Subgroups — #13263015

Order 1792.

310 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	C₇×#2922	C₂	1	1	yes	maximal, characteristic
2	#11727499	C₂	1	1	yes	maximal, characteristic, complemented
3	#11727498	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725524	C₂	1	2	yes	maximal, complemented
5	#11725495	C₂	1	2	yes	maximal
6	C₄×C₇×#41	C₄	1	1	yes	characteristic
7	C₇×#433	—	2	1	no	—
8	C₇×#433	—	2	1	no	—
9	C₇×#431	—	2	1	no	—
10	C₇×#431	—	2	1	no	—
11	C₇×#424	C₄	1	1	yes	characteristic
12	C₇×#424	C₂²	1	1	yes	characteristic
13	C₇×#408	C₄	1	1	yes	characteristic
14	C₇×#408	C₂²	1	1	yes	characteristic
15	C₄×C₇×M₁₆	C₄	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₂²⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
18	C₂²⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
19	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
20	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
21	#89256	—	2	1	no	—
22	C₄×C₇⋊C₁₆	—	2	1	no	—
23	#11464	—	7	1	no	maximal, Sylow
24	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
29	C₄×C₇×Q₈	C₈	1	1	yes	characteristic
30	C₄×C₇×Q₈	C₈	1	1	yes	characteristic
31	C₄×C₇×D₄	—	2	1	no	—
32	C₄×C₇×D₄	—	2	1	no	—
33	C₄×C₇×D₄	C₈	1	1	yes	characteristic
34	C₄×C₇×D₄	C₈	1	1	yes	characteristic
35	C₇×#117	—	2	1	no	—
36	C₇×#117	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	—	2	1	no	—
38	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
39	C₇×C₄⋊C₈	—	2	1	no	—
40	C₇×C₄⋊C₈	—	2	1	no	—
41	C₇×C₄⋊C₈	—	2	1	no	—
42	C₇×C₄⋊C₈	—	2	1	no	—
43	C₇×C₂²⋊C₈	—	2	1	no	—
44	C₇×C₂²⋊C₈	—	2	1	no	—
45	C₇×C₂²⋊C₈	—	2	1	no	—
46	C₇×C₂²⋊C₈	—	2	1	no	—
47	C₇×#97	—	2	1	no	—
48	C₄×C₇×C₈	—	2	1	no	—
49	C₄×C₇×C₈	D₄	1	1	yes	characteristic
50	C₄×C₇×C₈	D₄	1	1	yes	characteristic
51	C₄×C₇×C₈	D₄	1	1	yes	characteristic
52	C₄×C₇×C₈	D₄	1	1	yes	characteristic
53	C₂×C₇⋊C₁₆	—	2	2	no	—
54	C₂×C₇⋊C₁₆	—	2	2	no	—
55	#2922	D₇	1	1	yes	characteristic
56	#2163	—	7	1	no	—
57	#2162	—	7	1	no	—
58	#1323	—	7	2	no	—
59	#1315	—	7	2	no	—
60	C₇×#41	—	4	1	no	—
61	C₇×#41	—	4	1	no	—
62	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₇×D₄	—	2	1	no	—
64	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	—	2	1	no	—
66	C₂²×C₄×C₇	—	2	1	no	—
67	C₂²×C₄×C₇	—	2	1	no	—
68	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₇×M₁₆	—	4	1	no	—
72	C₇×M₁₆	—	4	1	no	—
73	C₂×C₇×C₈	—	2	1	no	—
74	C₂×C₇×C₈	—	2	1	no	—
75	C₂×C₇×C₈	—	4	1	no	—
76	C₂×C₇×C₈	—	4	1	no	—
77	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
81	C₇×C₄⋊C₄	—	2	1	no	—
82	C₇×C₄⋊C₄	—	2	1	no	—
83	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
84	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₇×C₂²⋊C₄	—	2	1	no	—
86	C₇×C₂²⋊C₄	—	2	1	no	—
87	C₇×C₂²⋊C₄	—	2	1	no	—
88	C₄²×C₇	—	2	1	no	—
89	C₄²×C₇	—	2	1	no	—
90	C₄²×C₇	—	2	1	no	—
91	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
95	C₇⋊C₁₆	—	4	2	no	—
96	C₇⋊C₁₆	—	4	2	no	—
97	C₄×#41	Dic₇	1	1	yes	characteristic
98	#433	—	2	1	no	—
99	#433	—	2	1	no	—
100	#431	—	2	1	no	—
101	#431	—	2	1	no	—
102	#424	Dic₇	1	1	yes	characteristic
103	#424	C₂×D₇	1	1	yes	characteristic
104	#408	Dic₇	1	1	yes	characteristic
105	#408	C₂×D₇	1	1	yes	characteristic
106	C₄×M₁₆	Dic₇	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
108	C₄⋊C₁₆	—	14	1	no	—
109	C₄⋊C₁₆	—	14	1	no	—
110	C₂²⋊C₁₆	—	7	2	no	—
111	C₂²⋊C₁₆	—	7	2	no	—
112	#346	—	14	1	no	—
113	C₄×C₁₆	—	14	1	no	—
114	C₂³×C₇	—	2	1	no	—
115	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
