Subgroups — #13244976

Order 1792.

470 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₇×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×#89275	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₇×#1277	C₂	1	1	yes	maximal, characteristic
5	#11725460	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇	—	2	2	no	—
9	C₂×C₈×Dic₇	C₄	1	2	yes	—
10	C₂×C₈×Dic₇	C₄	1	2	yes	—
11	C₄²×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	#89345	—	2	2	no	—
15	#89275	C₄	1	8	yes	complemented
16	C₄×#334	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	—	2	2	no	—
18	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇	Q₈	1	2	yes	—
24	C₂²×C₇⋊C₈	—	2	2	no	—
25	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₇×C₄⋊C₈	—	2	2	no	—
28	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₇×C₈	—	2	2	no	—
30	C₈×Dic₇	—	2	4	no	—
31	C₈×Dic₇	—	4	4	no	—
32	C₈×Dic₇	C₈	1	8	yes	complemented
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
35	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₄×C₄⋊C₈	—	7	2	no	—
37	C₄²×C₈	—	7	1	no	—
38	C₂×#334	—	7	2	no	—
39	#1277	—	7	1	no	—
40	#1277	D₇	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×Dic₇	C₄⋊C₄	1	2	yes	—
45	C₂²×Dic₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₇×C₈	—	2	2	no	—
47	C₂×C₇×C₈	—	2	2	no	—
48	C₂×C₇×C₈	—	2	4	no	—
49	C₂×C₇×C₈	—	4	2	no	—
50	C₂×C₇×C₈	—	4	2	no	—
51	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
52	C₄²×C₇	C₂²×C₄	1	2	yes	—
53	C₄²×C₇	#41	1	2	yes	—
54	C₄×Dic₇	—	2	1	no	—
55	C₄×Dic₇	—	2	1	no	—
56	C₄×Dic₇	—	2	4	no	—
57	C₄×Dic₇	C₂×C₈	1	4	yes	—
58	C₄×Dic₇	M₁₆	1	4	yes	—
59	C₄×Dic₇	SD₁₆	1	4	yes	—
60	C₂×C₇⋊C₈	—	2	2	no	—
61	C₂×C₇⋊C₈	—	2	4	no	—
62	C₂×C₇⋊C₈	—	4	2	no	—
63	C₂×C₇⋊C₈	—	4	2	no	—
64	C₂×C₇⋊C₈	C₄²	1	2	yes	—
65	C₂×C₇⋊C₈	C₄²	1	2	yes	—
66	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
67	C₂×C₄⋊C₈	—	7	2	no	—
68	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
69	C₂×C₄×C₈	—	7	2	no	—
70	C₂×C₄×C₈	—	7	1	no	—
71	C₂×C₄×C₈	—	7	2	no	—
72	C₂×C₄×C₈	—	14	2	no	—
73	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
74	C₄³	—	7	1	no	—
75	#336	—	14	2	no	—
76	#334	—	7	8	no	—
77	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₇	#117	1	1	yes	characteristic
81	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
83	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₇	#135	1	1	yes	characteristic
87	C₂×C₄×C₇	#135	1	1	yes	characteristic
88	C₂×Dic₇	—	2	2	no	—
89	C₂×Dic₇	—	2	2	no	—
90	C₂×Dic₇	C₄⋊C₈	1	4	yes	—
91	C₂×Dic₇	C₄⋊C₈	1	4	yes	—
92	C₂×Dic₇	#106	1	4	yes	—
93	C₂×Dic₇	#108	1	4	yes	—
94	C₇×C₈	—	2	4	no	—
95	C₇×C₈	—	4	4	no	—
96	C₇⋊C₈	—	2	2	no	—
97	C₇⋊C₈	—	4	4	no	—
98	C₇⋊C₈	C₄×C₈	1	4	yes	complemented
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	—	7	1	no	—
101	C₂²×C₈	—	7	1	no	—
102	C₂²×C₈	—	14	2	no	—
103	C₂²×C₈	C₄×D₇	1	2	yes	—
104	C₂×C₄²	—	7	2	no	—
105	C₂×C₄²	—	7	1	no	—
106	C₂×C₄²	—	7	1	no	—
107	C₂×C₄²	—	7	2	no	—
108	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	7	4	no	—
111	C₄⋊C₈	—	14	2	no	—
