Subgroups — #13240732

Order 1792.

274 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
2	C₄×#6392	C₂	1	1	yes	maximal, characteristic
3	C₂×#89252	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇×C₁₆	—	2	1	no	—
5	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×#6392	C₄	1	2	yes	—
7	C₄×C₄⋊Dic₇	—	2	1	no	—
8	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
9	C₇×#346	—	2	1	no	—
10	C₇×#346	C₄	1	2	yes	complemented
11	#89346	—	2	1	no	—
12	#89252	—	2	1	no	—
13	#89252	C₄	1	2	yes	complemented
14	#7178	—	7	1	no	maximal, Sylow
15	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₇	—	2	2	no	—
19	C₂×C₄×Dic₇	—	4	1	no	—
20	C₂²×C₇⋊C₈	—	4	1	no	—
21	C₂×C₇×C₁₆	—	2	4	no	—
22	C₂×C₇×C₁₆	—	2	2	no	—
23	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×C₈	D₄	1	1	yes	characteristic
26	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
27	C₇⋊#95	—	4	1	no	—
28	#6392	—	2	2	no	—
29	#6392	C₈	1	4	yes	complemented
30	C₄⋊(C₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇⋊C₈)	—	4	1	no	—
32	C₂×#346	D₇	1	1	yes	characteristic
33	C₄×#107	—	7	1	no	—
34	C₂×#334	—	7	1	no	—
35	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
38	C₂²×Dic₇	—	4	2	no	—
39	C₂×C₇×C₈	—	2	2	no	—
40	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
45	C₂×C₇×C₈	M₁₆	1	2	yes	—
46	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇	SD₁₆	1	1	yes	characteristic
49	C₄²×C₇	Q₁₆	1	1	yes	characteristic
50	C₄⋊Dic₇	—	2	4	no	—
51	C₄⋊Dic₇	—	4	2	no	—
52	C₄×Dic₇	—	8	2	no	—
53	C₂×C₇⋊C₈	—	4	2	no	—
54	C₂×C₇⋊C₈	—	8	2	no	—
55	C₇×C₁₆	—	2	4	no	—
56	C₂²×C₁₆	—	2	1	no	—
57	C₂×#107	—	7	2	no	—
58	C₂×C₄⋊C₈	—	14	1	no	—
59	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
60	C₄×C₄⋊C₄	—	14	1	no	—
61	#346	—	2	1	no	—
62	#346	Dic₇	1	2	yes	complemented
63	#340	—	14	1	no	—
64	#334	—	7	2	no	—
65	#334	—	14	1	no	—
66	C₂³×C₇	#102	1	1	yes	characteristic
67	C₂×C₄×C₇	—	2	2	no	—
68	C₂×C₄×C₇	—	2	1	no	—
69	C₂×C₄×C₇	—	2	1	no	—
70	C₂×C₄×C₇	#95	1	1	yes	characteristic
71	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₇	#102	1	1	yes	characteristic
73	C₂×C₄×C₇	#103	1	1	yes	characteristic
74	C₂×C₄×C₇	#103	1	1	yes	characteristic
75	C₂×C₄×C₇	#104	1	2	yes	—
76	C₂×C₄×C₇	#106	1	1	yes	characteristic
77	C₂×C₄×C₇	#107	1	1	yes	characteristic
78	C₂×Dic₇	—	4	4	no	—
79	C₂×Dic₇	—	8	2	no	—
80	C₂×Dic₇	—	8	2	no	—
81	C₇×C₈	—	2	1	no	—
82	C₇×C₈	—	2	1	no	—
83	C₇×C₈	C₄×C₈	1	1	yes	characteristic
84	C₇×C₈	#97	1	1	yes	characteristic
85	C₇×C₈	C₄⋊C₈	1	2	yes	—
86	C₇⋊C₈	—	8	2	no	—
87	C₂²×C₈	—	28	1	no	—
88	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
89	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	14	2	no	—
91	C₂×C₄²	—	28	1	no	—
92	C₂×C₄²	D₂₈	1	1	yes	characteristic
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	2	2	no	—
95	#107	—	7	4	no	—
96	#107	—	14	2	no	—
97	C₄⋊C₈	—	14	2	no	—
98	C₄⋊C₈	—	28	1	no	—
99	C₄×C₈	Dic₁₄	1	1	yes	characteristic
100	C₄×C₈	C₄×D₇	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
