Subgroups — C₇×#11272 #13188488

Order 1792.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#344	C₂	1	2	yes	maximal
3	C₇×#1738	C₂	1	2	yes	maximal
4	C₂×C₇×#341	C₂	1	1	yes	maximal, characteristic
5	C₇×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×#117	C₂²	1	1	yes	characteristic
11	C₂×C₇×#108	—	2	4	no	—
12	C₂×C₇×#108	—	2	2	no	—
13	C₇×#344	—	2	4	no	—
14	C₇×#341	—	2	4	no	—
15	C₇×#341	C₄	1	4	yes	complemented
16	C₇×#336	—	2	2	no	—
17	#11272	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂×C₇×M₁₆	—	2	1	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	—	2	2	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	—	2	2	no	—
31	C₂×C₇×M₁₆	—	2	2	no	—
32	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₇×M₁₆	D₄	1	2	yes	—
34	C₂×C₇×M₁₆	D₄	1	4	yes	—
35	C₂²×C₇×C₈	—	2	2	no	—
36	C₂²×C₇×C₈	—	2	2	no	—
37	C₂²×C₇×C₈	—	2	1	no	—
38	C₂²×C₇×C₈	—	2	2	no	—
39	C₂²×C₇×C₈	—	2	2	no	—
40	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₇×C₈	D₄	1	2	yes	—
42	C₂²×C₇×C₈	Q₈	1	2	yes	—
43	C₇×#117	—	2	2	no	—
44	C₇×#117	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	—	2	1	no	—
46	C₂×C₇×C₂²⋊C₄	Q₈	1	2	yes	—
47	C₂×C₄²×C₇	—	2	1	no	—
48	C₇×#108	—	2	8	no	—
49	C₂²×#108	C₂×C₇	1	1	yes	characteristic
50	C₂×#344	C₂×C₇	1	2	yes	—
51	#1738	C₂×C₇	1	2	yes	—
52	C₂×#341	C₂×C₇	1	1	yes	characteristic
53	#1729	C₂×C₇	1	1	yes	characteristic
54	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	2	2	no	—
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	#41	1	2	yes	—
62	C₂²×C₄×C₇	#41	1	2	yes	—
63	C₂²×C₄×C₇	#41	1	2	yes	—
64	C₂²×C₄×C₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₇	#41	1	1	yes	characteristic
66	C₇×M₁₆	—	2	4	no	—
67	C₇×M₁₆	—	2	4	no	—
68	C₇×M₁₆	—	2	4	no	—
69	C₇×M₁₆	—	2	4	no	—
70	C₇×M₁₆	—	2	4	no	—
71	C₇×M₁₆	—	4	2	no	—
72	C₇×M₁₆	—	4	2	no	—
73	C₇×M₁₆	—	4	2	no	—
74	C₇×M₁₆	—	4	2	no	—
75	C₇×M₁₆	—	4	2	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	2	no	—
78	C₂×C₇×C₈	—	2	4	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	—	2	4	no	—
83	C₂×C₇×C₈	—	4	2	no	—
84	C₂×C₇×C₈	—	4	2	no	—
85	C₂×C₇×C₈	—	4	2	no	—
86	C₂×C₇×C₈	—	4	2	no	—
87	C₂×C₇×C₈	—	4	1	no	—
88	C₂×C₇×C₈	—	4	1	no	—
89	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
90	C₇×C₄⋊C₄	—	4	2	no	—
91	C₇×C₂²⋊C₄	—	2	4	no	—
92	C₄²×C₇	—	4	1	no	—
93	C₄²×C₇	—	4	1	no	—
94	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
95	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
96	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
98	C₂×#117	C₂²×C₇	1	1	yes	characteristic
99	C₂×#108	—	2	4	no	—
100	C₂×#108	—	2	2	no	—
101	#344	—	2	4	no	—
102	#341	—	2	4	no	—
103	#341	C₄×C₇	1	4	yes	complemented
104	#336	—	2	2	no	—
105	C₂³×C₇	—	4	2	no	—
106	C₂³×C₇	C₄×Q₈	1	1	yes	characteristic
107	C₂³×C₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂³×C₇	C₂²⋊Q₈	1	2	yes	—
109	C₂³×C₇	#125	1	2	yes	—
110	C₂³×C₇	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇	—	4	4	no	—
112	C₂×C₄×C₇	—	4	2	no	—
113	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇	#117	1	2	yes	—
115	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇	#120	1	1	yes	characteristic
118	C₂×C₄×C₇	#121	1	2	yes	—
119	C₂×C₄×C₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₇	#123	1	1	yes	characteristic
122	C₂×C₄×C₇	#123	1	2	yes	—
123	C₂×C₄×C₇	#123	1	2	yes	—
124	C₂×C₄×C₇	#124	1	1	yes	characteristic
125	C₂×C₄×C₇	#124	1	1	yes	characteristic
126	C₂×C₄×C₇	#124	1	2	yes	—
127	C₂×C₄×C₇	#125	1	2	yes	—
128	C₂×C₄×C₇	#125	1	1	yes	characteristic
129	C₂×C₄×C₇	#126	1	2	yes	—
130	C₂×C₄×C₇	#126	1	2	yes	—
131	C₇×C₈	—	2	4	no	—
132	C₇×C₈	—	4	4	no	—
133	C₇×C₈	—	4	4	no	—
134	C₇×C₈	—	4	2	no	—
