Subgroups — #13253910

Order 1792.

440 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2862	C₂	1	1	yes	maximal, characteristic
2	#11727145	C₂	1	1	yes	maximal, characteristic
3	#11727143	C₂	1	1	yes	maximal, characteristic
4	#11725436	C₂	1	4	yes	maximal
5	C₂×C₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₇×#391	—	2	1	no	—
8	C₇×#391	—	2	1	no	—
9	C₇×#391	—	2	1	no	—
10	C₇×#391	—	2	1	no	—
11	C₇×#382	C₄	1	2	yes	—
12	C₇×#382	C₂²	1	2	yes	—
13	C₄³×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
15	C₂×#6377	C₂²	1	2	yes	—
16	C₂×#6377	C₂²	1	2	yes	—
17	#89319	—	2	1	no	—
18	#89319	—	2	1	no	—
19	#89319	—	2	1	no	—
20	#9895	—	7	1	no	maximal, Sylow
21	C₂²×C₇×Q₈	—	2	1	no	—
22	C₂²×C₇×Q₈	—	2	1	no	—
23	C₇×C₄⋊Q₈	—	4	2	no	—
24	C₇×C₄⋊Q₈	—	4	2	no	—
25	C₂×C₇×C₄⋊C₄	—	2	1	no	—
26	C₂×C₇×C₄⋊C₄	—	2	1	no	—
27	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₂²×C₇⋊C₈	—	2	2	no	—
37	C₂²×C₇⋊C₈	—	2	2	no	—
38	C₇×#95	—	2	2	no	—
39	C₇×#95	—	2	2	no	—
40	C₇×#95	—	2	2	no	—
41	C₇×#95	—	2	2	no	—
42	C₄⋊(C₇⋊C₈)	—	2	4	no	—
43	#6377	—	2	2	no	—
44	#6377	—	2	4	no	—
45	#6377	Q₈	1	4	yes	complemented
46	#2862	D₇	1	1	yes	characteristic
47	#1767	—	7	1	no	—
48	#1745	—	7	1	no	—
49	#1295	—	7	4	no	—
50	C₂×C₇×Q₈	—	4	2	no	—
51	C₂×C₇×Q₈	—	4	2	no	—
52	C₂×C₇×Q₈	—	4	2	no	—
53	C₂×C₇×Q₈	—	4	2	no	—
54	C₂×C₇×Q₈	—	4	2	no	—
55	C₂×C₇×Q₈	—	4	2	no	—
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	2	2	no	—
58	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
65	C₇×C₄⋊C₄	—	4	2	no	—
66	C₇×C₄⋊C₄	—	4	2	no	—
67	C₇×C₄⋊C₄	—	4	2	no	—
68	C₇×C₄⋊C₄	—	4	2	no	—
69	C₄²×C₇	—	2	2	no	—
70	C₄²×C₇	—	2	2	no	—
71	C₄²×C₇	—	2	2	no	—
72	C₄²×C₇	—	4	1	no	—
73	C₄²×C₇	—	4	1	no	—
74	C₄²×C₇	—	4	1	no	—
75	C₄²×C₇	C₂×Q₈	1	2	yes	—
76	C₄²×C₇	#41	1	2	yes	—
77	C₂×C₇⋊C₈	—	2	4	no	—
78	C₂×C₇⋊C₈	—	2	4	no	—
79	C₂×C₇⋊C₈	—	4	2	no	—
80	C₂×C₇⋊C₈	—	4	2	no	—
81	C₂×C₇⋊C₈	—	4	2	no	—
82	C₂×C₇⋊C₈	—	4	2	no	—
83	C₂×C₄⋊Q₈	Dic₇	1	1	yes	characteristic
84	C₂×C₄⋊Q₈	Dic₇	1	1	yes	characteristic
85	C₂×C₄⋊C₈	—	14	1	no	—
86	C₂×#97	—	7	2	no	—
87	C₂×#97	—	7	2	no	—
88	#391	—	2	1	no	—
89	#391	—	2	1	no	—
90	#391	—	2	1	no	—
91	#391	—	2	1	no	—
92	#382	Dic₇	1	2	yes	—
93	#382	C₂×D₇	1	2	yes	—
94	C₄³	C₂×D₇	1	1	yes	characteristic
95	#336	—	14	1	no	—
96	#336	—	14	1	no	—
97	#336	—	14	1	no	—
98	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₇×Q₈	—	8	4	no	—
100	C₇×Q₈	—	8	4	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	—	4	2	no	—
118	C₂×C₄×C₇	—	4	2	no	—
119	C₂×C₄×C₇	—	4	2	no	—
120	C₂×C₄×C₇	—	4	2	no	—
121	C₂×C₄×C₇	—	4	2	no	—
122	C₂×C₄×C₇	#100	1	2	yes	—
123	C₂×C₄×C₇	#101	1	2	yes	—
124	C₂×C₄×C₇	#104	1	2	yes	—
125	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₇	#124	1	1	yes	characteristic
128	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
129	C₇⋊C₈	—	4	4	no	—
130	C₇⋊C₈	—	4	4	no	—
131	C₂²×Q₈	—	2	1	no	—
132	C₂²×Q₈	—	2	1	no	—
133	C₂²×C₈	—	14	2	no	—
134	C₂²×C₈	—	14	2	no	—
135	C₄⋊Q₈	—	4	2	no	—
136	C₄⋊Q₈	—	4	2	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
