Subgroups — #13265283

Order 1792.

314 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#11727485	C₂	1	1	yes	maximal, characteristic
4	#11725608	C₂	1	2	yes	maximal, complemented
5	#11725599	C₂	1	2	yes	maximal, complemented
6	C₇×#533	—	2	1	no	—
7	C₇×#500	C₂²	1	1	yes	characteristic
8	C₇×#494	C₂²	1	1	yes	characteristic
9	C₇×#487	—	2	1	no	—
10	C₇×#467	—	2	1	no	—
11	C₇×#467	C₄	1	2	yes	—
12	C₄×C₇×Q₁₆	—	2	1	no	—
13	C₄×C₇×Q₁₆	C₄	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₇⋊C₁₆	—	2	1	no	—
16	#89696	—	2	1	no	—
17	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
18	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
19	#89253	—	2	1	no	—
20	#89253	C₂²	1	2	yes	—
21	#11983	—	7	1	no	maximal, Sylow
22	C₂×C₇×Q₁₆	—	2	1	no	—
23	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
26	C₇×C₄⋊Q₈	—	2	1	no	—
27	C₇×C₄⋊Q₈	—	2	1	no	—
28	C₇×#125	—	4	1	no	—
29	C₇×C₂²⋊Q₈	—	2	2	no	—
30	C₇×C₂²⋊Q₈	—	4	1	no	—
31	C₄×C₇×Q₈	—	2	2	no	—
32	C₄×C₇×Q₈	—	4	1	no	—
33	C₇×#117	—	4	1	no	—
34	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
35	C₂×#1136	—	4	1	no	—
36	C₇×#107	—	2	1	no	—
37	C₇×#107	C₂×C₄	1	2	yes	—
38	C₇×#103	—	2	2	no	—
39	C₇×#103	—	2	2	no	—
40	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₇⋊C₁₆	—	2	2	no	—
45	C₂×C₇⋊C₁₆	—	4	2	no	—
46	C₄⋊(C₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₇⋊C₈)	—	4	1	no	—
48	#3170	D₇	1	1	yes	characteristic
49	C₂×C₄⋊C₁₆	—	7	1	no	—
50	#1567	—	7	1	no	—
51	#1365	—	7	2	no	—
52	#1362	—	7	2	no	—
53	C₂×C₇×Q₈	—	2	2	no	—
54	C₂×C₇×Q₈	—	4	1	no	—
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
58	C₇×Q₁₆	—	4	2	no	—
59	C₇×Q₁₆	—	4	1	no	—
60	C₇×Q₁₆	—	4	1	no	—
61	C₂×C₇×C₈	—	2	1	no	—
62	C₂×C₇×C₈	—	2	1	no	—
63	C₂×C₇×C₈	—	4	1	no	—
64	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
68	C₇×C₄⋊C₄	—	2	2	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₇×C₄⋊C₄	—	4	2	no	—
71	C₇×C₄⋊C₄	—	4	1	no	—
72	C₇×C₄⋊C₄	—	4	1	no	—
73	C₇×C₄⋊C₄	—	4	1	no	—
74	C₇×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₇	—	4	2	no	—
76	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇	D₈	1	1	yes	characteristic
78	C₄²×C₇	SD₁₆	1	1	yes	characteristic
79	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
80	#1136	—	8	2	no	—
81	C₂×C₇⋊C₈	—	4	2	no	—
82	C₇⋊C₁₆	—	4	2	no	—
83	#533	—	2	1	no	—
84	C₂²×C₁₆	—	14	1	no	—
85	#500	C₂×D₇	1	1	yes	characteristic
86	#494	C₂×D₇	1	1	yes	characteristic
87	#487	—	2	1	no	—
88	#467	—	2	1	no	—
89	#467	Dic₇	1	2	yes	—
90	C₄×Q₁₆	—	2	1	no	—
91	C₄×Q₁₆	Dic₇	1	2	yes	—
92	#423	—	14	1	no	—
93	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
94	C₄⋊C₁₆	—	7	2	no	—
95	C₄⋊C₁₆	—	14	1	no	—
96	#335	—	7	2	no	—
97	#335	—	14	1	no	—
98	C₂³×C₇	#102	1	1	yes	characteristic
99	C₇×Q₈	—	4	2	no	—
100	C₇×Q₈	—	4	2	no	—
101	C₇×Q₈	—	8	2	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	4	1	no	—
105	C₂×C₄×C₇	—	4	2	no	—
106	C₂×C₄×C₇	—	4	2	no	—
107	C₂×C₄×C₇	#102	1	1	yes	characteristic
108	C₂×C₄×C₇	#102	1	1	yes	characteristic
109	C₂×C₄×C₇	#102	1	1	yes	characteristic
110	C₂×C₄×C₇	SD₃₂	1	2	yes	—
111	C₂×C₄×C₇	Q₃₂	1	2	yes	—
112	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₇×C₈	—	2	1	no	—
116	C₇×C₈	—	2	1	no	—
117	C₇×C₈	—	2	1	no	—
118	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₇×C₈	#137	1	1	yes	characteristic
