Subgroups — C₇×#22502 #13199718

Order 1792.

1154 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#436	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×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1894	C₂	1	2	yes	maximal, complemented
7	C₇×#1873	C₂	1	2	yes	maximal, complemented
8	C₇×#1823	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×#1736	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
13	C₇×#554	—	2	4	no	—
14	C₇×#554	C₄	1	8	yes	complemented
15	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×#125	—	2	1	no	—
17	C₂×C₇×C₂²⋊Q₈	—	2	2	no	—
18	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×D₄	—	2	1	no	—
21	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×#117	C₂²	1	2	yes	—
23	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
24	C₇×#436	—	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	#22502	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	—	2	2	no	—
49	C₂³×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₇×M₁₆	—	2	4	no	—
51	C₂×C₇×M₁₆	—	2	4	no	—
52	C₂×C₇×M₁₆	—	4	2	no	—
53	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₇×C₈	C₂³	1	2	yes	—
55	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₇×C₄⋊Q₈	—	2	2	no	—
57	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
58	C₇×#125	—	2	8	no	—
59	C₇×C₂²⋊Q₈	—	2	4	no	—
60	C₇×C₂²⋊Q₈	—	2	16	no	—
61	C₇×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
62	C₄×C₇×Q₈	—	2	2	no	—
63	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
64	C₄×C₇×D₄	—	2	8	no	—
65	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
66	C₄×C₇×D₄	D₄	1	4	yes	—
67	C₇×#117	—	2	4	no	—
68	C₇×#117	—	4	2	no	—
69	C₂×C₇×C₄⋊C₄	—	2	2	no	—
70	C₂×C₇×C₄⋊C₄	—	2	4	no	—
71	C₂×C₇×C₄⋊C₄	—	2	1	no	—
72	C₂×C₇×C₄⋊C₄	—	2	1	no	—
73	C₂×C₇×C₄⋊C₄	—	2	2	no	—
74	C₂×C₇×C₄⋊C₄	—	2	1	no	—
75	C₂×C₇×C₄⋊C₄	—	2	1	no	—
76	C₂×C₇×C₄⋊C₄	C₂×C₄	1	8	yes	—
77	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
78	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
79	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
82	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
84	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
86	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
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×#554	C₂×C₇	1	1	yes	characteristic
100	C₂×#436	C₂×C₇	1	1	yes	characteristic
101	#2354	C₂×C₇	1	1	yes	characteristic
102	#1990	C₂×C₇	1	1	yes	characteristic, complemented
103	#1987	C₂×C₇	1	1	yes	characteristic, complemented
104	#1894	C₂×C₇	1	2	yes	complemented
105	#1873	C₂×C₇	1	2	yes	complemented
106	#1823	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	#1736	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	—	8	1	no	—
118	C₂×C₇×Q₈	—	8	1	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	8	no	—
132	C₂²×C₄×C₇	—	2	1	no	—
133	C₂²×C₄×C₇	—	2	2	no	—
134	C₂²×C₄×C₇	—	8	2	no	—
135	C₂²×C₄×C₇	—	8	2	no	—
136	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
142	C₇×M₁₆	—	4	8	no	—
143	C₂×C₇×C₈	—	2	1	no	—
144	C₂×C₇×C₈	—	2	1	no	—
145	C₂×C₇×C₈	—	2	1	no	—
146	C₂×C₇×C₈	—	2	2	no	—
147	C₂×C₇×C₈	—	2	1	no	—
148	C₂×C₇×C₈	—	2	2	no	—
149	C₂×C₇×C₈	—	2	2	no	—
150	C₂×C₇×C₈	—	2	1	no	—
151	C₂×C₇×C₈	#41	1	2	yes	—
152	C₇×C₄⋊C₄	—	2	8	no	—
153	C₇×C₄⋊C₄	—	2	4	no	—
154	C₇×C₄⋊C₄	—	2	4	no	—