135	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	1	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
150	C₂×M₁₆	C₇×D₄	1	2	yes	—
151	C₂×M₁₆	C₇×D₄	1	4	yes	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
158	C₂²×C₈	C₇×D₄	1	2	yes	—
159	C₂²×C₈	C₇×Q₈	1	2	yes	—
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₂²⋊C₄	C₇×Q₈	1	2	yes	—
164	C₂×C₄²	—	2	1	no	—
165	#108	—	2	8	no	—
166	C₂²×C₇	—	2	2	no	—
167	C₂²×C₇	—	2	2	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	4	2	no	—
170	C₂²×C₇	—	4	2	no	—
171	C₂²×C₇	#387	1	2	yes	—
172	C₂²×C₇	#389	1	1	yes	characteristic
173	C₂²×C₇	#393	1	1	yes	characteristic
174	C₂²×C₇	#398	1	2	yes	—
175	C₂²×C₇	#400	1	1	yes	characteristic
176	C₄×C₇	—	8	2	no	—
177	C₄×C₇	#382	1	1	yes	characteristic
178	C₄×C₇	#385	1	1	yes	characteristic
179	C₄×C₇	#388	1	2	yes	—
180	C₄×C₇	#394	1	1	yes	characteristic
181	C₄×C₇	#397	1	1	yes	characteristic
182	C₄×C₇	#400	1	2	yes	—
183	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
188	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₇×#41	1	2	yes	—
191	C₂²×C₄	C₇×#41	1	2	yes	—
192	C₂²×C₄	C₇×#41	1	2	yes	—
193	C₂²×C₄	C₇×#41	1	1	yes	characteristic
194	C₂²×C₄	C₇×#41	1	1	yes	characteristic
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	4	2	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	4	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
219	C₄⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₂×C₇	—	2	1	no	—
224	C₂×C₇	—	2	2	no	—
225	C₂×C₇	—	2	2	no	—
226	C₂×C₇	—	2	1	no	—
227	C₂×C₇	#1442	1	1	yes	characteristic
228	C₂×C₇	#1953	1	2	yes	—
229	C₂³	—	4	2	no	—
230	C₂³	C₄×C₇×Q₈	1	1	yes	characteristic
231	C₂³	C₇×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂³	C₇×C₂²⋊Q₈	1	2	yes	—
233	C₂³	C₇×#125	1	2	yes	—
234	C₂³	C₇×C₄⋊Q₈	1	1	yes	characteristic
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₄	C₇×#117	1	2	yes	—
239	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
240	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
241	C₂×C₄	C₇×#120	1	1	yes	characteristic
242	C₂×C₄	C₇×#121	1	2	yes	—
243	C₂×C₄	C₇×#121	1	1	yes	characteristic
244	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
245	C₂×C₄	C₇×#123	1	1	yes	characteristic
246	C₂×C₄	C₇×#123	1	2	yes	—
247	C₂×C₄	C₇×#123	1	2	yes	—
248	C₂×C₄	C₇×#124	1	1	yes	characteristic
249	C₂×C₄	C₇×#124	1	1	yes	characteristic
250	C₂×C₄	C₇×#124	1	2	yes	—
251	C₂×C₄	C₇×#125	1	2	yes	—
252	C₂×C₄	C₇×#125	1	1	yes	characteristic
253	C₂×C₄	C₇×#126	1	2	yes	—
254	C₂×C₄	C₇×#126	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	—	4	4	no	—
257	C₈	—	4	4	no	—
258	C₈	—	4	2	no	—
259	C₇	#11272	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	C₇×#387	1	2	yes	—
266	C₂²	C₇×#389	1	1	yes	characteristic
267	C₂²	C₇×#393	1	1	yes	characteristic
268	C₂²	C₇×#398	1	2	yes	—
269	C₂²	C₇×#400	1	1	yes	characteristic
270	C₄	—	8	2	no	—
271	C₄	C₇×#382	1	1	yes	characteristic
272	C₄	C₇×#385	1	1	yes	characteristic
273	C₄	C₇×#388	1	2	yes	—
274	C₄	C₇×#394	1	1	yes	characteristic
275	C₄	C₇×#397	1	1	yes	characteristic
276	C₄	C₇×#400	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	1	no	—
281	C₂	C₇×#1442	1	1	yes	characteristic
282	C₂	C₇×#1953	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	#121
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂²×C₇	#389
prime radical	C₂⁴×C₇	C₂×Q₈
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×#341	C₄	semidirect
17. #11272	C₇	direct (N × H)
103. #341	C₄×C₇	semidirect
259. C₇	#11272	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#108	2
C₂×C₇×#344	2
C₇×#1738	2
C₂×C₇×#341	2
C₇×#1729	2
#11272	7

Subgroups — C7×#11272 #13188488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11272 #13188488