Subgroups — #13253952

Order 1792.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2868	C₂	1	1	yes	maximal, characteristic
2	#11727146	C₂	1	1	yes	maximal, characteristic
3	#11727145	C₂	1	1	yes	maximal, characteristic
4	#11725440	C₂	1	4	yes	maximal
5	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₇×C₄⋊Q₈	C₄	1	2	yes	—
8	C₇×#384	—	2	1	no	—
9	C₇×#384	—	2	1	no	—
10	C₇×#384	C₄	1	2	yes	—
11	C₇×#384	C₂²	1	2	yes	—
12	C₄³×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#6377	—	2	1	no	—
16	#89319	—	2	1	no	—
17	#89319	—	2	1	no	—
18	#89319	—	2	1	no	—
19	#9905	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₇×C₄⋊Q₈	—	4	4	no	—
22	C₇×C₄⋊Q₈	—	4	4	no	—
23	C₇×C₄⋊Q₈	—	4	2	no	—
24	C₇×C₄⋊Q₈	—	4	2	no	—
25	C₂×C₇×C₄⋊C₄	—	2	2	no	—
26	C₂×C₇×C₄⋊C₄	—	2	2	no	—
27	C₂×C₇×C₄⋊C₄	—	2	2	no	—
28	C₂×C₇×C₄⋊C₄	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	—	2	2	no	—
30	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂×C₄	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	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₂²×C₇⋊C₈	—	2	2	no	—
39	C₂²×C₇⋊C₈	—	2	2	no	—
40	C₄⋊(C₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₇⋊C₈)	—	2	4	no	—
42	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	complemented
43	#6377	—	4	2	no	—
44	#2868	D₇	1	1	yes	characteristic
45	#1831	—	7	1	no	—
46	#1767	—	7	1	no	—
47	#1296	—	7	4	no	—
48	C₂×C₇×Q₈	—	4	4	no	—
49	C₂×C₇×Q₈	—	4	4	no	—
50	C₂×C₇×Q₈	—	4	4	no	—
51	C₂²×C₄×C₇	—	2	4	no	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
59	C₇×C₄⋊C₄	—	4	4	no	—
60	C₇×C₄⋊C₄	—	4	4	no	—
61	C₇×C₄⋊C₄	—	4	4	no	—
62	C₇×C₄⋊C₄	—	4	4	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₇×C₄⋊C₄	—	4	2	no	—
65	C₇×C₄⋊C₄	—	4	4	no	—
66	C₄²×C₇	—	2	2	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	—	2	2	no	—
70	C₄²×C₇	—	2	2	no	—
71	C₄²×C₇	D₈	1	2	yes	—
72	C₄²×C₇	SD₁₆	1	2	yes	—
73	C₄²×C₇	C₂×Q₈	1	2	yes	—
74	C₄²×C₇	#41	1	2	yes	—
75	C₂×C₇⋊C₈	—	2	4	no	—
76	C₂×C₇⋊C₈	—	4	2	no	—
77	C₂×C₇⋊C₈	—	4	2	no	—
78	C₂×C₇⋊C₈	—	4	2	no	—
79	C₂×C₇⋊C₈	—	4	2	no	—
80	C₂×C₇⋊C₈	—	4	2	no	—
81	C₂×C₄⋊Q₈	—	2	1	no	—
82	C₂×C₄⋊Q₈	—	2	1	no	—
83	C₂×C₄⋊Q₈	Dic₇	1	2	yes	—
84	C₂×C₄⋊C₈	—	7	2	no	—
85	C₂×C₄⋊C₈	—	7	2	no	—
86	C₂×#97	—	14	1	no	—
87	#384	—	2	1	no	—
88	#384	—	2	1	no	—
89	#384	Dic₇	1	2	yes	—
90	#384	C₂×D₇	1	2	yes	—
91	C₄³	C₂×D₇	1	1	yes	characteristic
92	#336	—	14	1	no	—
93	#336	—	14	1	no	—
94	#336	—	14	1	no	—
95	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₇×Q₈	—	8	8	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	1	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	—	4	4	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	—	4	4	no	—
110	C₂×C₄×C₇	#100	1	2	yes	—
111	C₂×C₄×C₇	#101	1	2	yes	—
112	C₂×C₄×C₇	#102	1	2	yes	—
113	C₂×C₄×C₇	#102	1	2	yes	—
114	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇	#124	1	1	yes	characteristic
117	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₇	#137	1	2	yes	—
122	C₂×C₄×C₇	#137	1	2	yes	—
123	C₇⋊C₈	—	4	4	no	—
124	C₇⋊C₈	—	8	2	no	—
125	C₂²×Q₈	—	2	2	no	—
126	C₂²×C₈	—	14	2	no	—
127	C₂²×C₈	—	14	2	no	—
128	C₄⋊Q₈	—	4	4	no	—
129	C₄⋊Q₈	—	4	4	no	—
130	C₄⋊Q₈	—	4	2	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
