Subgroups — #30747130

Order 17152.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 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	#22558496	C₂	1	2	yes	maximal
3	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22558047	C₂	1	2	yes	maximal
5	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863149	C₄	1	2	yes	—
10	#16863147	C₄	1	2	yes	—
11	#16863118	—	2	2	no	—
12	#16863118	—	2	2	no	—
13	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
14	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
15	C₂×#108×C₆₇	C₄	1	2	yes	—
16	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#341×C₆₇	C₄	1	2	yes	—
19	C₂²×C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
21	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄×C₆₇⋊C₈	—	2	2	no	—
24	C₄×C₆₇⋊C₈	—	2	1	no	—
25	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	#108×C₆₇	—	2	4	no	—
34	#97×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	—	2	2	no	—
36	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₆₇⋊C₈	—	2	4	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	4	no	—
43	C₂×C₆₇⋊C₈	—	2	4	no	—
44	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
49	M₁₆×C₆₇	—	2	4	no	—
50	M₁₆×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	4	no	—
53	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
58	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₆₇	#41	1	1	yes	characteristic
62	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₆₇	#131	1	2	yes	—
73	C₈×C₆₇	—	2	1	no	—
74	C₈×C₆₇	—	2	1	no	—
75	C₈×C₆₇	—	4	4	no	—
76	C₈×C₆₇	#97	1	2	yes	—
77	C₈×C₆₇	#97	1	2	yes	—
78	C₆₇⋊C₈	—	2	4	no	—
79	C₆₇⋊C₈	—	2	2	no	—
80	C₆₇⋊C₈	—	4	4	no	—
81	C₆₇⋊C₈	#108	1	4	yes	complemented
82	C₂²×C₆₇	—	2	1	no	—
83	C₂²×C₆₇	—	2	1	no	—
84	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₆₇	#446	1	2	yes	—
86	C₄×C₆₇	—	2	1	no	—
87	C₄×C₆₇	—	2	1	no	—
88	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
89	C₄×C₆₇	#405	1	1	yes	characteristic
90	C₄×C₆₇	C₂×#108	1	2	yes	—
91	C₄×C₆₇	#436	1	2	yes	—
92	C₄×C₆₇	#443	1	2	yes	—
93	#8293	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	—	2	2	no	—
95	C₂×C₆₇	#1770	1	1	yes	characteristic
96	C₂×C₆₇	#1772	1	1	yes	characteristic
97	C₂×C₆₇	C₄×#108	1	1	yes	characteristic
98	C₄×#108	D₆₇	1	1	yes	characteristic
99	#1564	—	67	1	no	—
100	#1451	—	67	2	no	—
101	C₂×C₈²	—	67	1	no	—
102	#1278	—	67	2	no	—
103	C₆₇	#8293	1	1	yes	characteristic, complemented, Sylow
104	#431	—	67	2	no	—
105	C₂×#108	Dic₆₇	1	2	yes	—
106	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
107	C₂×C₄×C₈	—	67	1	no	—
108	C₂×C₄×C₈	—	67	1	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	#341	Dic₆₇	1	2	yes	—
111	#335	—	67	2	no	—
112	#334	—	67	2	no	—
113	#322	—	134	2	no	—
114	#322	—	134	2	no	—
115	C₈²	—	67	2	no	—
116	C₈²	—	67	2	no	—
117	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
118	C₂²×C₈	—	67	2	no	—
119	C₂²×C₈	—	134	1	no	—
120	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
123	#108	—	2	4	no	—
124	C₄⋊C₈	—	67	4	no	—
125	C₂²⋊C₈	—	134	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	67	2	no	—
129	C₄×C₈	—	67	2	no	—
130	C₄×C₈	—	67	1	no	—
131	C₄×C₈	—	67	1	no	—
132	C₄×C₈	—	134	2	no	—
133	C₄×C₈	—	134	1	no	—
134	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
135	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
136	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
137	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
139	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
140	C₂²×C₄	#11761629	1	1	yes	characteristic
141	M₁₆	—	2	4	no	—
142	M₁₆	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	67	2	no	—
146	C₂×C₈	—	67	2	no	—
147	C₂×C₈	—	134	4	no	—
148	C₂×C₈	—	134	4	no	—
149	C₂×C₈	—	134	4	no	—
150	C₂×C₈	—	134	2	no	—
151	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
155	C₂×C₈	#11761612	1	4	yes	—
156	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
158	C₄²	D₄×D₆₇	1	1	yes	characteristic
159	C₄²	#11761627	1	1	yes	characteristic
160	C₂³	Q₈×Dic₆₇	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
165	C₂×C₄	#14524001	1	1	yes	characteristic
166	C₂×C₄	#14524005	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
168	C₂×C₄	#14524008	1	1	yes	characteristic
169	C₂×C₄	C₂×#11761612	1	2	yes	—
170	C₂×C₄	#14524050	1	2	yes	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	4	no	—
174	C₈	—	268	4	no	—
175	C₈	—	67	4	no	—
176	C₈	—	134	4	no	—
177	C₈	—	134	2	no	—
178	C₈	#14523954	1	2	yes	—
179	C₈	#14523954	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
183	C₂²	#16863527	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×#14523954	1	1	yes	characteristic
187	C₄	#16863352	1	1	yes	characteristic
188	C₄	#108×D₆₇	1	2	yes	—
189	C₄	#16863474	1	2	yes	—
190	C₄	#16863512	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#22560078	1	1	yes	characteristic
193	C₂	#22560091	1	1	yes	characteristic
194	C₂	#108×Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₄×C₄⋊C₄
prime radical	C₂³×C₆₇	C₄×Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#108	D₆₇
hypocenter	C₆₇	#8293
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
81. C₆₇⋊C₈	#108	semidirect
103. C₆₇	#8293	semidirect

Maximal subgroups

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

subgroup	index
#22558957	2
#22558496	2
C₂×C₈×C₆₇⋊C₈	2
#22558047	2
C₄×#108×C₆₇	2
#8293	67