112	C₄⋊C₈	C₄×D₇	1	4	yes	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	7	8	no	—
115	C₄×C₈	—	7	4	no	—
116	C₄×C₈	—	14	4	no	—
117	C₄×C₈	—	28	4	no	—
118	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₇	C₂×C₄⋊C₈	1	2	yes	—
120	C₂²×C₇	C₂×#106	1	1	yes	characteristic
121	C₂²×C₇	#427	1	1	yes	characteristic
122	C₂²×C₇	C₂×#108	1	2	yes	—
123	C₄×C₇	—	2	2	no	—
124	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
125	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
126	C₄×C₇	#431	1	2	yes	—
127	C₄×C₇	C₄×SD₁₆	1	2	yes	—
128	C₄×C₇	C₄×SD₁₆	1	2	yes	—
129	Dic₇	—	2	4	no	—
130	Dic₇	#334	1	8	yes	complemented
131	C₂²×C₄	—	7	2	no	—
132	C₂²×C₄	—	7	2	no	—
133	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₇	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	7	2	no	—
142	C₂×C₈	—	7	4	no	—
143	C₂×C₈	—	7	2	no	—
144	C₂×C₈	—	14	2	no	—
145	C₂×C₈	—	14	4	no	—
146	C₂×C₈	—	28	2	no	—
147	C₂×C₈	—	28	2	no	—
148	C₂×C₈	C₈×D₇	1	4	yes	—
149	C₄²	—	7	4	no	—
150	C₄²	—	7	4	no	—
151	C₄²	—	7	4	no	—
152	C₄²	—	14	4	no	—
153	C₄²	—	14	1	no	—
154	C₄²	—	14	1	no	—
155	C₄²	C₂×C₄×D₇	1	2	yes	—
156	C₄²	#1159	1	2	yes	—
157	C₂×C₇	C₂×#334	1	2	yes	—
158	C₂×C₇	#1564	1	2	yes	—
159	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
160	C₂×C₇	C₄×#106	1	1	yes	characteristic
161	C₂×C₇	C₄×#108	1	1	yes	characteristic
162	C₂³	D₇×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	7	4	no	—
165	C₂×C₄	—	7	4	no	—
166	C₂×C₄	—	7	4	no	—
167	C₂×C₄	—	7	4	no	—
168	C₂×C₄	—	14	2	no	—
169	C₂×C₄	—	14	2	no	—
170	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
171	C₂×C₄	#6440	1	1	yes	characteristic
172	C₂×C₄	(C₂²⋊D₇)⋊C₄	1	2	yes	—
173	C₂×C₄	#6450	1	2	yes	—
174	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
175	C₂×C₄	D₇×M₁₆	1	2	yes	—
176	C₂×C₄	#6475	1	1	yes	characteristic
177	C₂×C₄	D₇×SD₁₆	1	2	yes	—
178	C₂×C₄	#6482	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	—	4	4	no	—
181	C₈	—	7	4	no	—
182	C₈	—	14	2	no	—
183	C₈	—	28	4	no	—
184	C₇	C₄×#334	1	1	yes	characteristic, complemented, Sylow
185	C₂²	#89421	1	1	yes	characteristic
186	C₂²	D₇×C₄⋊C₈	1	2	yes	—
187	C₂²	D₇×#106	1	1	yes	characteristic
188	C₂²	#89652	1	1	yes	characteristic
189	C₂²	D₇×#108	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	7	8	no	—
192	C₄	—	14	4	no	—
193	C₄	C₄×C₈×D₇	1	1	yes	characteristic
194	C₄	#89479	1	1	yes	characteristic
195	C₄	#89490	1	2	yes	—
196	C₄	#89530	1	2	yes	—
197	C₄	#89560	1	2	yes	—
198	C₂	D₇×#334	1	2	yes	—
199	C₂	#11726321	1	2	yes	—
200	C₂	#11726356	1	1	yes	characteristic
201	C₂	#11726538	1	1	yes	characteristic
202	C₂	#11726670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1277	C₂
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1277	D₇
hypocenter	C₇	C₄×#334
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
15. #89275	C₄	semidirect
32. C₈×Dic₇	C₈	semidirect
98. C₇⋊C₈	C₄×C₈	semidirect
130. Dic₇	#334	semidirect
184. C₇	C₄×#334	semidirect

Maximal subgroups

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

subgroup	index
Dic₇×C₄⋊C₈	2
C₂×#89275	2
C₄×C₈×Dic₇	2
C₇×#1277	2
#11725460	2
C₄×#334	7