102	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
103	#95	—	28	1	no	—
104	C₂²×C₇	D₄⋊C₈	1	2	yes	—
105	C₂²×C₇	#340	1	1	yes	characteristic
106	C₂²×C₇	#360	1	2	yes	—
107	C₂²×C₇	#361	1	1	yes	characteristic
108	C₂²×C₇	#362	1	1	yes	characteristic
109	C₄×C₇	—	2	1	no	—
110	C₄×C₇	—	2	1	no	—
111	C₄×C₇	—	2	1	no	—
112	C₄×C₇	—	2	1	no	—
113	C₄×C₇	Q₈⋊C₈	1	2	yes	—
114	C₄×C₇	#336	1	1	yes	characteristic
115	C₄×C₇	#339	1	1	yes	characteristic
116	Dic₇	—	8	4	no	—
117	C₂²×C₄	—	28	2	no	—
118	C₂²×C₄	#1132	1	1	yes	characteristic
119	C₂²×C₄	D₅₆	1	1	yes	characteristic
120	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	28	2	no	—
123	C₂×C₈	—	56	2	no	—
124	C₂×C₈	C₈×D₇	1	2	yes	—
125	C₂×C₈	#1131	1	2	yes	—
126	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
127	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
129	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
130	C₄⋊C₄	—	14	4	no	—
131	C₄⋊C₄	—	28	2	no	—
132	C₄²	—	56	2	no	—
133	C₄²	#1132	1	1	yes	characteristic
134	C₄²	Dic₂₈	1	1	yes	characteristic
135	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
136	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
137	C₁₆	—	2	4	no	—
138	C₂×C₇	#1279	1	1	yes	characteristic
139	C₂×C₇	#1333	1	2	yes	—
140	C₂×C₇	#1334	1	2	yes	—
141	C₂×C₇	#1386	1	1	yes	characteristic
142	C₂×C₇	#1387	1	1	yes	characteristic
143	C₂³	#6395	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	28	4	no	—
148	C₂×C₄	—	56	2	no	—
149	C₂×C₄	—	56	2	no	—
150	C₂×C₄	#6379	1	2	yes	—
151	C₂×C₄	#6390	1	1	yes	characteristic
152	C₂×C₄	#6390	1	1	yes	characteristic
153	C₂×C₄	#6391	1	1	yes	characteristic
154	C₂×C₄	#6392	1	1	yes	characteristic
155	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
156	C₂×C₄	#6395	1	1	yes	characteristic
157	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	56	2	no	—
161	C₈	C₈×Dic₇	1	1	yes	characteristic
162	C₈	Dic₇⋊C₈	1	2	yes	—
163	C₈	#6389	1	1	yes	characteristic
164	C₇	#7178	1	1	yes	characteristic, complemented, Sylow
165	C₂²	#89254	1	2	yes	—
166	C₂²	#89311	1	1	yes	characteristic
167	C₂²	#89312	1	1	yes	characteristic
168	C₂²	#89313	1	2	yes	—
169	C₂²	#89346	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	56	4	no	—
175	C₄	#89251	1	2	yes	—
176	C₄	#89318	1	1	yes	characteristic
177	C₄	#89345	1	1	yes	characteristic
178	C₂	#11725322	1	2	yes	—
179	C₂	#11725323	1	2	yes	—
180	C₂	#11725461	1	1	yes	characteristic
181	C₂	#11725623	1	1	yes	characteristic
182	C₂	#11725625	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6395
commutator	C₇×C₈	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇
Fitting	C₂×C₇×#346	C₂
socle, prime radical	C₂³×C₇	#102
cosocle	C₂×C₄×C₇×C₈	C₂²
hypercenter	C₂×#346	D₇
hypocenter	C₇	#7178
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
10. C₇×#346	C₄	semidirect
13. #89252	C₄	semidirect
29. #6392	C₈	semidirect
62. #346	Dic₇	semidirect
164. C₇	#7178	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#346	2
C₄×#6392	2
C₂×#89252	2
#7178	7