Subgroups — #30749572

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	2	yes	maximal
2	C₂×#16863149	C₂	1	4	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
6	#16863149	—	2	8	no	—
7	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
11	C₄³×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₈	—	2	4	no	—
13	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
14	C₄⋊(C₆₇⋊C₈)	—	4	8	no	—
15	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₆₇	D₄	1	2	yes	—
20	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
21	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₆₇	—	2	4	no	—
23	C₄×C₈×C₆₇	—	2	2	no	—
24	C₄×C₈×C₆₇	C₈	1	4	yes	complemented
25	C₄×C₈×C₆₇	D₄	1	4	yes	—
26	C₄×C₈×C₆₇	Q₈	1	4	yes	—
27	C₂×C₆₇⋊C₈	—	4	4	no	—
28	C₂×C₆₇⋊C₈	—	4	8	no	—
29	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
30	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₂×C₈×C₆₇	—	2	2	no	—
36	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
40	C₄²×C₆₇	—	2	4	no	—
41	C₄²×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	C₂×C₈	1	2	yes	—
43	C₄²×C₆₇	M₁₆	1	2	yes	—
44	C₄²×C₆₇	D₈	1	4	yes	—
45	C₄²×C₆₇	Q₁₆	1	4	yes	—
46	C₄²×C₆₇	C₂×D₄	1	2	yes	—
47	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
48	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
53	C₂×C₄×C₆₇	#107	1	4	yes	—
54	C₂×C₄×C₆₇	#107	1	4	yes	—
55	C₂×C₄×C₆₇	#108	1	4	yes	—
56	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
65	C₈×C₆₇	—	2	4	no	—
66	C₈×C₆₇	C₄⋊C₈	1	8	yes	complemented
67	C₆₇⋊C₈	—	8	4	no	—
68	C₂²×C₆₇	#384	1	1	yes	characteristic
69	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₂²×C₆₇	#423	1	1	yes	characteristic
71	C₂²×C₆₇	C₂×#107	1	2	yes	—
72	C₂²×C₆₇	C₂×#108	1	2	yes	—
73	C₄×C₆₇	—	2	4	no	—
74	C₄×C₆₇	—	2	2	no	—
75	C₄×C₆₇	#335	1	8	yes	complemented
76	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
77	C₄×C₆₇	#423	1	2	yes	—
78	C₄×C₆₇	#493	1	1	yes	characteristic
79	C₄×C₆₇	#494	1	1	yes	characteristic
80	C₄×C₆₇	#500	1	2	yes	—
81	#8806	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	C₂×#335	1	2	yes	—
83	C₂×C₆₇	#1567	1	2	yes	—
84	C₂×C₆₇	#1842	1	1	yes	characteristic
85	C₂×C₆₇	#1845	1	1	yes	characteristic
86	C₂×C₆₇	#1848	1	1	yes	characteristic
87	#1842	—	67	2	no	—
88	C₄²×C₈	D₆₇	1	1	yes	characteristic
89	C₂×#335	—	67	4	no	—
90	C₆₇	#8806	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₄⋊C₈	—	67	4	no	—
92	C₂×C₄⋊C₈	—	134	4	no	—
93	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
94	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
95	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
97	C₄³	C₂×D₆₇	1	1	yes	characteristic
98	#335	—	134	8	no	—
99	C₂²×C₈	—	134	4	no	—
100	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
101	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
102	C₂×C₄²	Dic₁₃₄	1	2	yes	—
103	C₂×C₄²	D₂₆₈	1	2	yes	—
104	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
106	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
107	C₄⋊C₈	—	268	8	no	—
108	C₄⋊C₈	—	134	8	no	—
109	C₄×C₈	—	2	4	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	C₆₇⋊C₈	1	4	yes	complemented
112	C₄×C₈	Dic₁₃₄	1	4	yes	—
113	C₄×C₈	D₂₆₈	1	4	yes	—
114	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
115	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
116	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
117	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
118	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	268	4	no	—
122	C₂×C₈	—	268	8	no	—
123	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
124	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
125	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
126	C₂×C₈	#11761612	1	4	yes	—
127	C₄²	—	2	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
130	C₄²	#11761612	1	2	yes	—
131	C₄²	D₅₃₆	1	4	yes	—
132	C₄²	Dic₂₆₈	1	4	yes	—
133	C₄²	C₂×D₂₆₈	1	2	yes	—
134	C₄²	C₂×Dic₁₃₄	1	2	yes	—
135	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
140	C₂×C₄	#14523977	1	4	yes	—
141	C₂×C₄	#14523977	1	4	yes	—
142	C₂×C₄	#14523979	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
144	C₂×C₄	#14524040	1	1	yes	characteristic
145	C₂×C₄	#14524043	1	2	yes	—
146	C₂×C₄	#14524043	1	1	yes	characteristic
147	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
148	C₂×C₄	C₂×#11761612	1	2	yes	—
149	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
150	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
151	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
152	C₈	—	2	4	no	—
153	C₈	—	536	4	no	—
154	C₈	C₄⋊(C₆₇⋊C₈)	1	8	yes	complemented
155	C₂²	#16863266	1	1	yes	characteristic
156	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
157	C₂²	#16863420	1	1	yes	characteristic
158	C₂²	C₂×#14523977	1	2	yes	—
159	C₂²	C₂×#14523979	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	4	no	—
162	C₄	#16863149	1	8	yes	complemented
163	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
164	C₄	#16863420	1	2	yes	—
165	C₄	#16863773	1	1	yes	characteristic
166	C₄	#16863776	1	1	yes	characteristic
167	C₄	#16863801	1	2	yes	—
168	C₂	C₂×#16863149	1	2	yes	—
169	C₂	#22558968	1	2	yes	—
170	C₂	#22560361	1	1	yes	characteristic
171	C₂	#22560369	1	1	yes	characteristic
172	C₂	#22560381	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#8806
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
24. C₄×C₈×C₆₇	C₈	semidirect
66. C₈×C₆₇	C₄⋊C₈	semidirect
75. C₄×C₆₇	#335	semidirect
90. C₆₇	#8806	semidirect
111. C₄×C₈	C₆₇⋊C₈	semidirect
154. C₈	C₄⋊(C₆₇⋊C₈)	semidirect
162. C₄	#16863149	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₂×#16863149	2
C₄²×C₈×C₆₇	2
#8806	67