Subgroups — C₇×#11294 #13188510

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2911	C₂	1	1	yes	maximal, characteristic
2	C₇×#1742	C₂	1	2	yes	maximal
3	C₂×C₇×#339	C₂	1	2	yes	maximal
4	C₂×C₇×#338	C₂	1	1	yes	maximal, characteristic
5	C₇×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₇×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₇×#428	—	2	4	no	—
12	C₇×#427	—	2	2	no	—
13	C₇×#343	—	2	4	no	—
14	C₇×#339	—	2	4	no	—
15	C₇×#338	C₄	1	4	yes	complemented
16	C₇×#336	—	2	2	no	—
17	#11294	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	—	2	1	no	—
19	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	—	2	1	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	—	2	2	no	—
27	C₂×C₇×M₁₆	—	2	2	no	—
28	C₂×C₇×M₁₆	—	2	2	no	—
29	C₂×C₇×M₁₆	—	2	2	no	—
30	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₇×M₁₆	Q₈	1	2	yes	—
33	C₂×C₇×M₁₆	Q₈	1	2	yes	—
34	C₂²×C₇×C₈	—	2	1	no	—
35	C₂²×C₇×C₈	—	2	2	no	—
36	C₂²×C₇×C₈	D₄	1	2	yes	—
37	C₇×#117	—	2	2	no	—
38	C₇×#117	—	2	2	no	—
39	C₇×#117	—	2	2	no	—
40	C₇×#117	—	2	2	no	—
41	C₂×C₇×C₄⋊C₄	D₄	1	4	yes	—
42	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₄²×C₇	—	2	2	no	—
44	C₂×C₄²×C₇	—	2	2	no	—
45	C₂×C₄²×C₇	—	2	1	no	—
46	C₂×C₄²×C₇	—	2	2	no	—
47	C₂×C₄²×C₇	D₄	1	2	yes	—
48	C₇×#107	—	2	4	no	—
49	C₇×#106	—	2	4	no	—
50	#2911	C₂×C₇	1	1	yes	characteristic
51	#1742	C₂×C₇	1	2	yes	—
52	C₂×#339	C₂×C₇	1	2	yes	—
53	C₂×#338	C₂×C₇	1	1	yes	characteristic
54	#1728	C₂×C₇	1	1	yes	characteristic
55	C₂⁴×C₇	#41	1	1	yes	characteristic
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	2	4	no	—
58	C₂²×C₄×C₇	—	4	4	no	—
59	C₂²×C₄×C₇	—	4	2	no	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇	#41	1	2	yes	—
67	C₂²×C₄×C₇	#41	1	2	yes	—
68	C₂²×C₄×C₇	#41	1	2	yes	—
69	C₂²×C₄×C₇	#41	1	1	yes	characteristic
70	C₇×M₁₆	—	2	4	no	—
71	C₇×M₁₆	—	2	4	no	—
72	C₇×M₁₆	—	2	4	no	—
73	C₇×M₁₆	—	2	4	no	—
74	C₇×M₁₆	—	2	4	no	—
75	C₇×M₁₆	—	4	2	no	—
76	C₇×M₁₆	—	4	2	no	—
77	C₇×M₁₆	—	4	2	no	—
78	C₂×C₇×C₈	—	2	2	no	—
79	C₂×C₇×C₈	—	2	2	no	—
80	C₂×C₇×C₈	—	2	4	no	—
81	C₂×C₇×C₈	—	2	2	no	—
82	C₂×C₇×C₈	—	4	1	no	—
83	C₂×C₇×C₈	—	4	2	no	—
84	C₂×C₇×C₈	—	4	1	no	—
85	C₂×C₇×C₈	—	4	2	no	—
86	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
87	C₇×C₄⋊C₄	—	2	4	no	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₂²⋊C₄	—	4	2	no	—
90	C₇×C₂²⋊C₄	—	4	2	no	—
91	C₄²×C₇	—	2	4	no	—
92	C₄²×C₇	—	4	2	no	—
93	C₄²×C₇	—	4	1	no	—
94	C₄²×C₇	—	4	1	no	—
95	C₄²×C₇	—	4	2	no	—
96	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
97	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
98	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
99	C₂×#117	C₂²×C₇	1	2	yes	—
100	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
101	#428	—	2	4	no	—
102	#427	—	2	2	no	—
103	#343	—	2	4	no	—
104	#339	—	2	4	no	—
105	#338	C₄×C₇	1	4	yes	complemented
106	#336	—	2	2	no	—
107	C₂³×C₇	—	4	2	no	—
108	C₂³×C₇	C₄×Q₈	1	1	yes	characteristic
109	C₂³×C₇	#121	1	1	yes	characteristic
110	C₂³×C₇	#123	1	2	yes	—
111	C₂³×C₇	#125	1	1	yes	characteristic
112	C₂³×C₇	#126	1	2	yes	—
113	C₂×C₄×C₇	—	2	4	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	—	4	8	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	4	4	no	—
118	C₂×C₄×C₇	—	8	4	no	—
119	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇	#117	1	2	yes	—
121	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇	#120	1	1	yes	characteristic
124	C₂×C₄×C₇	#121	1	2	yes	—
125	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇	#123	1	1	yes	characteristic
129	C₂×C₄×C₇	#123	1	2	yes	—
130	C₂×C₄×C₇	#124	1	1	yes	characteristic
131	C₂×C₄×C₇	#124	1	2	yes	—
132	C₂×C₄×C₇	#124	1	1	yes	characteristic
133	C₂×C₄×C₇	#125	1	2	yes	—
134	C₂×C₄×C₇	#125	1	2	yes	—
135	C₂×C₄×C₇	#126	1	2	yes	—
