
76	C₂×C₄²×C₇	—	2	2	no	—
77	C₂×C₄²×C₇	—	2	2	no	—
78	C₂×C₄²×C₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
80	C₇×#106	C₂×C₄	1	4	yes	—
81	C₇×C₄⋊C₈	—	2	2	no	—
82	C₇×#103	—	2	4	no	—
83	C₇×#103	C₂×C₄	1	4	yes	—
84	C₇×#102	—	4	2	no	—
85	C₇×#102	C₂×C₄	1	4	yes	—
86	C₇×#97	—	4	2	no	—
87	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
88	C₇×#95	—	2	2	no	—
89	C₇×#95	—	2	2	no	—
90	C₇×#95	D₄	1	2	yes	—
91	C₂×C₄×SD₁₆	C₂×C₇	1	1	yes	characteristic
92	#2354	C₂×C₇	1	1	yes	characteristic
93	Q₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
94	#1936	C₂×C₇	1	2	yes	—
95	#1919	C₂×C₇	1	2	yes	complemented
96	#1880	C₂×C₇	1	2	yes	—
97	#1834	C₂×C₇	1	1	yes	characteristic, complemented
98	#1803	C₂×C₇	1	1	yes	characteristic, complemented
99	#1802	C₂×C₇	1	1	yes	characteristic
100	#1770	C₂×C₇	1	1	yes	characteristic, complemented
101	#1763	C₂×C₇	1	2	yes	complemented
102	C₂⁴×C₇	—	2	1	no	—
103	C₂×C₇×Q₈	—	2	1	no	—
104	C₂×C₇×Q₈	—	2	1	no	—
105	C₂×C₇×Q₈	—	2	1	no	—
106	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
107	C₂×C₇×Q₈	#41	1	2	yes	—
108	C₂×C₇×Q₈	#41	1	2	yes	—
109	C₂×C₇×D₄	—	2	1	no	—
110	C₂×C₇×D₄	—	2	1	no	—
111	C₂×C₇×D₄	—	2	1	no	—
112	C₂×C₇×D₄	—	2	1	no	—
113	C₂×C₇×D₄	—	2	1	no	—
114	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₇	—	2	1	no	—
116	C₂²×C₄×C₇	—	2	1	no	—
117	C₂²×C₄×C₇	—	2	1	no	—
118	C₂²×C₄×C₇	—	2	4	no	—
119	C₂²×C₄×C₇	—	2	2	no	—
120	C₂²×C₄×C₇	—	2	2	no	—
121	C₂²×C₄×C₇	—	4	2	no	—
122	C₂²×C₄×C₇	—	4	2	no	—
123	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
128	C₇×SD₁₆	—	2	8	no	—
129	C₂×C₇×C₈	—	2	1	no	—
130	C₂×C₇×C₈	—	2	1	no	—
131	C₂×C₇×C₈	—	2	1	no	—
132	C₂×C₇×C₈	—	2	1	no	—
133	C₂×C₇×C₈	—	2	2	no	—
134	C₂×C₇×C₈	—	2	2	no	—
135	C₂×C₇×C₈	—	2	2	no	—
136	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
137	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
138	C₇×C₄⋊C₄	—	2	4	no	—
139	C₇×C₄⋊C₄	—	2	2	no	—
140	C₇×C₄⋊C₄	—	2	2	no	—
141	C₇×C₄⋊C₄	—	2	1	no	—
142	C₇×C₄⋊C₄	—	2	1	no	—
143	C₇×C₄⋊C₄	—	2	2	no	—
144	C₇×C₄⋊C₄	—	2	2	no	—
145	C₇×C₄⋊C₄	—	2	2	no	—
146	C₇×C₄⋊C₄	—	2	2	no	—
147	C₇×C₄⋊C₄	—	4	4	no	—
148	C₇×C₄⋊C₄	—	4	2	no	—
149	C₇×C₄⋊C₄	—	4	2	no	—
150	C₇×C₄⋊C₄	—	8	1	no	—
151	C₇×C₄⋊C₄	—	8	1	no	—
152	C₇×C₄⋊C₄	—	8	2	no	—
153	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
154	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
155	C₇×C₂²⋊C₄	—	2	4	no	—
156	C₇×C₂²⋊C₄	—	4	4	no	—
157	C₇×C₂²⋊C₄	—	8	2	no	—
158	C₄²×C₇	—	2	4	no	—
159	C₄²×C₇	—	2	2	no	—
160	C₄²×C₇	—	2	2	no	—
161	C₄²×C₇	—	4	4	no	—
162	C₄²×C₇	—	4	4	no	—
163	C₄²×C₇	C₂²×C₄	1	2	yes	—
164	C₄²×C₇	#41	1	2	yes	—
165	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
166	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
167	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
