Subgroups — C₇×#22490 #13199706

Order 1792.

1142 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#2354	C₂	1	1	yes	maximal, characteristic
4	C₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1892	C₂	1	2	yes	maximal, complemented
7	C₇×#1873	C₂	1	2	yes	maximal, complemented
8	C₇×#1825	C₂	1	2	yes	maximal, complemented
9	C₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1735	C₂	1	2	yes	maximal, complemented
12	C₂³×C₇×C₈	C₂²	1	2	yes	—
13	C₇×#556	—	2	4	no	—
14	C₇×#556	C₄	1	8	yes	complemented
15	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×#126	—	2	2	no	—
17	C₂×C₇×#124	—	2	1	no	—
18	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×Q₈	—	2	1	no	—
20	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×#117	C₂²	1	2	yes	—
23	C₂×C₇×#117	C₂²	1	2	yes	—
24	C₇×C₈×D₄	—	2	4	no	—
25	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
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×C₂²⋊C₈	C₂²	1	2	yes	—
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#390	—	2	1	no	—
32	C₇×#388	—	2	2	no	—
33	C₇×#384	C₂²	1	1	yes	characteristic
34	C₇×#380	C₂²	1	2	yes	—
35	C₄×C₇×C₄⋊C₄	—	2	1	no	—
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	#22490	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₇×Q₈	—	2	1	no	—
46	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₇	C₂³	1	2	yes	—
49	C₂²×C₇×C₈	—	2	4	no	—
50	C₂²×C₇×C₈	—	2	4	no	—
51	C₂²×C₇×C₈	—	4	2	no	—
52	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₇×C₈	C₂³	1	2	yes	—
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₇×C₄⋊Q₈	—	2	2	no	—
56	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
57	C₇×#126	—	2	16	no	—
58	C₇×#124	—	2	8	no	—
59	C₇×C₂²⋊Q₈	—	2	4	no	—
60	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
61	C₄×C₇×Q₈	—	2	8	no	—
62	C₄×C₇×Q₈	—	2	2	no	—
63	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
64	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
65	C₄×C₇×D₄	D₄	1	4	yes	—
66	C₇×#117	—	2	4	no	—
67	C₇×#117	—	2	4	no	—
68	C₇×#117	—	4	2	no	—
69	C₇×#117	C₂×C₄	1	8	yes	—
70	C₂×C₇×C₄⋊C₄	—	2	2	no	—
71	C₂×C₇×C₄⋊C₄	—	2	1	no	—
72	C₂×C₇×C₄⋊C₄	—	2	1	no	—
73	C₂×C₇×C₄⋊C₄	—	2	1	no	—
74	C₂×C₇×C₄⋊C₄	—	2	1	no	—
75	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
76	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
77	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₂×C₇×C₄⋊C₄	C₂³	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×C₂²⋊C₄	—	2	2	no	—
82	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
83	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
84	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
86	C₂×C₄²×C₇	—	2	2	no	—
87	C₂×C₄²×C₇	—	2	1	no	—
88	C₂×C₄²×C₇	—	2	2	no	—
89	C₂×C₄²×C₇	—	2	1	no	—
90	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
