Subgroups — C₇×#22110 #13199326

Order 1792.

848 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#474	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2293	C₂	1	1	yes	maximal, characteristic
4	C₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1942	C₂	1	2	yes	maximal, complemented
7	C₇×#1934	C₂	1	2	yes	maximal
8	C₇×#1925	C₂	1	2	yes	maximal
9	C₇×#1918	C₂	1	2	yes	maximal, complemented
10	C₇×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×Q₈	—	2	1	no	—
14	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×#117	—	2	1	no	—
16	C₇×#474	C₄	1	8	yes	complemented
17	C₂×C₇×#107	C₂²	1	2	yes	—
18	C₂×C₇×#107	C₂²	1	2	yes	—
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₇×#102	C₂²	1	2	yes	—
22	C₂×C₇×#102	C₂²	1	2	yes	—
23	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
24	C₇×#391	—	2	1	no	—
25	C₇×#390	C₂²	1	1	yes	characteristic
26	C₇×#390	C₂²	1	1	yes	characteristic
27	C₇×#389	—	2	1	no	—
28	C₇×#389	C₂²	1	1	yes	characteristic
29	C₇×#389	C₂²	1	1	yes	characteristic
30	C₇×#389	C₂²	1	2	yes	—
31	C₇×#388	—	2	1	no	—
32	C₇×#388	—	2	1	no	—
33	C₇×#384	C₂²	1	1	yes	characteristic
34	C₄×C₇×C₄⋊C₄	—	2	1	no	—
35	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
37	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
38	C₄³×C₇	C₂²	1	1	yes	characteristic
39	C₇×#340	C₂²	1	2	yes	—
40	C₇×#340	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	2	yes	—
42	C₇×#340	C₂²	1	2	yes	—
43	C₇×#336	C₂²	1	2	yes	—
44	#22110	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×Q₈	—	2	1	no	—
46	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₇	—	2	1	no	—
48	C₂²×C₇×C₈	C₂³	1	2	yes	—
49	C₂²×C₇×C₈	C₂³	1	2	yes	—
50	C₇×C₄⋊Q₈	—	2	2	no	—
51	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
52	C₄×C₇×Q₈	—	4	4	no	—
53	C₄×C₇×D₄	—	2	2	no	—
54	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
55	C₇×#117	—	4	4	no	—
56	C₂×C₇×C₄⋊C₄	—	2	1	no	—
57	C₂×C₇×C₄⋊C₄	—	2	1	no	—
58	C₂×C₇×C₄⋊C₄	—	2	1	no	—
59	C₂×C₇×C₄⋊C₄	—	2	1	no	—
60	C₂×C₇×C₄⋊C₄	—	2	1	no	—
61	C₂×C₇×C₄⋊C₄	—	2	2	no	—
62	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
63	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
64	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
66	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
68	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
69	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
70	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
71	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
72	C₂×C₄²×C₇	—	2	1	no	—
73	C₂×C₄²×C₇	—	2	2	no	—
74	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
81	C₇×#107	—	4	2	no	—
82	C₇×#107	C₂×C₄	1	8	yes	—
83	C₇×C₄⋊C₈	—	2	2	no	—
84	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
85	C₇×#102	—	4	2	no	—
86	C₇×#102	C₂×C₄	1	8	yes	—
87	C₄×C₇×C₈	—	4	2	no	—
88	C₇×#95	—	2	1	no	—
89	C₇×#95	—	2	1	no	—
90	C₇×#95	—	2	2	no	—
91	C₇×#95	—	2	1	no	—
92	C₇×#95	—	2	2	no	—
93	C₂×#474	C₂×C₇	1	1	yes	characteristic
94	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
95	#2293	C₂×C₇	1	1	yes	characteristic
96	#1988	C₂×C₇	1	1	yes	characteristic, complemented
97	#1947	C₂×C₇	1	1	yes	characteristic, complemented
98	#1942	C₂×C₇	1	2	yes	complemented
99	#1934	C₂×C₇	1	2	yes	—
100	#1925	C₂×C₇	1	2	yes	—
101	#1918	C₂×C₇	1	2	yes	complemented
102	#1802	C₂×C₇	1	1	yes	characteristic
103	C₄×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
104	C₂⁴×C₇	—	2	1	no	—
105	C₂×C₇×Q₈	—	2	4	no	—
106	C₂×C₇×Q₈	—	4	2	no	—