168	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
169	C₂×#117	—	2	1	no	—
170	C₄×SD₁₆	C₄×C₇	1	8	yes	complemented
171	C₂×#106	C₂²×C₇	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
173	C₂×#103	C₂²×C₇	1	2	yes	—
174	C₂×#103	C₂²×C₇	1	1	yes	characteristic
175	C₂×#102	C₂²×C₇	1	2	yes	—
176	C₂×#102	C₂²×C₇	1	1	yes	characteristic
177	C₂×#97	C₂²×C₇	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
179	#390	C₂²×C₇	1	2	yes	—
180	#389	—	2	2	no	—
181	#389	C₂²×C₇	1	2	yes	—
182	#388	—	2	2	no	—
183	#384	—	2	1	no	—
184	#384	C₂²×C₇	1	1	yes	characteristic
185	#384	C₂²×C₇	1	1	yes	characteristic
186	#380	—	2	1	no	—
187	C₄×C₄⋊C₄	—	2	1	no	—
188	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
189	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
190	#340	C₂²×C₇	1	2	yes	—
191	#340	C₂²×C₇	1	2	yes	—
192	#340	C₂²×C₇	1	2	yes	—
193	#340	C₂²×C₇	1	2	yes	—
194	#336	C₂²×C₇	1	2	yes	—
195	C₂³×C₇	—	2	2	no	—
196	C₂³×C₇	—	4	1	no	—
197	C₂³×C₇	—	4	1	no	—
198	C₂³×C₇	—	4	1	no	—
199	C₂³×C₇	—	4	1	no	—
200	C₂³×C₇	—	4	1	no	—
201	C₂³×C₇	—	4	1	no	—
202	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
203	C₇×Q₈	—	2	2	no	—
204	C₇×Q₈	—	2	2	no	—
205	C₇×Q₈	—	2	2	no	—
206	C₇×Q₈	#117	1	4	yes	complemented
207	C₇×D₄	—	2	2	no	—
208	C₇×D₄	—	2	2	no	—
209	C₇×D₄	—	4	1	no	—
210	C₇×D₄	—	4	1	no	—
211	C₂×C₄×C₇	—	2	2	no	—
212	C₂×C₄×C₇	—	2	2	no	—
213	C₂×C₄×C₇	—	2	2	no	—
214	C₂×C₄×C₇	—	2	1	no	—
215	C₂×C₄×C₇	—	2	1	no	—
216	C₂×C₄×C₇	—	2	2	no	—
217	C₂×C₄×C₇	—	2	2	no	—
218	C₂×C₄×C₇	—	2	2	no	—
219	C₂×C₄×C₇	—	2	2	no	—
220	C₂×C₄×C₇	—	2	2	no	—
221	C₂×C₄×C₇	—	2	2	no	—
222	C₂×C₄×C₇	—	2	2	no	—
223	C₂×C₄×C₇	—	2	2	no	—
224	C₂×C₄×C₇	—	4	1	no	—
225	C₂×C₄×C₇	—	4	1	no	—
226	C₂×C₄×C₇	—	4	1	no	—
227	C₂×C₄×C₇	—	4	1	no	—
228	C₂×C₄×C₇	—	4	8	no	—
229	C₂×C₄×C₇	—	4	2	no	—
230	C₂×C₄×C₇	—	4	2	no	—
231	C₂×C₄×C₇	—	4	2	no	—
232	C₂×C₄×C₇	—	4	2	no	—
233	C₂×C₄×C₇	—	4	2	no	—
234	C₂×C₄×C₇	—	4	2	no	—
235	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
236	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
237	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
238	C₂×C₄×C₇	#137	1	2	yes	—
239	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
241	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
242	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
245	C₇×C₈	—	2	2	no	—
246	C₇×C₈	—	2	2	no	—
247	C₇×C₈	—	4	2	no	—
248	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
249	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
250	C₂³×C₄	—	2	1	no	—
251	C₂×SD₁₆	—	2	2	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×SD₁₆	C₂×C₄×C₇	1	4	yes	—
254	C₂²×C₈	C₂³×C₇	1	2	yes	—
255	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
257	C₄×Q₈	—	2	2	no	—
258	C₄×Q₈	—	2	4	no	—
259	C₄×Q₈	—	4	2	no	—
260	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