91	C₇×C₄⋊C₈	—	2	2	no	—
92	C₇×#103	—	2	2	no	—
93	C₇×#103	—	2	2	no	—
94	C₇×#103	—	4	2	no	—
95	C₇×C₂²⋊C₈	—	2	4	no	—
96	C₄×C₇×C₈	—	2	2	no	—
97	C₇×#95	—	2	2	no	—
98	C₇×#95	—	2	1	no	—
99	C₂×#556	C₂×C₇	1	1	yes	characteristic
100	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
101	#2354	C₂×C₇	1	1	yes	characteristic
102	#1990	C₂×C₇	1	1	yes	characteristic, complemented
103	#1988	C₂×C₇	1	1	yes	characteristic, complemented
104	#1892	C₂×C₇	1	2	yes	complemented
105	#1873	C₂×C₇	1	2	yes	complemented
106	#1825	C₂×C₇	1	2	yes	complemented
107	#1803	C₂×C₇	1	1	yes	characteristic, complemented
108	C₄×#103	C₂×C₇	1	1	yes	characteristic, complemented
109	#1735	C₂×C₇	1	2	yes	complemented
110	C₂⁴×C₇	C₂×D₄	1	2	yes	—
111	C₂×C₇×Q₈	—	2	4	no	—
112	C₂×C₇×Q₈	—	2	1	no	—
113	C₂×C₇×Q₈	—	2	1	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	—	4	2	no	—
118	C₂×C₇×Q₈	—	4	2	no	—
119	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
120	C₂×C₇×D₄	—	2	2	no	—
121	C₂×C₇×D₄	—	2	2	no	—
122	C₂×C₇×D₄	C₂²⋊C₄	1	4	yes	—
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	1	no	—
128	C₂²×C₄×C₇	—	2	1	no	—
129	C₂²×C₄×C₇	—	2	1	no	—
130	C₂²×C₄×C₇	—	2	2	no	—
131	C₂²×C₄×C₇	—	2	1	no	—
132	C₂²×C₄×C₇	—	2	2	no	—
133	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
139	C₂×C₇×C₈	—	2	1	no	—
140	C₂×C₇×C₈	—	2	1	no	—
141	C₂×C₇×C₈	—	2	1	no	—
142	C₂×C₇×C₈	—	2	2	no	—
143	C₂×C₇×C₈	—	2	1	no	—
144	C₂×C₇×C₈	—	2	2	no	—
145	C₂×C₇×C₈	—	2	2	no	—
146	C₂×C₇×C₈	—	2	1	no	—
147	C₂×C₇×C₈	—	4	8	no	—
148	C₂×C₇×C₈	#41	1	2	yes	—
149	C₇×C₄⋊C₄	—	2	8	no	—
150	C₇×C₄⋊C₄	—	2	4	no	—
151	C₇×C₄⋊C₄	—	2	4	no	—
152	C₇×C₄⋊C₄	—	2	1	no	—
153	C₇×C₄⋊C₄	—	2	2	no	—
154	C₇×C₄⋊C₄	—	2	1	no	—
155	C₇×C₄⋊C₄	—	2	1	no	—
156	C₇×C₄⋊C₄	—	2	2	no	—
157	C₇×C₄⋊C₄	—	2	2	no	—
158	C₇×C₄⋊C₄	—	2	1	no	—
159	C₇×C₄⋊C₄	—	4	2	no	—
160	C₇×C₄⋊C₄	—	8	1	no	—
161	C₇×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
162	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
163	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
164	C₇×C₄⋊C₄	#41	1	2	yes	—
165	C₇×C₂²⋊C₄	—	2	8	no	—
166	C₇×C₂²⋊C₄	—	2	8	no	—
167	C₇×C₂²⋊C₄	—	4	4	no	—
168	C₇×C₂²⋊C₄	—	8	2	no	—
169	C₇×C₂²⋊C₄	C₂²⋊C₄	1	8	yes	—
170	C₄²×C₇	—	2	8	no	—
171	C₄²×C₇	—	2	4	no	—
172	C₄²×C₇	—	4	4	no	—
173	C₄²×C₇	—	4	2	no	—
174	C₄²×C₇	C₂²×C₄	1	2	yes	—
175	C₄²×C₇	C₂×D₄	1	2	yes	—
176	C₂³×C₈	C₂²×C₇	1	2	yes	—
177	#556	—	2	4	no	—
178	#556	C₄×C₇	1	8	yes	complemented
179	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
180	C₂×#126	—	2	2	no	—
181	C₂×#124	—	2	1	no	—
182	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
183	C₂×C₄×Q₈	—	2	1	no	—
184	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
185	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
186	C₂×#117	C₂²×C₇	1	2	yes	—
187	C₂×#117	C₂²×C₇	1	2	yes	—