155	C₇×C₄⋊C₄	—	2	8	no	—
156	C₇×C₄⋊C₄	—	2	1	no	—
157	C₇×C₄⋊C₄	—	2	2	no	—
158	C₇×C₄⋊C₄	—	2	1	no	—
159	C₇×C₄⋊C₄	—	2	1	no	—
160	C₇×C₄⋊C₄	—	2	2	no	—
161	C₇×C₄⋊C₄	—	2	2	no	—
162	C₇×C₄⋊C₄	—	2	1	no	—
163	C₇×C₄⋊C₄	—	4	2	no	—
164	C₇×C₄⋊C₄	—	8	1	no	—
165	C₇×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
166	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
167	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
168	C₇×C₄⋊C₄	#41	1	2	yes	—
169	C₇×C₂²⋊C₄	—	2	8	no	—
170	C₇×C₂²⋊C₄	—	4	4	no	—
171	C₇×C₂²⋊C₄	—	8	2	no	—
172	C₇×C₂²⋊C₄	C₂²⋊C₄	1	8	yes	—
173	C₄²×C₇	—	2	4	no	—
174	C₄²×C₇	—	4	4	no	—
175	C₄²×C₇	—	4	2	no	—
176	C₄²×C₇	C₂²×C₄	1	2	yes	—
177	C₄²×C₇	C₂×D₄	1	2	yes	—
178	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
179	#554	—	2	4	no	—
180	#554	C₄×C₇	1	8	yes	complemented
181	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
182	C₂×#125	—	2	1	no	—
183	C₂×C₂²⋊Q₈	—	2	2	no	—
184	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
185	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
186	C₂×C₄×D₄	—	2	1	no	—
187	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
188	C₂×#117	C₂²×C₇	1	2	yes	—
189	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
190	#436	—	2	4	no	—
191	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
192	C₂×#103	C₂²×C₇	1	1	yes	characteristic
193	C₂×#103	C₂²×C₇	1	1	yes	characteristic
194	C₂×#103	C₂²×C₇	1	2	yes	—
195	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
196	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
197	#390	—	2	1	no	—
198	#388	—	2	2	no	—
199	#384	C₂²×C₇	1	1	yes	characteristic
200	#380	C₂²×C₇	1	2	yes	—
201	C₄×C₄⋊C₄	—	2	1	no	—
202	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
203	#340	C₂²×C₇	1	2	yes	—
204	#340	C₂²×C₇	1	1	yes	characteristic
205	#340	C₂²×C₇	1	2	yes	—
206	#340	C₂²×C₇	1	2	yes	—
207	#340	C₂²×C₇	1	2	yes	—
208	#340	C₂²×C₇	1	2	yes	—
209	#340	C₂²×C₇	1	1	yes	characteristic
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	—	2	2	no	—
215	C₂³×C₇	—	2	2	no	—
216	C₂³×C₇	C₂×C₂²⋊C₄	1	4	yes	—
217	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
218	C₇×Q₈	—	2	2	no	—
219	C₇×Q₈	—	2	2	no	—
220	C₇×Q₈	—	4	1	no	—
221	C₇×Q₈	—	4	1	no	—
222	C₇×Q₈	—	8	4	no	—
223	C₇×D₄	—	2	8	no	—
224	C₂×C₄×C₇	—	2	2	no	—
225	C₂×C₄×C₇	—	2	2	no	—
226	C₂×C₄×C₇	—	2	1	no	—
227	C₂×C₄×C₇	—	2	1	no	—
228	C₂×C₄×C₇	—	2	4	no	—
229	C₂×C₄×C₇	—	2	4	no	—
230	C₂×C₄×C₇	—	2	4	no	—
231	C₂×C₄×C₇	—	2	4	no	—
232	C₂×C₄×C₇	—	2	2	no	—
233	C₂×C₄×C₇	—	2	2	no	—
234	C₂×C₄×C₇	—	2	4	no	—
235	C₂×C₄×C₇	—	2	2	no	—
236	C₂×C₄×C₇	—	4	1	no	—
237	C₂×C₄×C₇	—	4	1	no	—
238	C₂×C₄×C₇	—	4	1	no	—
239	C₂×C₄×C₇	—	4	1	no	—
240	C₂×C₄×C₇	—	4	2	no	—
241	C₂×C₄×C₇	—	4	1	no	—
242	C₂×C₄×C₇	—	4	1	no	—
243	C₂×C₄×C₇	—	4	2	no	—
244	C₂×C₄×C₇	—	4	2	no	—
245	C₂×C₄×C₇	—	4	1	no	—
246	C₂×C₄×C₇	—	4	1	no	—
247	C₂×C₄×C₇	—	4	2	no	—
248	C₂×C₄×C₇	—	4	1	no	—
249	C₂×C₄×C₇	—	4	2	no	—
250	C₂×C₄×C₇	—	8	8	no	—
251	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
252	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
253	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
254	C₂×C₄×C₇	#137	1	2	yes	—
255	C₂×C₄×C₇	#137	1	2	yes	—
256	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
261	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
262	C₇×C₈	—	2	2	no	—
263	C₇×C₈	—	4	2	no	—