116	C₇×Q₈	—	4	1	no	—
117	C₇×D₄	—	4	1	no	—
118	C₇×D₄	—	4	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	1	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	4	1	no	—
126	C₂×C₄×C₇	—	4	2	no	—
127	C₂×C₄×C₇	—	4	2	no	—
128	C₂×C₄×C₇	—	4	1	no	—
129	C₂×C₄×C₇	—	4	1	no	—
130	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
136	C₇×C₈	—	2	1	no	—
137	C₇×C₈	—	2	1	no	—
138	C₇×C₈	—	2	1	no	—
139	C₇×C₈	—	2	1	no	—
140	C₇×C₈	—	4	1	no	—
141	C₇×C₈	—	4	1	no	—
142	C₂×#41	C₂×Dic₇	1	1	yes	characteristic
143	C₂×M₁₆	C₂×Dic₇	1	1	yes	characteristic
144	C₂×M₁₆	C₂×Dic₇	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
147	C₄×Q₈	C₇⋊C₈	1	1	yes	characteristic
148	C₄×Q₈	C₇⋊C₈	1	1	yes	characteristic
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	—	2	1	no	—
151	C₄×D₄	C₇⋊C₈	1	1	yes	characteristic
152	C₄×D₄	C₇⋊C₈	1	1	yes	characteristic
153	#117	—	2	1	no	—
154	#117	C₂×Dic₇	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
157	C₂×C₁₆	—	14	2	no	—
158	C₂×C₁₆	—	14	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₂²⋊C₈	—	2	1	no	—
164	C₂²⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₂²⋊C₈	—	2	1	no	—
167	#97	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
173	C₂²×C₇	—	2	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	—	4	1	no	—
177	C₂²×C₇	—	4	1	no	—
178	C₂²×C₇	—	4	1	no	—
179	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	4	1	no	—
185	C₄×C₇	—	4	1	no	—
186	C₄×C₇	—	4	1	no	—
187	C₄×C₇	#350	1	2	yes	—
188	C₄×C₇	#410	1	1	yes	characteristic
189	C₄×C₇	#413	1	1	yes	characteristic
190	#41	—	4	1	no	—
191	#41	—	4	1	no	—
192	C₂×Q₈	C₂×C₇⋊C₈	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₂×C₇⋊C₈	1	1	yes	characteristic
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	#1136	1	1	yes	characteristic
199	C₂²×C₄	#1136	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
201	M₁₆	—	4	1	no	—
202	M₁₆	—	4	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
214	C₄⋊C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
222	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
223	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
224	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
225	C₁₆	—	28	2	no	—
226	C₁₆	—	28	2	no	—
227	C₂×C₇	—	4	1	no	—
228	C₂×C₇	—	8	1	no	—
229	C₂×C₇	#1458	1	1	yes	characteristic
230	C₂×C₇	C₂×#350	1	1	yes	characteristic
231	C₂×C₇	#2117	1	1	yes	characteristic
232	C₂³	—	2	1	no	—
233	C₂³	C₂×#1136	1	1	yes	characteristic
234	Q₈	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
249	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₇	#11464	1	1	yes	characteristic, complemented, Sylow
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	#89356	1	2	yes	—
276	C₄	#90009	1	1	yes	characteristic
277	C₄	#90012	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	#11729325	1	1	yes	characteristic
281	C₂	C₂×#89356	1	1	yes	characteristic
282	C₂	#11729760	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#2922	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂×C₇×#41	C₂×C₄
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#2922	D₇
hypocenter	C₇	#11464
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

Complements

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

N	H ≅ G/N	type
2. #11727499	C₂	semidirect
3. #11727498	C₂	semidirect
4. #11725524	C₂	semidirect
260. C₇	#11464	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2922	2
#11727499	2
#11727498	2
#11725524	2
#11725495	2
#11464	7