136	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
137	C₇×C₈	—	2	4	no	—
138	C₇×C₈	—	4	4	no	—
139	C₇×C₈	—	4	2	no	—
140	C₂³×C₄	—	2	1	no	—
141	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	1	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
153	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
154	C₂×M₁₆	C₇×Q₈	1	2	yes	—
155	C₂×M₁₆	C₇×Q₈	1	2	yes	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	C₇×D₄	1	2	yes	—
159	#117	—	2	2	no	—
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	C₂×C₄⋊C₄	C₇×D₄	1	4	yes	—
164	C₂×C₂²⋊C₄	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₇×D₄	1	2	yes	—
170	#107	—	2	4	no	—
171	#106	—	2	4	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	4	no	—
174	C₂²×C₇	—	4	2	no	—
175	C₂²×C₇	—	4	1	no	—
176	C₂²×C₇	—	4	1	no	—
177	C₂²×C₇	#387	1	2	yes	—
178	C₂²×C₇	#389	1	1	yes	characteristic
179	C₂²×C₇	#394	1	1	yes	characteristic
180	C₂²×C₇	#397	1	1	yes	characteristic
181	C₂²×C₇	#400	1	2	yes	—
182	C₄×C₇	—	4	4	no	—
183	C₄×C₇	—	8	4	no	—
184	C₄×C₇	#382	1	1	yes	characteristic
185	C₄×C₇	#385	1	1	yes	characteristic
186	C₄×C₇	#388	1	2	yes	—
187	C₄×C₇	#393	1	1	yes	characteristic
188	C₄×C₇	#398	1	2	yes	—
189	C₄×C₇	#400	1	1	yes	characteristic
190	C₂⁴	C₇×#41	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
198	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₇×#41	1	2	yes	—
202	C₂²×C₄	C₇×#41	1	2	yes	—
203	C₂²×C₄	C₇×#41	1	2	yes	—
204	C₂²×C₄	C₇×#41	1	1	yes	characteristic
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	4	2	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	4	2	no	—
231	C₂×C₇	—	2	1	no	—
232	C₂×C₇	—	2	2	no	—
233	C₂×C₇	—	2	1	no	—
234	C₂×C₇	—	2	2	no	—
235	C₂×C₇	#1442	1	1	yes	characteristic
236	C₂×C₇	#2050	1	2	yes	—
237	C₂³	—	4	2	no	—
238	C₂³	C₄×C₇×Q₈	1	1	yes	characteristic
239	C₂³	C₇×#121	1	1	yes	characteristic
240	C₂³	C₇×#123	1	2	yes	—
241	C₂³	C₇×#125	1	1	yes	characteristic
242	C₂³	C₇×#126	1	2	yes	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₇×#117	1	2	yes	—
251	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
252	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
253	C₂×C₄	C₇×#120	1	1	yes	characteristic
254	C₂×C₄	C₇×#121	1	2	yes	—
255	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
257	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
258	C₂×C₄	C₇×#123	1	1	yes	characteristic
259	C₂×C₄	C₇×#123	1	2	yes	—
260	C₂×C₄	C₇×#124	1	1	yes	characteristic
261	C₂×C₄	C₇×#124	1	1	yes	characteristic
262	C₂×C₄	C₇×#124	1	2	yes	—
263	C₂×C₄	C₇×#125	1	2	yes	—
264	C₂×C₄	C₇×#125	1	2	yes	—
265	C₂×C₄	C₇×#126	1	2	yes	—
266	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	4	4	no	—
269	C₈	—	4	2	no	—
270	C₇	#11294	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	C₇×#387	1	2	yes	—
277	C₂²	C₇×#389	1	1	yes	characteristic
278	C₂²	C₇×#394	1	1	yes	characteristic
279	C₂²	C₇×#397	1	1	yes	characteristic
280	C₂²	C₇×#400	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	8	4	no	—
283	C₄	C₇×#382	1	1	yes	characteristic
284	C₄	C₇×#385	1	1	yes	characteristic
285	C₄	C₇×#388	1	2	yes	—
286	C₄	C₇×#393	1	1	yes	characteristic
287	C₄	C₇×#398	1	2	yes	—
288	C₄	C₇×#400	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	C₇×#1442	1	1	yes	characteristic
294	C₂	C₇×#2050	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	#124
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂²×C₇	#394
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₇×#338	C₄	semidirect
17. #11294	C₇	direct (N × H)
105. #338	C₄×C₇	semidirect
270. C₇	#11294	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#2911	2
C₇×#1742	2
C₂×C₇×#339	2
C₂×C₇×#338	2
C₇×#1728	2
#11294	7

Subgroups — C7×#11294 #13188510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11294 #13188510