264	C₇×C₈	—	4	1	no	—
265	C₂²×Q₈	—	2	1	no	—
266	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
267	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
268	C₂³×C₄	—	2	2	no	—
269	C₂³×C₄	C₂³×C₇	1	2	yes	—
270	C₂×M₁₆	—	2	4	no	—
271	C₂×M₁₆	—	2	4	no	—
272	C₂×M₁₆	—	4	2	no	—
273	C₂²×C₈	C₂³×C₇	1	2	yes	—
274	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
276	C₄⋊Q₈	—	2	2	no	—
277	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	—
278	#125	—	2	8	no	—
279	C₂²⋊Q₈	—	2	4	no	—
280	C₂²⋊Q₈	—	2	16	no	—
281	C₂²⋊Q₈	C₂×C₄×C₇	1	8	yes	—
282	C₄×Q₈	—	2	2	no	—
283	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
284	C₄×D₄	—	2	8	no	—
285	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
286	C₄×D₄	C₇×D₄	1	4	yes	—
287	#117	—	2	4	no	—
288	#117	—	4	2	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	4	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	C₂×C₄×C₇	1	8	yes	—
297	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
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	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
307	C₂×C₄²	—	2	1	no	—
308	C₂×C₄²	—	2	2	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
311	C₄⋊C₈	—	2	2	no	—
312	#103	—	2	2	no	—
313	#103	—	2	2	no	—
314	#103	—	4	2	no	—
315	C₂²⋊C₈	—	2	4	no	—
316	C₄×C₈	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	1	no	—
319	C₂²×C₇	—	2	4	no	—
320	C₂²×C₇	—	2	4	no	—
321	C₂²×C₇	—	2	4	no	—
322	C₂²×C₇	—	4	2	no	—
323	C₂²×C₇	—	4	2	no	—
324	C₂²×C₇	—	4	2	no	—
325	C₂²×C₇	—	4	2	no	—
326	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
327	C₂²×C₇	#546	1	1	yes	characteristic
328	C₂²×C₇	#547	1	1	yes	characteristic
329	C₂²×C₇	C₂×#137	1	1	yes	characteristic
330	C₂²×C₇	C₂×#137	1	1	yes	characteristic
331	C₂²×C₇	#576	1	1	yes	characteristic
332	C₂²×C₇	#577	1	1	yes	characteristic
333	C₄×C₇	—	2	4	no	—
334	C₄×C₇	—	2	2	no	—
335	C₄×C₇	—	4	2	no	—
336	C₄×C₇	—	4	2	no	—
337	C₄×C₇	—	4	2	no	—
338	C₄×C₇	—	4	4	no	—
339	C₄×C₇	—	8	2	no	—
340	C₄×C₇	—	8	1	no	—
341	C₄×C₇	#419	1	4	yes	—
342	C₄×C₇	C₄×#41	1	1	yes	characteristic
343	C₄×C₇	#520	1	1	yes	characteristic
344	C₄×C₇	#546	1	1	yes	characteristic
345	C₄×C₇	#547	1	1	yes	characteristic
346	C₂⁴	C₂×C₇×D₄	1	2	yes	—
347	C₂×Q₈	—	2	4	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×Q₈	—	2	1	no	—
353	C₂×Q₈	—	8	1	no	—
354	C₂×Q₈	—	8	1	no	—
355	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	C₇×C₂²⋊C₄	1	4	yes	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	8	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	8	2	no	—
371	C₂²×C₄	—	8	2	no	—
372	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
373	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
374	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
375	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
376	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
378	M₁₆	—	4	8	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	C₇×#41	1	2	yes	—
388	C₄⋊C₄	—	2	8	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	8	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	4	2	no	—
400	C₄⋊C₄	—	8	1	no	—
401	C₄⋊C₄	C₇×C₂²⋊C₄	1	4	yes	—
402	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
403	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
404	C₄⋊C₄	C₇×#41	1	2	yes	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₂²⋊C₄	—	4	4	no	—