261	C₄×D₄	—	2	2	no	—
262	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
263	#117	—	4	4	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	—	2	1	no	—
278	C₂×C₂²⋊C₄	—	2	2	no	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₄²	—	2	1	no	—
281	C₂×C₄²	—	2	2	no	—
282	C₂×C₄²	—	2	2	no	—
283	C₂×C₄²	C₂³×C₇	1	2	yes	—
284	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
285	#106	C₂×C₄×C₇	1	4	yes	—
286	C₄⋊C₈	—	2	2	no	—
287	#103	—	2	4	no	—
288	#103	C₂×C₄×C₇	1	4	yes	—
289	#102	—	4	2	no	—
290	#102	C₂×C₄×C₇	1	4	yes	—
291	#97	—	4	2	no	—
292	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	C₇×D₄	1	2	yes	—
296	C₂²×C₇	—	4	2	no	—
297	C₂²×C₇	—	4	2	no	—
298	C₂²×C₇	—	4	2	no	—
299	C₂²×C₇	—	8	1	no	—
300	C₂²×C₇	—	8	1	no	—
301	C₂²×C₇	—	8	1	no	—
302	C₂²×C₇	—	8	1	no	—
303	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
304	C₂²×C₇	#546	1	1	yes	characteristic
305	C₂²×C₇	#548	1	1	yes	characteristic
306	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₂²×C₇	C₂×#137	1	1	yes	characteristic
308	C₂²×C₇	#577	1	1	yes	characteristic
309	C₂²×C₇	#577	1	1	yes	characteristic
310	C₄×C₇	—	2	4	no	—
311	C₄×C₇	—	2	2	no	—
312	C₄×C₇	—	4	2	no	—
313	C₄×C₇	—	4	2	no	—
314	C₄×C₇	—	4	2	no	—
315	C₄×C₇	—	4	2	no	—
316	C₄×C₇	—	8	2	no	—
317	C₄×C₇	—	8	2	no	—
318	C₄×C₇	#440	1	4	yes	—
319	C₄×C₇	C₂×#117	1	1	yes	characteristic
320	C₄×C₇	#520	1	1	yes	characteristic
321	C₄×C₇	#552	1	1	yes	characteristic
322	C₄×C₇	#553	1	1	yes	characteristic
323	C₂⁴	—	2	1	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
328	C₂×Q₈	C₇×#41	1	2	yes	—
329	C₂×Q₈	C₇×#41	1	2	yes	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
345	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
346	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
347	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
348	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
349	SD₁₆	—	2	8	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
358	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
359	C₄⋊C₄	—	2	4	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	4	4	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	8	1	no	—
372	C₄⋊C₄	—	8	1	no	—
373	C₄⋊C₄	—	8	2	no	—
374	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
375	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₂²⋊C₄	—	8	2	no	—
379	C₄²	—	2	4	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	4	4	no	—
383	C₄²	—	4	4	no	—
384	C₄²	C₂²×C₄×C₇	1	2	yes	—
385	C₄²	C₇×#41	1	2	yes	—
386	C₂×C₇	—	8	2	no	—
387	C₂×C₇	#2352	1	1	yes	characteristic
388	C₂×C₇	C₂×#440	1	1	yes	characteristic
389	C₂×C₇	#2952	1	1	yes	characteristic
390	C₂×C₇	#3329	1	1	yes	characteristic
391	C₂×C₇	#3332	1	1	yes	characteristic
392	C₂×C₇	#3368	1	1	yes	characteristic
393	C₂×C₇	#3369	1	1	yes	characteristic