140	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
141	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
142	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
145	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
148	C₄⋊C₈	—	14	4	no	—
149	#97	—	7	4	no	—
150	#97	—	14	2	no	—
151	#97	—	14	4	no	—
152	#95	—	2	2	no	—
153	#95	—	2	2	no	—
154	#95	—	2	2	no	—
155	#95	—	2	2	no	—
156	C₂²×C₇	#391	1	1	yes	characteristic
157	C₂²×C₇	C₂×#100	1	1	yes	characteristic
158	C₂²×C₇	C₂×#101	1	1	yes	characteristic
159	C₂²×C₇	C₂×#104	1	1	yes	characteristic
160	C₂²×C₇	#421	1	1	yes	characteristic
161	C₂²×C₇	#421	1	1	yes	characteristic
162	C₂²×C₇	#421	1	1	yes	characteristic
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	2	2	no	—
166	C₄×C₇	—	2	2	no	—
167	C₄×C₇	—	2	2	no	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	2	2	no	—
170	C₄×C₇	—	4	1	no	—
171	C₄×C₇	—	8	2	no	—
172	C₄×C₇	—	8	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
188	C₂×C₈	—	14	4	no	—
189	C₂×C₈	—	14	4	no	—
190	C₂×C₈	—	28	2	no	—
191	C₂×C₈	—	28	2	no	—
192	C₂×C₈	—	28	2	no	—
193	C₂×C₈	—	28	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	1	no	—
202	C₄²	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	Q₈×D₇	1	2	yes	—
205	C₄²	#1161	1	2	yes	—
206	C₂×C₇	#1527	1	1	yes	characteristic
207	C₂×C₇	#1531	1	1	yes	characteristic
208	C₂×C₇	#1533	1	1	yes	characteristic
209	C₂×C₇	#1537	1	1	yes	characteristic
210	C₂×C₇	#1986	1	1	yes	characteristic
211	C₂×C₇	#1995	1	1	yes	characteristic
212	C₂×C₇	#1997	1	1	yes	characteristic
213	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
214	Q₈	—	8	4	no	—
215	Q₈	—	8	4	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	#6407	1	2	yes	—
238	C₂×C₄	#6410	1	2	yes	—
239	C₂×C₄	#6411	1	2	yes	—
240	C₂×C₄	#6507	1	1	yes	characteristic
241	C₂×C₄	Q₈×Dic₇	1	2	yes	—
242	C₂×C₄	#6509	1	2	yes	—
243	C₂×C₄	#6510	1	1	yes	characteristic
244	C₈	—	28	4	no	—
245	C₈	—	28	4	no	—
246	C₇	#9895	1	1	yes	characteristic, complemented, Sylow
247	C₂²	C₂×#6407	1	1	yes	characteristic
248	C₂²	C₂×#6410	1	1	yes	characteristic
249	C₂²	#90002	1	1	yes	characteristic
250	C₂²	C₂×#6411	1	1	yes	characteristic
251	C₂²	#90008	1	1	yes	characteristic
252	C₂²	#90008	1	1	yes	characteristic
253	C₂²	#90008	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	2	no	—
264	C₂	#11729472	1	1	yes	characteristic
265	C₂	#11729473	1	1	yes	characteristic
266	C₂	#11729544	1	1	yes	characteristic
267	C₂	#11729576	1	1	yes	characteristic
268	C₂	#11729577	1	1	yes	characteristic
269	C₂	#11729594	1	1	yes	characteristic
270	C₂	#11729595	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2862	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2862	D₇
hypocenter	C₇	#9895
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
45. #6377	Q₈	semidirect
246. C₇	#9895	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2862	2
#11727145	2
#11727143	2
#11725436	2
#9895	7