Subgroups — #30749495

Order 17152.

382 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560326	C₂	1	2	yes	maximal
2	#22558044	C₂	1	4	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
5	#16863151	—	2	4	no	—
6	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	C₄³×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₈	—	2	4	no	—
12	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
13	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	D₄	1	2	yes	—
18	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₆₇	—	2	4	no	—
21	C₄×C₈×C₆₇	—	2	2	no	—
22	C₄×C₈×C₆₇	—	2	4	no	—
23	C₄×C₈×C₆₇	C₈	1	4	yes	complemented
24	C₂×C₆₇⋊C₈	—	4	4	no	—
25	C₂×C₆₇⋊C₈	—	4	8	no	—
26	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	—	2	2	no	—
33	C₂×C₈×C₆₇	—	2	2	no	—
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
37	C₄²×C₆₇	—	2	4	no	—
38	C₄²×C₆₇	—	2	2	no	—
39	C₄²×C₆₇	—	2	4	no	—
40	C₄²×C₆₇	C₂×C₈	1	2	yes	—
41	C₄²×C₆₇	M₁₆	1	2	yes	—
42	C₄²×C₆₇	#41	1	4	yes	—
43	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	2	2	no	—
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
50	C₂×C₄×C₆₇	#108	1	4	yes	—
51	C₂×C₄×C₆₇	#117	1	2	yes	—
52	C₂×C₄×C₆₇	#124	1	2	yes	—
53	C₂×C₄×C₆₇	#125	1	2	yes	—
54	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#135	1	4	yes	—
58	C₈×C₆₇	—	2	4	no	—
59	C₈×C₆₇	—	2	4	no	—
60	C₆₇⋊C₈	—	8	4	no	—
61	C₂²×C₆₇	#382	1	1	yes	characteristic
62	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²×C₆₇	#423	1	1	yes	characteristic
64	C₂²×C₆₇	#427	1	2	yes	—
65	C₂²×C₆₇	C₂×#108	1	2	yes	—
66	C₄×C₆₇	—	2	4	no	—
67	C₄×C₆₇	—	2	4	no	—
68	C₄×C₆₇	—	2	2	no	—
69	C₄×C₆₇	#431	1	2	yes	—
70	C₄×C₆₇	#432	1	2	yes	—
71	C₄×C₆₇	#488	1	4	yes	—
72	#8785	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	#1564	1	2	yes	—
74	C₂×C₆₇	#1568	1	2	yes	—
75	C₂×C₆₇	#1831	1	1	yes	characteristic
76	C₂×C₆₇	#1835	1	1	yes	characteristic
77	C₂×C₆₇	#1837	1	1	yes	characteristic
78	#1831	—	67	2	no	—
79	C₄²×C₈	D₆₇	1	1	yes	characteristic
80	#1277	—	67	4	no	—
81	C₆₇	#8785	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊C₈	—	67	4	no	—
83	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
84	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
85	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
87	C₄³	C₂×D₆₇	1	1	yes	characteristic
88	#336	—	134	4	no	—
89	C₂²×C₈	—	134	4	no	—
90	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
91	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
92	C₂×C₄²	Dic₁₃₄	1	2	yes	—
93	C₂×C₄²	D₂₆₈	1	2	yes	—
94	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
96	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
97	C₄⋊C₈	—	134	8	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	2	4	no	—
101	C₄×C₈	C₆₇⋊C₈	1	4	yes	complemented
102	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
103	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
104	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	268	4	no	—
112	C₂×C₈	—	268	8	no	—
113	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
114	C₂×C₈	#11761612	1	4	yes	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
119	C₄²	#11761612	1	2	yes	—
120	C₄²	#11761628	1	4	yes	—
121	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
128	C₂×C₄	#14523979	1	4	yes	—
129	C₂×C₄	#14524002	1	2	yes	—
130	C₂×C₄	#14524031	1	2	yes	—
131	C₂×C₄	#14524035	1	2	yes	—
132	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
133	C₂×C₄	C₂×#11761612	1	2	yes	—
134	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
135	C₂×C₄	#14524065	1	4	yes	—
136	C₈	—	2	4	no	—
137	C₈	—	2	4	no	—
138	C₈	—	536	4	no	—
139	C₂²	#16863262	1	1	yes	characteristic
140	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
141	C₂²	#16863420	1	1	yes	characteristic
142	C₂²	#16863434	1	2	yes	—
143	C₂²	C₂×#14523979	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#16863449	1	2	yes	—
148	C₄	#16863453	1	2	yes	—
149	C₄	#16863751	1	4	yes	—
150	C₂	#22558957	1	2	yes	—
151	C₂	#22558972	1	2	yes	—
152	C₂	#22560326	1	1	yes	characteristic
153	C₂	#22560340	1	1	yes	characteristic
154	C₂	#22560347	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	#8785
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
23. C₄×C₈×C₆₇	C₈	semidirect
81. C₆₇	#8785	semidirect
101. C₄×C₈	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22560326	2
#22558044	2
C₄²×C₈×C₆₇	2
#8785	67