138	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
139	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
141	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
145	C₄⋊C₈	—	7	4	no	—
146	C₄⋊C₈	—	14	2	no	—
147	C₄⋊C₈	—	14	4	no	—
148	#97	—	28	2	no	—
149	C₂²×C₇	#391	1	1	yes	characteristic
150	C₂²×C₇	C₂×#100	1	1	yes	characteristic
151	C₂²×C₇	C₂×#101	1	1	yes	characteristic
152	C₂²×C₇	C₂×#102	1	1	yes	characteristic
153	C₂²×C₇	#418	1	1	yes	characteristic
154	C₂²×C₇	#419	1	1	yes	characteristic
155	C₂²×C₇	#419	1	1	yes	characteristic
156	C₄×C₇	—	2	2	no	—
157	C₄×C₇	—	2	2	no	—
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	—	2	2	no	—
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	—	8	4	no	—
162	C₄×C₇	#474	1	2	yes	—
163	C₄×C₇	#475	1	2	yes	—
164	C₄×C₇	#489	1	2	yes	—
165	C₄×C₇	#501	1	2	yes	—
166	C₂×Q₈	—	4	4	no	—
167	C₂×Q₈	—	4	4	no	—
168	C₂×Q₈	—	4	4	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
177	C₂×C₈	—	14	4	no	—
178	C₂×C₈	—	28	2	no	—
179	C₂×C₈	—	28	2	no	—
180	C₂×C₈	—	28	2	no	—
181	C₂×C₈	—	28	2	no	—
182	C₂×C₈	—	28	2	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	#1141	1	2	yes	—
196	C₄²	#1142	1	2	yes	—
197	C₄²	Q₈×D₇	1	2	yes	—
198	C₄²	#1161	1	2	yes	—
199	C₂×C₇	#1545	1	1	yes	characteristic
200	C₂×C₇	#1546	1	1	yes	characteristic
201	C₂×C₇	#1556	1	1	yes	characteristic
202	C₂×C₇	#1558	1	1	yes	characteristic
203	C₂×C₇	#1988	1	1	yes	characteristic
204	C₂×C₇	#1993	1	1	yes	characteristic
205	C₂×C₇	#1997	1	1	yes	characteristic
206	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
207	Q₈	—	8	8	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	#6406	1	2	yes	—
222	C₂×C₄	#6406	1	2	yes	—
223	C₂×C₄	#6407	1	2	yes	—
224	C₂×C₄	#6410	1	2	yes	—
225	C₂×C₄	C₂×#1141	1	1	yes	characteristic
226	C₂×C₄	#6495	1	2	yes	—
227	C₂×C₄	C₂×#1142	1	1	yes	characteristic
228	C₂×C₄	#6505	1	2	yes	—
229	C₂×C₄	#6505	1	2	yes	—
230	C₂×C₄	#6507	1	1	yes	characteristic
231	C₂×C₄	Q₈×Dic₇	1	2	yes	—
232	C₂×C₄	#6509	1	2	yes	—
233	C₂×C₄	#6510	1	1	yes	characteristic
234	C₈	—	28	4	no	—
235	C₈	—	56	2	no	—
236	C₇	#9905	1	1	yes	characteristic, complemented, Sylow
237	C₂²	C₂×#6406	1	1	yes	characteristic
238	C₂²	#89987	1	1	yes	characteristic
239	C₂²	C₂×#6407	1	1	yes	characteristic
240	C₂²	#89997	1	1	yes	characteristic
241	C₂²	#89997	1	1	yes	characteristic
242	C₂²	C₂×#6410	1	1	yes	characteristic
243	C₂²	#90002	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	8	4	no	—
250	C₄	#89852	1	2	yes	—
251	C₄	#89859	1	2	yes	—
252	C₄	#89861	1	2	yes	—
253	C₄	#89866	1	2	yes	—
254	C₂	#11729513	1	1	yes	characteristic
255	C₂	#11729527	1	1	yes	characteristic
256	C₂	#11729530	1	1	yes	characteristic
257	C₂	#11729531	1	1	yes	characteristic
258	C₂	#11729534	1	1	yes	characteristic
259	C₂	#11729542	1	1	yes	characteristic
260	C₂	#11729544	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×#2868	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2868	D₇
hypocenter	C₇	#9905
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
42. C₄⋊(C₇⋊C₈)	Q₈	semidirect
236. C₇	#9905	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2868	2
#11727146	2
#11727145	2
#11725440	2
#9905	7