407	C₂²⋊C₄	—	8	2	no	—
408	C₂²⋊C₄	C₇×C₂²⋊C₄	1	8	yes	—
409	C₄²	—	2	4	no	—
410	C₄²	—	4	4	no	—
411	C₄²	—	4	2	no	—
412	C₄²	C₂²×C₄×C₇	1	2	yes	—
413	C₄²	C₂×C₇×D₄	1	2	yes	—
414	C₂×C₇	—	4	4	no	—
415	C₂×C₇	#2339	1	1	yes	characteristic
416	C₂×C₇	C₂×#419	1	1	yes	characteristic
417	C₂×C₇	#2897	1	1	yes	characteristic
418	C₂×C₇	#3331	1	1	yes	characteristic
419	C₂×C₇	#3333	1	1	yes	characteristic
420	C₂×C₇	#3356	1	1	yes	characteristic
421	C₂×C₇	#3357	1	1	yes	characteristic
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	C₂×C₇×C₂²⋊C₄	1	4	yes	—
429	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
430	Q₈	—	2	2	no	—
431	Q₈	—	2	2	no	—
432	Q₈	—	4	1	no	—
433	Q₈	—	4	1	no	—
434	Q₈	—	8	4	no	—
435	D₄	—	2	8	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	8	8	no	—
463	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
464	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
465	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
466	C₂×C₄	C₇×#137	1	2	yes	—
467	C₂×C₄	C₇×#137	1	2	yes	—
468	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
469	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
470	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
471	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
472	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
473	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	4	2	no	—
476	C₈	—	4	1	no	—
477	C₇	#22502	1	1	yes	characteristic, direct product factor, Sylow
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
486	C₂²	C₇×#546	1	1	yes	characteristic
487	C₂²	C₇×#547	1	1	yes	characteristic
488	C₂²	C₂×C₇×#137	1	1	yes	characteristic
489	C₂²	C₂×C₇×#137	1	1	yes	characteristic
490	C₂²	C₇×#576	1	1	yes	characteristic
491	C₂²	C₇×#577	1	1	yes	characteristic
492	C₄	—	2	4	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	4	no	—
498	C₄	—	8	2	no	—
499	C₄	—	8	1	no	—
500	C₄	C₇×#419	1	4	yes	—
501	C₄	C₄×C₇×#41	1	1	yes	characteristic
502	C₄	C₇×#520	1	1	yes	characteristic
503	C₄	C₇×#546	1	1	yes	characteristic
504	C₄	C₇×#547	1	1	yes	characteristic
505	C₂	—	4	4	no	—
506	C₂	C₇×#2339	1	1	yes	characteristic
507	C₂	C₂×C₇×#419	1	1	yes	characteristic
508	C₂	C₇×#2897	1	1	yes	characteristic
509	C₂	C₇×#3331	1	1	yes	characteristic
510	C₂	C₇×#3333	1	1	yes	characteristic
511	C₂	C₇×#3356	1	1	yes	characteristic
512	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	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×#1987	C₂	semidirect
6. C₇×#1894	C₂	semidirect
7. C₇×#1873	C₂	semidirect
8. C₇×#1823	C₂	semidirect
9. C₇×#1803	C₂	semidirect
10. C₄×C₇×#103	C₂	semidirect
11. C₇×#1736	C₂	semidirect
14. C₇×#554	C₄	semidirect
44. #22502	C₇	direct (N × H)
102. #1990	C₂×C₇	semidirect
103. #1987	C₂×C₇	semidirect
104. #1894	C₂×C₇	semidirect
105. #1873	C₂×C₇	semidirect
106. #1823	C₂×C₇	semidirect
107. #1803	C₂×C₇	semidirect
108. C₄×#103	C₂×C₇	semidirect
109. #1736	C₂×C₇	semidirect
180. #554	C₄×C₇	semidirect
477. C₇	#22502	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×#554	2
C₂×C₇×#436	2
C₇×#2354	2
C₇×#1990	2
C₇×#1987	2
C₇×#1894	2
C₇×#1873	2
C₇×#1823	2
C₇×#1803	2
C₄×C₇×#103	2
C₇×#1736	2
#22502	7

Subgroups — C7×#22502 #13199718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22502 #13199718