188	C₈×D₄	—	2	4	no	—
189	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
190	C₂×#103	C₂²×C₇	1	1	yes	characteristic
191	C₂×#103	C₂²×C₇	1	1	yes	characteristic
192	C₂×#103	C₂²×C₇	1	2	yes	—
193	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
194	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
195	#390	—	2	1	no	—
196	#388	—	2	2	no	—
197	#384	C₂²×C₇	1	1	yes	characteristic
198	#380	C₂²×C₇	1	2	yes	—
199	C₄×C₄⋊C₄	—	2	1	no	—
200	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
201	#340	C₂²×C₇	1	2	yes	—
202	#340	C₂²×C₇	1	1	yes	characteristic
203	#340	C₂²×C₇	1	2	yes	—
204	#340	C₂²×C₇	1	2	yes	—
205	#340	C₂²×C₇	1	2	yes	—
206	#340	C₂²×C₇	1	2	yes	—
207	#340	C₂²×C₇	1	1	yes	characteristic
208	C₂³×C₇	—	2	2	no	—
209	C₂³×C₇	—	2	2	no	—
210	C₂³×C₇	—	2	2	no	—
211	C₂³×C₇	—	2	2	no	—
212	C₂³×C₇	—	2	2	no	—
213	C₂³×C₇	—	2	2	no	—
214	C₂³×C₇	C₂×C₂²⋊C₄	1	4	yes	—
215	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
216	C₇×Q₈	—	2	2	no	—
217	C₇×Q₈	—	2	2	no	—
218	C₇×Q₈	—	4	1	no	—
219	C₇×Q₈	—	4	1	no	—
220	C₇×Q₈	—	4	8	no	—
221	C₇×D₄	—	2	8	no	—
222	C₂×C₄×C₇	—	2	2	no	—
223	C₂×C₄×C₇	—	2	2	no	—
224	C₂×C₄×C₇	—	2	1	no	—
225	C₂×C₄×C₇	—	2	1	no	—
226	C₂×C₄×C₇	—	2	4	no	—
227	C₂×C₄×C₇	—	2	4	no	—
228	C₂×C₄×C₇	—	2	4	no	—
229	C₂×C₄×C₇	—	2	4	no	—
230	C₂×C₄×C₇	—	2	2	no	—
231	C₂×C₄×C₇	—	2	2	no	—
232	C₂×C₄×C₇	—	2	4	no	—
233	C₂×C₄×C₇	—	2	2	no	—
234	C₂×C₄×C₇	—	4	1	no	—
235	C₂×C₄×C₇	—	4	1	no	—
236	C₂×C₄×C₇	—	4	1	no	—
237	C₂×C₄×C₇	—	4	2	no	—
238	C₂×C₄×C₇	—	4	2	no	—
239	C₂×C₄×C₇	—	4	1	no	—
240	C₂×C₄×C₇	—	4	1	no	—
241	C₂×C₄×C₇	—	4	1	no	—
242	C₂×C₄×C₇	—	4	2	no	—
243	C₂×C₄×C₇	—	4	1	no	—
244	C₂×C₄×C₇	—	4	1	no	—
245	C₂×C₄×C₇	—	4	2	no	—
246	C₂×C₄×C₇	—	4	1	no	—
247	C₂×C₄×C₇	—	4	2	no	—
248	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
249	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
250	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
251	C₂×C₄×C₇	#135	1	2	yes	—
252	C₂×C₄×C₇	#135	1	2	yes	—
253	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
257	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
259	C₇×C₈	—	2	2	no	—
260	C₇×C₈	—	4	2	no	—
261	C₇×C₈	—	4	1	no	—
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
264	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₇	1	2	yes	—
266	C₂²×C₈	—	2	4	no	—
267	C₂²×C₈	—	2	4	no	—
268	C₂²×C₈	—	4	2	no	—
269	C₂²×C₈	C₂³×C₇	1	2	yes	—
270	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
272	C₄⋊Q₈	—	2	2	no	—
273	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	—
274	#126	—	2	16	no	—
275	#124	—	2	8	no	—
276	C₂²⋊Q₈	—	2	4	no	—
277	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	—
278	C₄×Q₈	—	2	8	no	—
279	C₄×Q₈	—	2	2	no	—
280	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
281	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