107	C₂×C₇×Q₈	—	4	2	no	—
108	C₂×C₇×D₄	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₇	—	2	2	no	—
115	C₂²×C₄×C₇	—	2	1	no	—
116	C₂²×C₄×C₇	—	2	4	no	—
117	C₂²×C₄×C₇	—	2	1	no	—
118	C₂²×C₄×C₇	—	2	1	no	—
119	C₂²×C₄×C₇	—	2	2	no	—
120	C₂²×C₄×C₇	—	4	2	no	—
121	C₂²×C₄×C₇	—	4	2	no	—
122	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
129	C₂×C₇×C₈	—	2	2	no	—
130	C₂×C₇×C₈	—	2	2	no	—
131	C₂×C₇×C₈	—	2	2	no	—
132	C₂×C₇×C₈	—	2	2	no	—
133	C₂×C₇×C₈	—	2	2	no	—
134	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
135	C₇×C₄⋊C₄	—	2	4	no	—
136	C₇×C₄⋊C₄	—	2	1	no	—
137	C₇×C₄⋊C₄	—	2	2	no	—
138	C₇×C₄⋊C₄	—	2	2	no	—
139	C₇×C₄⋊C₄	—	2	1	no	—
140	C₇×C₄⋊C₄	—	2	1	no	—
141	C₇×C₄⋊C₄	—	2	1	no	—
142	C₇×C₄⋊C₄	—	2	2	no	—
143	C₇×C₄⋊C₄	—	2	1	no	—
144	C₇×C₄⋊C₄	—	4	2	no	—
145	C₇×C₄⋊C₄	—	4	2	no	—
146	C₇×C₄⋊C₄	—	4	2	no	—
147	C₇×C₄⋊C₄	—	4	4	no	—
148	C₇×C₄⋊C₄	—	4	2	no	—
149	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
150	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
151	C₇×C₂²⋊C₄	—	2	4	no	—
152	C₇×C₂²⋊C₄	—	4	2	no	—
153	C₇×C₂²⋊C₄	—	4	2	no	—
154	C₇×C₂²⋊C₄	—	4	2	no	—
155	C₇×C₂²⋊C₄	—	4	2	no	—
156	C₄²×C₇	—	2	2	no	—
157	C₄²×C₇	—	2	2	no	—
158	C₄²×C₇	—	2	1	no	—
159	C₄²×C₇	—	2	1	no	—
160	C₄²×C₇	—	2	1	no	—
161	C₄²×C₇	—	2	1	no	—
162	C₄²×C₇	—	4	1	no	—
163	C₄²×C₇	—	4	2	no	—
164	C₄²×C₇	—	4	4	no	—
165	C₄²×C₇	D₈	1	4	yes	—
166	C₄²×C₇	C₂²×C₄	1	2	yes	—
167	C₄²×C₇	#41	1	2	yes	—
168	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
169	C₂×C₄×Q₈	—	2	1	no	—
170	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
171	C₂×#117	—	2	1	no	—
172	#474	C₄×C₇	1	8	yes	complemented
173	C₂×#107	C₂²×C₇	1	2	yes	—
174	C₂×#107	C₂²×C₇	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
177	C₂×#102	C₂²×C₇	1	2	yes	—
178	C₂×#102	C₂²×C₇	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
180	#391	—	2	1	no	—
181	#390	C₂²×C₇	1	1	yes	characteristic
182	#390	C₂²×C₇	1	1	yes	characteristic
183	#389	—	2	1	no	—
184	#389	C₂²×C₇	1	2	yes	—
185	#389	C₂²×C₇	1	1	yes	characteristic
186	#389	C₂²×C₇	1	1	yes	characteristic
187	#388	—	2	1	no	—
188	#388	—	2	1	no	—
189	#384	C₂²×C₇	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	2	1	no	—
191	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
192	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
193	C₄×C₂²⋊C₄	—	2	1	no	—
194	C₄³	C₂²×C₇	1	1	yes	characteristic
195	#340	C₂²×C₇	1	2	yes	—
196	#340	C₂²×C₇	1	2	yes	—
197	#340	C₂²×C₇	1	2	yes	—
198	#340	C₂²×C₇	1	2	yes	—
199	#336	C₂²×C₇	1	2	yes	—
200	C₂³×C₇	—	2	2	no	—
201	C₂³×C₇	—	4	1	no	—
202	C₂³×C₇	—	4	1	no	—
203	C₂³×C₇	—	4	1	no	—
204	C₂³×C₇	—	4	1	no	—
205	C₂³×C₇	—	4	1	no	—
206	C₂³×C₇	—	4	1	no	—
207	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
208	C₇×Q₈	—	4	8	no	—
209	C₇×D₄	—	2	2	no	—
210	C₇×D₄	—	2	2	no	—
211	C₇×D₄	—	4	1	no	—
212	C₇×D₄	—	4	1	no	—
213	C₂×C₄×C₇	—	2	1	no	—
214	C₂×C₄×C₇	—	2	1	no	—
215	C₂×C₄×C₇	—	2	1	no	—
216	C₂×C₄×C₇	—	2	1	no	—
217	C₂×C₄×C₇	—	2	1	no	—
218	C₂×C₄×C₇	—	2	1	no	—
219	C₂×C₄×C₇	—	2	1	no	—
220	C₂×C₄×C₇	—	2	1	no	—
221	C₂×C₄×C₇	—	2	2	no	—
222	C₂×C₄×C₇	—	2	4	no	—
223	C₂×C₄×C₇	—	2	1	no	—
224	C₂×C₄×C₇	—	2	1	no	—
225	C₂×C₄×C₇	—	2	1	no	—
226	C₂×C₄×C₇	—	4	2	no	—
227	C₂×C₄×C₇	—	4	1	no	—
228	C₂×C₄×C₇	—	4	1	no	—
229	C₂×C₄×C₇	—	4	2	no	—
230	C₂×C₄×C₇	—	4	8	no	—
231	C₂×C₄×C₇	—	4	1	no	—
232	C₂×C₄×C₇	—	4	2	no	—
233	C₂×C₄×C₇	—	4	1	no	—
234	C₂×C₄×C₇	—	4	1	no	—
235	C₂×C₄×C₇	—	4	1	no	—
236	C₂×C₄×C₇	—	4	2	no	—
237	C₂×C₄×C₇	—	4	1	no	—
238	C₂×C₄×C₇	—	4	1	no	—
239	C₂×C₄×C₇	—	4	2	no	—