394	C₂³	—	2	2	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
402	Q₈	—	2	2	no	—
403	Q₈	—	2	2	no	—
404	Q₈	—	2	2	no	—
405	Q₈	C₇×#117	1	4	yes	complemented
406	D₄	—	2	2	no	—
407	D₄	—	2	2	no	—
408	D₄	—	4	1	no	—
409	D₄	—	4	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	8	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
435	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
436	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
437	C₂×C₄	C₇×#137	1	2	yes	—
438	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
439	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
440	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
441	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
442	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
443	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
444	C₈	—	2	2	no	—
445	C₈	—	2	2	no	—
446	C₈	—	4	2	no	—
447	C₇	#22759	1	1	yes	characteristic, direct product factor, Sylow
448	C₂²	—	4	2	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
456	C₂²	C₇×#546	1	1	yes	characteristic
457	C₂²	C₇×#548	1	1	yes	characteristic
458	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
459	C₂²	C₂×C₇×#137	1	1	yes	characteristic
460	C₂²	C₇×#577	1	1	yes	characteristic
461	C₂²	C₇×#577	1	1	yes	characteristic
462	C₄	—	2	2	no	—
463	C₄	—	2	4	no	—
464	C₄	—	4	2	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	8	2	no	—
469	C₄	—	8	2	no	—
470	C₄	C₇×#440	1	4	yes	—
471	C₄	C₂×C₇×#117	1	1	yes	characteristic
472	C₄	C₇×#520	1	1	yes	characteristic
473	C₄	C₇×#552	1	1	yes	characteristic
474	C₄	C₇×#553	1	1	yes	characteristic
475	C₂	—	8	2	no	—
476	C₂	C₇×#2352	1	1	yes	characteristic
477	C₂	C₂×C₇×#440	1	1	yes	characteristic
478	C₂	C₇×#2952	1	1	yes	characteristic
479	C₂	C₇×#3329	1	1	yes	characteristic
480	C₂	C₇×#3332	1	1	yes	characteristic
481	C₂	C₇×#3368	1	1	yes	characteristic
482	C₂	C₇×#3369	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	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×Q₈×C₄⋊C₄	C₂	semidirect
5. C₇×#1919	C₂	semidirect
7. C₇×#1834	C₂	semidirect
8. C₇×#1803	C₂	semidirect
10. C₇×#1770	C₂	semidirect
11. C₇×#1763	C₂	semidirect
17. C₄×C₇×SD₁₆	C₄	semidirect
42. #22759	C₇	direct (N × H)
93. Q₈×C₄⋊C₄	C₂×C₇	semidirect
95. #1919	C₂×C₇	semidirect
97. #1834	C₂×C₇	semidirect
98. #1803	C₂×C₇	semidirect
100. #1770	C₂×C₇	semidirect
101. #1763	C₂×C₇	semidirect
170. C₄×SD₁₆	C₄×C₇	semidirect
206. C₇×Q₈	#117	semidirect
405. Q₈	C₇×#117	semidirect
447. C₇	#22759	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×C₄×C₇×SD₁₆	2
C₇×#2354	2
C₇×Q₈×C₄⋊C₄	2
C₇×#1936	2
C₇×#1919	2
C₇×#1880	2
C₇×#1834	2
C₇×#1803	2
C₇×#1802	2
C₇×#1770	2
C₇×#1763	2
#22759	7

Subgroups — C7×#22759 #13199975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22759 #13199975