282	C₄×D₄	C₇×D₄	1	4	yes	—
283	#117	—	2	4	no	—
284	#117	—	2	4	no	—
285	#117	—	4	2	no	—
286	#117	C₂×C₄×C₇	1	8	yes	—
287	C₂×C₄⋊C₄	—	2	2	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	—	2	1	no	—
305	C₂×C₄²	—	2	2	no	—
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
308	C₄⋊C₈	—	2	2	no	—
309	#103	—	2	2	no	—
310	#103	—	2	2	no	—
311	#103	—	4	2	no	—
312	C₂²⋊C₈	—	2	4	no	—
313	C₄×C₈	—	2	2	no	—
314	#95	—	2	2	no	—
315	#95	—	2	1	no	—
316	C₂²×C₇	—	2	4	no	—
317	C₂²×C₇	—	2	4	no	—
318	C₂²×C₇	—	2	4	no	—
319	C₂²×C₇	—	4	2	no	—
320	C₂²×C₇	—	4	2	no	—
321	C₂²×C₇	—	4	2	no	—
322	C₂²×C₇	—	4	2	no	—
323	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
324	C₂²×C₇	#546	1	1	yes	characteristic
325	C₂²×C₇	#547	1	1	yes	characteristic
326	C₂²×C₇	C₂×#135	1	1	yes	characteristic
327	C₂²×C₇	C₂×#135	1	1	yes	characteristic
328	C₂²×C₇	#577	1	1	yes	characteristic
329	C₂²×C₇	#578	1	1	yes	characteristic
330	C₄×C₇	—	2	4	no	—
331	C₄×C₇	—	2	2	no	—
332	C₄×C₇	—	4	2	no	—
333	C₄×C₇	—	4	4	no	—
334	C₄×C₇	—	4	2	no	—
335	C₄×C₇	—	4	2	no	—
336	C₄×C₇	—	8	1	no	—
337	C₄×C₇	—	8	2	no	—
338	C₄×C₇	#416	1	4	yes	—
339	C₄×C₇	C₄×#41	1	1	yes	characteristic
340	C₄×C₇	#520	1	1	yes	characteristic
341	C₄×C₇	#546	1	1	yes	characteristic
342	C₄×C₇	#547	1	1	yes	characteristic
343	C₂⁴	C₂×C₇×D₄	1	2	yes	—
344	C₂×Q₈	—	2	4	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×Q₈	—	4	2	no	—
352	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	C₇×C₂²⋊C₄	1	4	yes	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
367	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
368	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
369	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
370	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
371	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	4	8	no	—
381	C₂×C₈	C₇×#41	1	2	yes	—
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	8	1	no	—
394	C₄⋊C₄	C₇×C₂²⋊C₄	1	4	yes	—
395	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
396	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
397	C₄⋊C₄	C₇×#41	1	2	yes	—
398	C₂²⋊C₄	—	2	8	no	—
399	C₂²⋊C₄	—	2	8	no	—
400	C₂²⋊C₄	—	4	4	no	—
401	C₂²⋊C₄	—	8	2	no	—
402	C₂²⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
403	C₄²	—	2	8	no	—
404	C₄²	—	2	4	no	—
405	C₄²	—	4	4	no	—
406	C₄²	—	4	2	no	—
407	C₄²	C₂²×C₄×C₇	1	2	yes	—
408	C₄²	C₂×C₇×D₄	1	2	yes	—
409	C₂×C₇	—	4	4	no	—
410	C₂×C₇	#2339	1	1	yes	characteristic
411	C₂×C₇	C₂×#416	1	1	yes	characteristic
412	C₂×C₇	#2898	1	1	yes	characteristic
413	C₂×C₇	#3302	1	1	yes	characteristic
414	C₂×C₇	#3304	1	1	yes	characteristic
415	C₂×C₇	#3340	1	1	yes	characteristic
416	C₂×C₇	#3342	1	1	yes	characteristic
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	C₂×C₇×C₂²⋊C₄	1	4	yes	—
