Subgroups — #30749569

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 168 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×#16863147	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	#16863147	—	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	SD₁₆	1	8	yes	—
45	C₄²×C₆₇	C₂×D₄	1	2	yes	—
46	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
47	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₆₇	#106	1	4	yes	—
53	C₂×C₄×C₆₇	#106	1	4	yes	—
54	C₂×C₄×C₆₇	#108	1	4	yes	—
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
58	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
63	C₈×C₆₇	—	2	4	no	—
64	C₈×C₆₇	C₄⋊C₈	1	8	yes	complemented
65	C₆₇⋊C₈	—	8	4	no	—
66	C₂²×C₆₇	#384	1	1	yes	characteristic
67	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₆₇	#423	1	1	yes	characteristic
69	C₂²×C₆₇	C₂×#106	1	2	yes	—
70	C₂²×C₆₇	C₂×#108	1	2	yes	—
71	C₄×C₆₇	—	2	4	no	—
72	C₄×C₆₇	—	2	2	no	—
73	C₄×C₆₇	#334	1	8	yes	complemented
74	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
75	C₄×C₆₇	#423	1	2	yes	—
76	C₄×C₆₇	#492	1	2	yes	—
77	C₄×C₆₇	#498	1	2	yes	—
78	#8805	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	C₂×#334	1	2	yes	—
80	C₂×C₆₇	#1566	1	2	yes	—
81	C₂×C₆₇	#1842	1	1	yes	characteristic
82	C₂×C₆₇	#1844	1	1	yes	characteristic
83	C₂×C₆₇	#1848	1	1	yes	characteristic
84	#1842	—	67	2	no	—
85	C₄²×C₈	D₆₇	1	1	yes	characteristic
86	C₂×#334	—	67	4	no	—
87	C₆₇	#8805	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊C₈	—	67	4	no	—
89	C₂×C₄⋊C₈	—	134	4	no	—
90	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
91	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
92	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
94	C₄³	C₂×D₆₇	1	1	yes	characteristic
95	#334	—	134	8	no	—
96	C₂²×C₈	—	134	4	no	—
97	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
98	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
99	C₂×C₄²	Dic₁₃₄	1	2	yes	—
100	C₂×C₄²	D₂₆₈	1	2	yes	—
101	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
104	C₄⋊C₈	—	268	8	no	—
105	C₄⋊C₈	—	134	8	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	C₆₇⋊C₈	1	4	yes	complemented
109	C₄×C₈	Dic₁₃₄	1	4	yes	—
110	C₄×C₈	D₂₆₈	1	4	yes	—
111	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
112	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
113	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	268	4	no	—
119	C₂×C₈	—	268	8	no	—
120	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
121	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
122	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
123	C₂×C₈	#11761612	1	4	yes	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
127	C₄²	#11761612	1	2	yes	—
128	C₄²	#11761617	1	8	yes	—
129	C₄²	C₂×D₂₆₈	1	2	yes	—
130	C₄²	C₂×Dic₁₃₄	1	2	yes	—
131	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
136	C₂×C₄	#14523975	1	4	yes	—
137	C₂×C₄	#14523975	1	4	yes	—
138	C₂×C₄	#14523979	1	4	yes	—
139	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
140	C₂×C₄	#14524040	1	1	yes	characteristic
141	C₂×C₄	#14524043	1	2	yes	—
142	C₂×C₄	#14524043	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
144	C₂×C₄	C₂×#11761612	1	2	yes	—
145	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
146	C₂×C₄	C₂×#11761617	1	4	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	536	4	no	—
149	C₈	C₄⋊(C₆₇⋊C₈)	1	8	yes	complemented
150	C₂²	#16863266	1	1	yes	characteristic
151	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
152	C₂²	#16863420	1	1	yes	characteristic
153	C₂²	C₂×#14523975	1	2	yes	—
154	C₂²	C₂×#14523979	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	2	4	no	—
157	C₄	#16863147	1	8	yes	complemented
158	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
159	C₄	#16863420	1	2	yes	—
160	C₄	#16863769	1	2	yes	—
161	C₄	#16863794	1	2	yes	—
162	C₂	C₂×#16863147	1	2	yes	—
163	C₂	#22558965	1	2	yes	—
164	C₂	#22560361	1	1	yes	characteristic
165	C₂	#22560366	1	1	yes	characteristic
166	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	#8805
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
64. C₈×C₆₇	C₄⋊C₈	semidirect
73. C₄×C₆₇	#334	semidirect
87. C₆₇	#8805	semidirect
108. C₄×C₈	C₆₇⋊C₈	semidirect
149. C₈	C₄⋊(C₆₇⋊C₈)	semidirect
157. C₄	#16863147	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×#16863147	2
C₄²×C₈×C₆₇	2
#8805	67