Subgroups — #13380386

Order 1792.

664 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11729388	C₂	1	4	yes	maximal
3	#11727236	C₂	1	8	yes	maximal
4	#11727147	C₂	1	2	yes	maximal
5	C₂×C₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
7	C₇×#391	C₄	1	2	yes	—
8	C₇×#389	C₄	1	4	yes	—
9	C₇×#384	C₄	1	1	yes	characteristic
10	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₄³×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₄×C₇⋊C₈	C₂²	1	4	yes	—
14	#89319	C₂²	1	8	yes	—
15	#89319	C₂²	1	4	yes	—
16	#89319	C₂²	1	8	yes	—
17	#23363	—	7	1	no	maximal, Sylow
18	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
19	C₇×C₄⋊Q₈	C₈	1	8	yes	complemented
20	C₄×C₇×Q₈	—	4	2	no	—
21	C₄×C₇×Q₈	—	4	2	no	—
22	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
23	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
25	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂³	1	4	yes	—
30	C₂×C₄²×C₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
32	C₂²×C₇⋊C₈	C₂³	1	8	yes	—
33	C₇×#95	C₂×C₄	1	8	yes	—
34	C₄⋊(C₇⋊C₈)	—	4	4	no	—
35	C₄×C₇⋊C₈	—	4	4	no	—
36	C₄×C₄⋊Q₈	D₇	1	1	yes	characteristic
37	#1984	—	7	4	no	—
38	#1916	—	7	8	no	—
39	#1839	—	7	2	no	—
40	C₂×C₇×Q₈	—	4	2	no	—
41	C₂×C₇×Q₈	—	4	2	no	—
42	C₂×C₇×Q₈	C₂×C₈	1	8	yes	—
43	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
50	C₇×C₄⋊C₄	—	4	4	no	—
51	C₇×C₄⋊C₄	—	4	2	no	—
52	C₇×C₄⋊C₄	—	4	4	no	—
53	C₇×C₄⋊C₄	C₂×C₈	1	16	yes	—
54	C₄²×C₇	—	2	2	no	—
55	C₄²×C₇	—	2	4	no	—
56	C₄²×C₇	—	2	4	no	—
57	C₄²×C₇	—	2	4	no	—
58	C₄²×C₇	—	4	2	no	—
59	C₄²×C₇	—	4	1	no	—
60	C₄²×C₇	C₂×C₈	1	4	yes	—
61	C₂×C₇⋊C₈	—	2	8	no	—
62	C₂×C₇⋊C₈	—	2	16	no	—
63	C₂×C₄⋊Q₈	Dic₇	1	1	yes	characteristic
64	C₂×C₄×Q₈	C₂×D₇	1	2	yes	—
65	C₂×C₄⋊C₈	—	7	4	no	—
66	C₂×C₄×C₈	—	7	4	no	—
67	#391	Dic₇	1	2	yes	—
68	#389	Dic₇	1	4	yes	—
69	#384	Dic₇	1	1	yes	characteristic
70	C₄×C₄⋊C₄	C₂×D₇	1	4	yes	—
71	C₄³	C₂×D₇	1	1	yes	characteristic
72	#336	—	7	8	no	—
73	#336	—	7	4	no	—
74	#336	—	7	8	no	—
75	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
76	C₇×Q₈	—	4	8	no	—
77	C₂×C₄×C₇	—	2	1	no	—
78	C₂×C₄×C₇	—	2	4	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	1	no	—
82	C₂×C₄×C₇	—	2	4	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	4	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	8	no	—
90	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
93	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
94	C₂×C₄×C₇	2₋¹⁺⁴	1	4	yes	—
95	C₇⋊C₈	—	4	8	no	—
96	C₂²×Q₈	C₂×Dic₇	1	2	yes	—
97	C₂²×C₈	—	7	8	no	—
98	C₄⋊Q₈	C₇⋊C₈	1	8	yes	complemented
99	C₄×Q₈	—	4	2	no	—
100	C₄×Q₈	—	4	2	no	—
101	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
103	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
104	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
105	C₂×C₄²	C₂×Dic₇	1	2	yes	—
106	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
107	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
108	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
109	C₂×C₄²	C₂²×D₇	1	4	yes	—
110	C₂×C₄²	C₂²×D₇	1	2	yes	—
111	C₄⋊C₈	—	28	4	no	—
112	C₄×C₈	—	28	4	no	—
113	#95	C₂×Dic₇	1	8	yes	—
114	C₂²×C₇	#518	1	1	yes	characteristic
115	C₂²×C₇	#519	1	2	yes	—
116	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
117	C₂²×C₇	#568	1	1	yes	characteristic
118	C₂²×C₇	#568	1	2	yes	—
119	C₄×C₇	—	2	8	no	—
120	C₄×C₇	—	2	2	no	—
121	C₄×C₇	—	2	4	no	—
122	C₄×C₇	—	4	1	no	—
123	C₄×C₇	—	4	4	no	—
124	C₄×C₇	—	4	2	no	—
125	C₄×C₇	#563	1	4	yes	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	2	no	—
128	C₂×Q₈	C₂×C₇⋊C₈	1	8	yes	—
129	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
130	C₂²×C₄	C₂²×Dic₇	1	4	yes	—
131	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
132	C₂²×C₄	C₂²×Dic₇	1	4	yes	—
133	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
134	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
135	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
136	C₂×C₈	—	14	8	no	—
137	C₂×C₈	—	14	16	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	C₂×C₇⋊C₈	1	16	yes	—
142	C₄²	—	2	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	4	2	no	—
147	C₄²	—	4	1	no	—
148	C₄²	C₂×C₇⋊C₈	1	4	yes	—
149	C₂×C₇	#2381	1	1	yes	characteristic
150	C₂×C₇	#2972	1	1	yes	characteristic
151	C₂×C₇	#2980	1	1	yes	characteristic
152	C₂×C₇	#2982	1	2	yes	—
153	C₂×C₇	#2987	1	2	yes	—
154	C₂³	C₂³×Dic₇	1	1	yes	characteristic
155	Q₈	—	4	8	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
170	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
171	C₂×C₄	C₂²×C₇⋊C₈	1	8	yes	—
172	C₂×C₄	#6548	1	2	yes	—
173	C₂×C₄	#6551	1	4	yes	—
174	C₈	—	28	8	no	—
175	C₇	#23363	1	1	yes	characteristic, complemented, Sylow
176	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
177	C₂²	#90489	1	1	yes	characteristic
178	C₂²	#90503	1	2	yes	—
179	C₂²	#90510	1	2	yes	—
180	C₂²	#90510	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	8	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	2	no	—
187	C₄	#90426	1	4	yes	—
188	C₂	#11739746	1	1	yes	characteristic
189	C₂	#11739880	1	2	yes	—
190	C₂	#11740676	1	1	yes	characteristic
191	C₂	#11740724	1	1	yes	characteristic
192	C₂	#11740778	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇
commutator	C₂²×C₇	C₂³×C₈
Fitting	C₄×C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₇	C₂³×C₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₇
hypocenter	C₇	#23363
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
19. C₇×C₄⋊Q₈	C₈	semidirect
98. C₄⋊Q₈	C₇⋊C₈	semidirect
175. C₇	#23363	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×C₄⋊Q₈	2
#11729388	2
#11727236	2
#11727147	2
#23363	7