424	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
425	Q₈	—	2	2	no	—
426	Q₈	—	2	2	no	—
427	Q₈	—	4	1	no	—
428	Q₈	—	4	1	no	—
429	Q₈	—	4	8	no	—
430	D₄	—	2	8	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
458	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
459	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
460	C₂×C₄	C₇×#135	1	2	yes	—
461	C₂×C₄	C₇×#135	1	2	yes	—
462	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
463	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
464	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
465	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
466	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
467	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	4	2	no	—
470	C₈	—	4	1	no	—
471	C₇	#22490	1	1	yes	characteristic, direct product factor, Sylow
472	C₂²	—	2	4	no	—
473	C₂²	—	2	4	no	—
474	C₂²	—	2	4	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
480	C₂²	C₇×#546	1	1	yes	characteristic
481	C₂²	C₇×#547	1	1	yes	characteristic
482	C₂²	C₂×C₇×#135	1	1	yes	characteristic
483	C₂²	C₂×C₇×#135	1	1	yes	characteristic
484	C₂²	C₇×#577	1	1	yes	characteristic
485	C₂²	C₇×#578	1	1	yes	characteristic
486	C₄	—	2	4	no	—
487	C₄	—	2	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	4	no	—
491	C₄	—	4	2	no	—
492	C₄	—	8	2	no	—
493	C₄	—	8	1	no	—
494	C₄	C₇×#416	1	4	yes	—
495	C₄	C₄×C₇×#41	1	1	yes	characteristic
496	C₄	C₇×#520	1	1	yes	characteristic
497	C₄	C₇×#546	1	1	yes	characteristic
498	C₄	C₇×#547	1	1	yes	characteristic
499	C₂	—	4	4	no	—
500	C₂	C₇×#2339	1	1	yes	characteristic
501	C₂	C₂×C₇×#416	1	1	yes	characteristic
502	C₂	C₇×#2898	1	1	yes	characteristic
503	C₂	C₇×#3302	1	1	yes	characteristic
504	C₂	C₇×#3304	1	1	yes	characteristic
505	C₂	C₇×#3340	1	1	yes	characteristic
506	C₂	C₇×#3342	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	D₄
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
4. C₇×#1990	C₂	semidirect
5. C₇×#1988	C₂	semidirect
6. C₇×#1892	C₂	semidirect
7. C₇×#1873	C₂	semidirect
8. C₇×#1825	C₂	semidirect
9. C₇×#1803	C₂	semidirect
10. C₄×C₇×#103	C₂	semidirect
11. C₇×#1735	C₂	semidirect
14. C₇×#556	C₄	semidirect
44. #22490	C₇	direct (N × H)
102. #1990	C₂×C₇	semidirect
103. #1988	C₂×C₇	semidirect
104. #1892	C₂×C₇	semidirect
105. #1873	C₂×C₇	semidirect
106. #1825	C₂×C₇	semidirect
107. #1803	C₂×C₇	semidirect
108. C₄×#103	C₂×C₇	semidirect
109. #1735	C₂×C₇	semidirect
178. #556	C₄×C₇	semidirect
471. C₇	#22490	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×#556	2
C₂×C₇×C₈×D₄	2
C₇×#2354	2
C₇×#1990	2
C₇×#1988	2
C₇×#1892	2
C₇×#1873	2
C₇×#1825	2
C₇×#1803	2
C₄×C₇×#103	2
C₇×#1735	2
#22490	7

Subgroups — C7×#22490 #13199706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22490 #13199706