120	C₇⋊C₈	—	8	2	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	C₂×Dic₇	1	2	yes	—
123	C₂×M₁₆	—	28	1	no	—
124	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	1	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	#125	—	4	1	no	—
129	C₂²⋊Q₈	—	2	2	no	—
130	C₂²⋊Q₈	—	4	1	no	—
131	C₄×Q₈	—	2	2	no	—
132	C₄×Q₈	—	4	1	no	—
133	#117	—	4	1	no	—
134	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
135	C₂×C₁₆	—	14	2	no	—
136	C₂×C₁₆	—	28	2	no	—
137	#107	—	2	1	no	—
138	#107	C₂×Dic₇	1	2	yes	—
139	C₄⋊C₈	—	14	2	no	—
140	C₄⋊C₈	—	28	1	no	—
141	#103	—	2	2	no	—
142	#103	—	2	2	no	—
143	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
147	C₂²×C₇	—	2	1	no	—
148	C₂²×C₇	—	2	1	no	—
149	C₂²×C₇	#358	1	2	yes	—
150	C₂²×C₇	C₂×#102	1	1	yes	characteristic
151	C₄×C₇	—	2	1	no	—
152	C₄×C₇	—	2	1	no	—
153	C₄×C₇	—	2	1	no	—
154	C₄×C₇	—	8	2	no	—
155	C₄×C₇	—	8	2	no	—
156	C₄×C₇	#358	1	2	yes	—
157	C₄×C₇	#413	1	1	yes	characteristic
158	C₄×C₇	#417	1	1	yes	characteristic
159	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
160	C₄×C₇	C₂×Q₃₂	1	1	yes	characteristic
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	#1141	1	1	yes	characteristic
164	C₂²×C₄	#1142	1	1	yes	characteristic
165	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	1	no	—
168	Q₁₆	—	4	1	no	—
169	M₁₆	—	56	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	28	2	no	—
174	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
176	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	#1141	1	1	yes	characteristic
187	C₄²	#1142	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
189	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
190	C₁₆	—	28	2	no	—
191	C₂×C₇	—	2	2	no	—
192	C₂×C₇	#1547	1	1	yes	characteristic
193	C₂×C₇	C₂×#358	1	1	yes	characteristic
194	C₂×C₇	#2148	1	1	yes	characteristic
195	C₂³	#6406	1	1	yes	characteristic
196	Q₈	—	4	2	no	—
197	Q₈	—	4	2	no	—
198	Q₈	—	8	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	#6402	1	2	yes	—
205	C₂×C₄	#6403	1	2	yes	—
206	C₂×C₄	#6406	1	1	yes	characteristic
207	C₂×C₄	#6406	1	1	yes	characteristic
208	C₂×C₄	#6406	1	1	yes	characteristic
209	C₂×C₄	C₂×#1141	1	1	yes	characteristic
210	C₂×C₄	C₂×#1142	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	56	2	no	—
216	C₈	#6512	1	1	yes	characteristic
217	C₈	#6514	1	1	yes	characteristic
218	C₇	#11983	1	1	yes	characteristic, complemented, Sylow
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#89359	1	2	yes	—
222	C₂²	C₂×#6406	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	#89359	1	2	yes	—
229	C₄	C₂×#6402	1	1	yes	characteristic
230	C₄	C₂×#6403	1	1	yes	characteristic
231	C₄	#90004	1	1	yes	characteristic
232	C₄	#90009	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	#11729560	1	1	yes	characteristic
235	C₂	C₂×#89359	1	1	yes	characteristic
236	C₂	#11729835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6406
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#3170	C₂
socle	C₂²×C₇	C₂×#102
prime radical	C₂³×C₇	#102
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#3170	D₇
hypocenter	C₇	#11983
LCS	G ▷ C₂×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #11725608	C₂	semidirect
5. #11725599	C₂	semidirect
218. C₇	#11983	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3170	2
C₂×C₄⋊(C₇⋊C₁₆)	2
#11727485	2
#11725608	2
#11725599	2
#11983	7