Subgroups — #30767459

Order 17152.

236 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562008	C₂	1	1	yes	maximal, characteristic
2	#22561965	C₂	1	1	yes	maximal, characteristic
3	#22558279	C₂	1	2	yes	maximal, complemented
4	#22558273	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	—	2	1	no	—
7	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
10	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
12	C₄×C₆₇⋊C₁₆	—	2	1	no	—
13	#431×C₆₇	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#322×C₆₇	C₄	1	2	yes	—
17	C₈²×C₆₇	C₂²	1	2	yes	—
18	#14523983	—	4	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	C₂×M₁₆×C₆₇	—	2	1	no	—
23	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₆₇	C₈	1	2	yes	—
27	C₂²⋊C₈×C₆₇	C₈	1	2	yes	—
28	#97×C₆₇	—	2	1	no	—
29	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₆₇⋊C₁₆	—	4	1	no	—
36	C₆₇⋊C₁₆	—	4	2	no	—
37	C₆₇⋊C₁₆	—	4	1	no	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
41	M₁₆×C₆₇	—	2	4	no	—
42	C₂×C₈×C₆₇	—	2	2	no	—
43	C₂×C₈×C₆₇	—	2	1	no	—
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	4	1	no	—
46	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
54	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	1	no	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₆₇	—	2	4	no	—
67	C₈×C₆₇	—	2	1	no	—
68	C₈×C₆₇	—	2	1	no	—
69	C₈×C₆₇	—	2	1	no	—
70	C₈×C₆₇	—	2	4	no	—
71	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₈×C₆₇	#137	1	1	yes	characteristic
73	C₂²×C₆₇	—	2	1	no	—
74	C₂²×C₆₇	—	2	1	no	—
75	C₂²×C₆₇	—	2	1	no	—
76	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₄×C₆₇	—	2	1	no	—
78	C₄×C₆₇	—	2	1	no	—
79	C₄×C₆₇	—	2	1	no	—
80	C₄×C₆₇	#364	1	2	yes	—
81	C₄×C₆₇	#428	1	1	yes	characteristic
82	C₄×C₆₇	#430	1	1	yes	characteristic
83	C₄×C₆₇	#504	1	2	yes	—
84	#12018	—	67	1	no	maximal, Sylow
85	C₂×C₆₇	—	4	1	no	—
86	C₂×C₆₇	#1565	1	1	yes	characteristic
87	C₂×C₆₇	#2151	1	1	yes	characteristic
88	C₂×C₆₇	C₂×#364	1	1	yes	characteristic
89	#2162	—	67	1	no	—
90	#2150	—	67	1	no	—
91	C₈×M₁₆	D₆₇	1	1	yes	characteristic
92	#1370	—	67	2	no	—
93	#1367	—	67	2	no	—
94	C₆₇	#12018	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₃₂	—	134	1	no	—
96	#431	Dic₆₇	1	1	yes	characteristic
97	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
99	C₄⋊C₁₆	—	67	1	no	—
100	C₄⋊C₁₆	—	67	2	no	—
101	C₄⋊C₁₆	—	67	1	no	—
102	C₄⋊C₁₆	—	134	1	no	—
103	C₂²⋊C₁₆	—	134	1	no	—
104	C₄×C₁₆	—	134	1	no	—
105	#322	Dic₆₇	1	2	yes	—
106	C₈²	C₂×D₆₇	1	2	yes	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
109	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
111	M₃₂	—	268	2	no	—
112	C₂×C₁₆	—	134	1	no	—
113	C₂×C₁₆	—	134	1	no	—
114	C₂×C₁₆	—	134	2	no	—
115	C₄⋊C₈	C₆₇⋊C₈	1	2	yes	—
116	C₂²⋊C₈	C₆₇⋊C₈	1	2	yes	—
117	#97	—	2	1	no	—
118	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
119	C₄×C₈	D₂₆₈	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
121	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
122	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
123	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
126	C₂²×C₄	#11761612	1	1	yes	characteristic
127	M₁₆	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
134	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
135	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
137	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
138	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
139	C₄²	#11761612	1	1	yes	characteristic
140	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
141	C₁₆	—	268	2	no	—
142	C₁₆	—	268	1	no	—
143	C₁₆	—	268	1	no	—
144	C₂³	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
150	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
151	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
154	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	4	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	#14524071	1	1	yes	characteristic
161	C₈	#14524076	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	#16863217	1	2	yes	—
170	C₄	#16863438	1	1	yes	characteristic
171	C₄	#16863445	1	1	yes	characteristic
172	C₄	#16863813	1	2	yes	—
173	C₂	—	4	1	no	—
174	C₂	#22558961	1	1	yes	characteristic
175	C₂	#22561968	1	1	yes	characteristic
176	C₂	C₂×#16863217	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#12018
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22558279	C₂	semidirect
4. #22558273	C₂	semidirect
94. C₆₇	#12018	semidirect

Maximal subgroups

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

subgroup	index
#22562008	2
#22561965	2
#22558279	2
#22558273	2
C₈×M₁₆×C₆₇	2
#12018	67