240	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
241	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
242	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
243	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
244	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
245	C₂×C₄×C₇	#135	1	2	yes	—
246	C₂×C₄×C₇	#137	1	2	yes	—
247	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
251	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
252	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
253	C₇×C₈	—	2	4	no	—
254	C₇×C₈	—	4	2	no	—
255	C₂²×Q₈	—	2	1	no	—
256	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
257	C₂³×C₄	—	2	1	no	—
258	C₂²×C₈	C₂³×C₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₇	1	2	yes	—
260	C₄⋊Q₈	—	2	2	no	—
261	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	—
262	C₄×Q₈	—	4	4	no	—
263	C₄×D₄	—	2	2	no	—
264	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
265	#117	—	4	4	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
275	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
277	C₂×C₂²⋊C₄	—	2	1	no	—
278	C₂×C₂²⋊C₄	—	2	1	no	—
279	C₂×C₂²⋊C₄	—	2	1	no	—
280	C₂×C₂²⋊C₄	—	2	1	no	—
281	C₂×C₂²⋊C₄	—	2	1	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	—	2	2	no	—
284	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
285	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
286	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
287	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
290	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
291	#107	—	4	2	no	—
292	#107	C₂×C₄×C₇	1	8	yes	—
293	C₄⋊C₈	—	2	2	no	—
294	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
295	#102	—	4	2	no	—
296	#102	C₂×C₄×C₇	1	8	yes	—
297	C₄×C₈	—	4	2	no	—
298	#95	—	2	1	no	—
299	#95	—	2	1	no	—
300	#95	—	2	2	no	—
301	#95	—	2	1	no	—
302	#95	—	2	2	no	—
303	C₂²×C₇	—	4	2	no	—
304	C₂²×C₇	—	4	2	no	—
305	C₂²×C₇	—	4	2	no	—
306	C₂²×C₇	—	8	1	no	—
307	C₂²×C₇	—	8	1	no	—
308	C₂²×C₇	—	8	1	no	—
309	C₂²×C₇	—	8	1	no	—
310	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
311	C₂²×C₇	#536	1	1	yes	characteristic
312	C₂²×C₇	#537	1	1	yes	characteristic
313	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
314	C₂²×C₇	C₂×#135	1	1	yes	characteristic
315	C₂²×C₇	C₂×#137	1	1	yes	characteristic
316	C₂²×C₇	#575	1	1	yes	characteristic
317	C₄×C₇	—	2	2	no	—
318	C₄×C₇	—	2	2	no	—
319	C₄×C₇	—	2	2	no	—
320	C₄×C₇	—	2	2	no	—
321	C₄×C₇	—	4	2	no	—
322	C₄×C₇	—	4	4	no	—
323	C₄×C₇	—	4	1	no	—
324	C₄×C₇	—	8	1	no	—
325	C₄×C₇	—	8	1	no	—
326	C₄×C₇	—	8	2	no	—
327	C₄×C₇	C₄×D₈	1	4	yes	—
328	C₄×C₇	#519	1	1	yes	characteristic
329	C₄×C₇	#520	1	1	yes	characteristic
330	C₄×C₇	#541	1	1	yes	characteristic
331	C₄×C₇	#542	1	1	yes	characteristic
332	C₂⁴	—	2	1	no	—
333	C₂×Q₈	—	2	4	no	—
334	C₂×Q₈	—	4	2	no	—
335	C₂×Q₈	—	4	2	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	—	4	2	no	—
350	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
351	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
352	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
353	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
354	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
355	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
356	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	4	2	no	—
373	C₄⋊C₄	—	4	2	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	—	4	4	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
