Subgroups — #30767389

Order 17152.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558957	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
3	#22558275	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863449	—	2	1	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	#16863149	—	2	1	no	—
8	#16863147	—	2	1	no	—
9	#16863147	C₂²	1	2	yes	—
10	#16863147	C₂²	1	2	yes	—
11	C₂×M₃₂×C₆₇	C₄	1	1	yes	characteristic
12	C₂×M₃₂×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#346×C₆₇	C₂²	1	2	yes	—
15	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
16	C₂²×C₆₇⋊C₈	—	4	1	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
21	C₄×C₆₇⋊C₈	—	4	1	no	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
25	M₃₂×C₆₇	—	2	2	no	—
26	M₃₂×C₆₇	C₈	1	4	yes	complemented
27	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	8	2	no	—
36	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
39	C₂×C₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
44	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
48	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
49	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₆₇	—	2	2	no	—
51	C₁₆×C₆₇	C₂×C₈	1	4	yes	complemented
52	C₂³×C₆₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	4	1	no	—
55	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#108	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₆₇	—	2	1	no	—
64	C₈×C₆₇	—	2	1	no	—
65	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
66	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
68	C₆₇⋊C₈	—	8	2	no	—
69	C₆₇⋊C₈	—	8	2	no	—
70	C₂²×C₆₇	—	2	1	no	—
71	C₂²×C₆₇	—	2	1	no	—
72	C₂²×C₆₇	#334	1	2	yes	—
73	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
74	C₄×C₆₇	—	2	1	no	—
75	C₄×C₆₇	—	2	1	no	—
76	C₄×C₆₇	—	2	1	no	—
77	C₄×C₆₇	#334	1	2	yes	—
78	C₄×C₆₇	#365	1	2	yes	—
79	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
81	#12000	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	2	no	—
83	C₂×C₆₇	C₂×#334	1	1	yes	characteristic
84	C₂×C₆₇	C₂×#365	1	1	yes	characteristic
85	C₂×C₆₇	#2155	1	1	yes	characteristic
86	C₄×M₃₂	D₆₇	1	1	yes	characteristic
87	#1564	—	67	1	no	—
88	C₂×#334	—	67	1	no	—
89	#1368	—	67	4	no	—
90	C₆₇	#12000	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	Dic₆₇	1	1	yes	characteristic
92	C₂×M₃₂	Dic₆₇	1	1	yes	characteristic
93	#431	—	134	1	no	—
94	C₂×C₄⋊C₈	—	134	1	no	—
95	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
96	#346	C₂×D₆₇	1	2	yes	—
97	C₄×C₁₆	Dic₆₇	1	2	yes	—
98	#335	—	134	1	no	—
99	#334	—	67	2	no	—
100	#334	—	67	2	no	—
101	#334	—	134	1	no	—
102	C₂²×C₈	—	268	1	no	—
103	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
105	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₆₇⋊C₈	1	4	yes	complemented
108	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
110	C₄⋊C₈	—	268	1	no	—
111	C₄⋊C₈	—	134	2	no	—
112	C₄⋊C₈	—	134	2	no	—
113	C₂²⋊C₈	—	268	1	no	—
114	C₄×C₈	—	268	1	no	—
115	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
116	C₄×C₈	D₂₆₈	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
120	C₂²×C₄	#11761617	1	1	yes	characteristic
121	C₂²×C₄	#11761617	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	268	2	no	—
124	C₂×C₈	—	268	2	no	—
125	C₂×C₈	—	536	2	no	—
126	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
128	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
129	C₂×C₈	#11761612	1	2	yes	—
130	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
132	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
133	C₄²	#11761617	1	1	yes	characteristic
134	C₄²	#11761617	1	1	yes	characteristic
135	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	C₂×C₆₇⋊C₈	1	4	yes	complemented
138	C₂³	#14523975	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
142	C₂×C₄	#14523975	1	1	yes	characteristic
143	C₂×C₄	#14523975	1	1	yes	characteristic
144	C₂×C₄	#14523975	1	1	yes	characteristic
145	C₂×C₄	#14523979	1	2	yes	—
146	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
147	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
148	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	536	2	no	—
152	C₈	—	536	2	no	—
153	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
154	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
155	C₈	C₂×#11761612	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#16863147	1	2	yes	—
159	C₂²	C₂×#14523975	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#16863147	1	2	yes	—
164	C₄	#16863219	1	2	yes	—
165	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
166	C₄	C₂×#14523979	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#16863147	1	1	yes	characteristic
169	C₂	C₂×#16863219	1	1	yes	characteristic
170	C₂	#22561982	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523975
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂³×C₆₇	#106
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#12000
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22558275	C₂	semidirect
26. M₃₂×C₆₇	C₈	semidirect
51. C₁₆×C₆₇	C₂×C₈	semidirect
90. C₆₇	#12000	semidirect
107. M₃₂	C₆₇⋊C₈	semidirect
137. C₁₆	C₂×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22558957	2
C₂×#16863147	2
#22558275	2
C₄×M₃₂×C₆₇	2
#12000	67