Subgroups — #30767401

Order 17152.

270 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561965	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22558278	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	—	2	1	no	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
10	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
11	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂×M₃₂×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₆₇	—	2	1	no	—
15	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₆₇⋊C₈	—	2	1	no	—
17	C₂²×C₆₇⋊C₈	C₈	1	2	yes	—
18	#14523983	—	4	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₄×C₆₇⋊C₈	—	2	1	no	—
21	C₄×C₆₇⋊C₈	C₈	1	1	yes	characteristic
22	C₄×C₆₇⋊C₈	C₈	1	1	yes	characteristic
23	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₆₇	—	4	2	no	—
29	C₂×C₁₆×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₆₇⋊C₈	—	2	4	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	4	no	—
37	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
38	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
39	C₆₇⋊C₁₆	—	4	2	no	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	D₈	1	2	yes	—
47	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
48	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
53	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₁₆×C₆₇	—	4	2	no	—
55	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#107	1	2	yes	—
62	C₂×C₄×C₆₇	#108	1	2	yes	—
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	1	no	—
67	C₈×C₆₇	—	2	1	no	—
68	C₈×C₆₇	#107	1	2	yes	—
69	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
70	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₆₇⋊C₈	—	2	4	no	—
72	C₆₇⋊C₈	—	2	2	no	—
73	C₆₇⋊C₈	M₃₂	1	4	yes	complemented
74	C₂²×C₆₇	—	2	1	no	—
75	C₂²×C₆₇	—	2	1	no	—
76	C₂²×C₆₇	#335	1	2	yes	—
77	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₄×C₆₇	—	2	1	no	—
79	C₄×C₆₇	—	2	1	no	—
80	C₄×C₆₇	#335	1	2	yes	—
81	C₄×C₆₇	#364	1	2	yes	—
82	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
83	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
84	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
85	#12004	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	—	2	2	no	—
87	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
88	C₂×C₆₇	#2150	1	1	yes	characteristic
89	C₂×C₆₇	C₂×#364	1	1	yes	characteristic
90	#2150	—	67	1	no	—
91	#2150	D₆₇	1	1	yes	characteristic
92	C₂×C₈²	—	67	1	no	—
93	#1370	—	67	4	no	—
94	C₆₇	#12004	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₃₂	—	2	1	no	—
96	C₂×M₃₂	—	134	1	no	—
97	C₂×C₄×C₈	—	67	1	no	—
98	C₂×C₄×C₈	—	67	1	no	—
99	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
100	C₄⋊C₁₆	—	2	1	no	—
101	C₄⋊C₁₆	—	67	2	no	—
102	C₄⋊C₁₆	—	134	1	no	—
103	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
104	C₈²	—	67	2	no	—
105	C₈²	—	67	2	no	—
106	C₂²×C₈	—	67	2	no	—
107	C₂²×C₈	—	134	1	no	—
108	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
111	M₃₂	—	4	2	no	—
112	M₃₂	—	268	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	134	2	no	—
115	C₄×C₈	—	67	1	no	—
116	C₄×C₈	—	67	1	no	—
117	C₄×C₈	—	67	2	no	—
118	C₄×C₈	—	67	2	no	—
119	C₄×C₈	—	134	1	no	—
120	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
121	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
125	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
126	C₂²×C₄	#11761611	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	67	2	no	—
129	C₂×C₈	—	67	2	no	—
130	C₂×C₈	—	134	4	no	—
131	C₂×C₈	—	134	4	no	—
132	C₂×C₈	—	134	2	no	—
133	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
134	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
135	C₂×C₈	#11761613	1	2	yes	—
136	C₂×C₈	#11761618	1	2	yes	—
137	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
138	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
139	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
140	C₄²	C₈×D₆₇	1	1	yes	characteristic
141	C₄²	#11761611	1	1	yes	characteristic
142	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
143	C₁₆	—	4	2	no	—
144	C₁₆	—	268	2	no	—
145	C₂³	Dic₆₇⋊C₈	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
149	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
150	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
151	C₂×C₄	#14523976	1	2	yes	—
152	C₂×C₄	#14523978	1	2	yes	—
153	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
155	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	67	4	no	—
159	C₈	—	134	4	no	—
160	C₈	—	134	2	no	—
161	C₈	#14523976	1	2	yes	—
162	C₈	C₂×#11761613	1	1	yes	characteristic
163	C₈	C₂×#11761618	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#16863148	1	2	yes	—
167	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#16863148	1	2	yes	—
171	C₄	#16863216	1	2	yes	—
172	C₄	C₂×#14523976	1	1	yes	characteristic
173	C₄	C₂×#14523978	1	1	yes	characteristic
174	C₄	M₃₂×D₆₇	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	C₂×#16863148	1	1	yes	characteristic
177	C₂	#22561964	1	1	yes	characteristic
178	C₂	C₂×#16863216	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#12004
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #22558278	C₂	semidirect
73. C₆₇⋊C₈	M₃₂	semidirect
94. C₆₇	#12004	semidirect

Maximal subgroups

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

subgroup	index
#22561965	2
C₂×C₈×C₆₇⋊C₈	2
#22558278	2
C₆₇×#2150	2
#12004	67