378	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	4	1	no	—
391	C₄²	—	4	2	no	—
392	C₄²	—	4	4	no	—
393	C₄²	C₇×D₈	1	4	yes	—
394	C₄²	C₂²×C₄×C₇	1	2	yes	—
395	C₄²	C₇×#41	1	2	yes	—
396	C₂×C₇	—	8	2	no	—
397	C₂×C₇	#2323	1	1	yes	characteristic
398	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
399	C₂×C₇	C₄×#137	1	1	yes	characteristic
400	C₂×C₇	#3226	1	1	yes	characteristic
401	C₂×C₇	#3233	1	1	yes	characteristic
402	C₂×C₇	#3245	1	1	yes	characteristic
403	C₂×C₇	#3248	1	1	yes	characteristic
404	C₂³	—	2	2	no	—
405	C₂³	—	4	1	no	—
406	C₂³	—	4	1	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	1	no	—
411	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
412	Q₈	—	4	8	no	—
413	D₄	—	2	2	no	—
414	D₄	—	2	2	no	—
415	D₄	—	4	1	no	—
416	D₄	—	4	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	8	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
445	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
446	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
447	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
448	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
449	C₂×C₄	C₇×#135	1	2	yes	—
450	C₂×C₄	C₇×#137	1	2	yes	—
451	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
452	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
453	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
454	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
455	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
456	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
457	C₈	—	2	4	no	—
458	C₈	—	4	2	no	—
459	C₇	#22110	1	1	yes	characteristic, direct product factor, Sylow
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	8	1	no	—
465	C₂²	—	8	1	no	—
466	C₂²	—	8	1	no	—
467	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
468	C₂²	C₇×#536	1	1	yes	characteristic
469	C₂²	C₇×#537	1	1	yes	characteristic
470	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
471	C₂²	C₂×C₇×#135	1	1	yes	characteristic
472	C₂²	C₂×C₇×#137	1	1	yes	characteristic
473	C₂²	C₇×#575	1	1	yes	characteristic
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	4	no	—
480	C₄	—	4	1	no	—
481	C₄	—	8	1	no	—
482	C₄	—	8	1	no	—
483	C₄	—	8	2	no	—
484	C₄	C₄×C₇×D₈	1	4	yes	—
485	C₄	C₇×#519	1	1	yes	characteristic
486	C₄	C₇×#520	1	1	yes	characteristic
487	C₄	C₇×#541	1	1	yes	characteristic
488	C₄	C₇×#542	1	1	yes	characteristic
489	C₂	—	8	2	no	—
490	C₂	C₇×#2323	1	1	yes	characteristic
491	C₂	C₂×C₄×C₇×D₈	1	1	yes	characteristic
492	C₂	C₄×C₇×#137	1	1	yes	characteristic
493	C₂	C₇×#3226	1	1	yes	characteristic
494	C₂	C₇×#3233	1	1	yes	characteristic
495	C₂	C₇×#3245	1	1	yes	characteristic
496	C₂	C₇×#3248	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 ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×C₄⋊Q₈	C₂	semidirect
4. C₇×#1988	C₂	semidirect
5. C₇×#1947	C₂	semidirect
6. C₇×#1942	C₂	semidirect
9. C₇×#1918	C₂	semidirect
11. C₄×C₇×C₄⋊C₈	C₂	semidirect
16. C₇×#474	C₄	semidirect
44. #22110	C₇	direct (N × H)
94. C₄×C₄⋊Q₈	C₂×C₇	semidirect
96. #1988	C₂×C₇	semidirect
97. #1947	C₂×C₇	semidirect
98. #1942	C₂×C₇	semidirect
101. #1918	C₂×C₇	semidirect
103. C₄×C₄⋊C₈	C₂×C₇	semidirect
172. #474	C₄×C₇	semidirect
459. C₇	#22110	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×#474	2
C₄×C₇×C₄⋊Q₈	2
C₇×#2293	2
C₇×#1988	2
C₇×#1947	2
C₇×#1942	2
C₇×#1934	2
C₇×#1925	2
C₇×#1918	2
C₇×#1802	2
C₄×C₇×C₄⋊C₈	2
#22110	7

Subgroups — C7×#22110 #13199326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22110 #13199326