Subgroups — #30747140

Order 17152.

334 conjugacy classes of subgroups.

The subgroup lattice has 188 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×#16863117	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	4	no	—
12	#16863117	C₄	1	2	yes	—
13	#16863117	C₄	1	2	yes	—
14	C₂×#108×C₆₇	C₄	1	2	yes	—
15	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#341×C₆₇	C₄	1	2	yes	—
18	C₂²×C₆₇⋊C₈	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
20	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄×C₆₇⋊C₈	—	2	2	no	—
23	C₄×C₆₇⋊C₈	—	2	1	no	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
27	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	#108×C₆₇	—	2	4	no	—
32	#97×C₆₇	—	2	2	no	—
33	C₄×C₈×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₆₇⋊C₈	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
46	M₁₆×C₆₇	—	2	4	no	—
47	M₁₆×C₆₇	—	2	4	no	—
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	2	4	no	—
50	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
55	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₇	#41	1	1	yes	characteristic
59	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₆₇	#131	1	2	yes	—
70	C₈×C₆₇	—	2	1	no	—
71	C₈×C₆₇	—	2	1	no	—
72	C₈×C₆₇	—	4	4	no	—
73	C₈×C₆₇	#97	1	2	yes	—
74	C₈×C₆₇	#97	1	2	yes	—
75	C₆₇⋊C₈	—	2	4	no	—
76	C₆₇⋊C₈	—	4	2	no	—
77	C₆₇⋊C₈	—	4	4	no	—
78	C₂²×C₆₇	—	2	1	no	—
79	C₂²×C₆₇	—	2	1	no	—
80	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₆₇	#446	1	2	yes	—
82	C₄×C₆₇	—	2	1	no	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
85	C₄×C₆₇	#405	1	1	yes	characteristic
86	C₄×C₆₇	#430	1	2	yes	—
87	C₄×C₆₇	#436	1	2	yes	—
88	C₄×C₆₇	#444	1	2	yes	—
89	#8295	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	#1770	1	1	yes	characteristic
92	C₂×C₆₇	#1773	1	1	yes	characteristic
93	C₂×C₆₇	#1778	1	1	yes	characteristic
94	C₄×#108	D₆₇	1	1	yes	characteristic
95	#1564	—	67	1	no	—
96	#1451	—	67	2	no	—
97	C₂×#322	—	67	1	no	—
98	#1278	—	67	2	no	—
99	C₆₇	#8295	1	1	yes	characteristic, complemented, Sylow
100	#431	—	67	2	no	—
101	C₂×#108	Dic₆₇	1	2	yes	—
102	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
103	C₂×C₄×C₈	—	67	1	no	—
104	C₂×C₄×C₈	—	67	1	no	—
105	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
106	#341	Dic₆₇	1	2	yes	—
107	#335	—	67	2	no	—
108	#334	—	67	2	no	—
109	#322	—	67	2	no	—
110	#322	—	67	2	no	—
111	#322	—	134	4	no	—
112	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
113	C₂²×C₈	—	67	2	no	—
114	C₂²×C₈	—	134	1	no	—
115	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
117	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	#108	—	2	4	no	—
119	C₄⋊C₈	—	67	4	no	—
120	C₂²⋊C₈	—	134	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	67	2	no	—
124	C₄×C₈	—	67	2	no	—
125	C₄×C₈	—	67	2	no	—
126	C₄×C₈	—	134	2	no	—
127	C₄×C₈	—	134	1	no	—
128	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
129	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
130	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
131	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
132	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
134	C₂²×C₄	#11761629	1	1	yes	characteristic
135	M₁₆	—	2	4	no	—
136	M₁₆	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	268	2	no	—
140	C₂×C₈	—	268	2	no	—
141	C₂×C₈	—	67	4	no	—
142	C₂×C₈	—	134	2	no	—
143	C₂×C₈	—	134	4	no	—
144	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
145	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
146	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
147	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
148	C₂×C₈	#11761612	1	4	yes	—
149	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
150	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
151	C₄²	D₄×D₆₇	1	1	yes	characteristic
152	C₄²	#11761627	1	1	yes	characteristic
153	C₂³	Q₈×Dic₆₇	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
158	C₂×C₄	#14524001	1	1	yes	characteristic
159	C₂×C₄	#14524005	1	1	yes	characteristic
160	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
161	C₂×C₄	#14524008	1	1	yes	characteristic
162	C₂×C₄	C₂×#11761612	1	2	yes	—
163	C₂×C₄	#14524050	1	2	yes	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	4	no	—
167	C₈	—	268	2	no	—
168	C₈	—	268	4	no	—
169	C₈	—	134	4	no	—
170	C₈	#14523954	1	2	yes	—
171	C₈	#14523954	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
175	C₂²	#16863527	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	C₂×#14523954	1	1	yes	characteristic
179	C₄	#16863352	1	1	yes	characteristic
180	C₄	#16863443	1	2	yes	—
181	C₄	#16863474	1	2	yes	—
182	C₄	#16863517	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#22560078	1	1	yes	characteristic
185	C₂	#22560096	1	1	yes	characteristic
186	C₂	#22560117	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	#8295
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
99. C₆₇	#8295	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×#16863117	2
#22558047	2
C₄×#108×C₆₇	2
#8295	67