Subgroups — #30747138

Order 17152.

334 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523979	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558498	C₂	1	2	yes	maximal
4	#22558046	C₂	1	2	yes	maximal
5	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523979	C₄	1	2	yes	—
8	C₄×#11761612	C₂²	1	2	yes	—
9	#16863161	C₄	1	2	yes	—
10	#16863149	C₄	1	2	yes	—
11	#16863147	C₄	1	2	yes	—
12	#16863118	—	2	2	no	—
13	#16863117	—	2	2	no	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#322×C₆₇	C₄	1	2	yes	—
18	#322×C₆₇	C₄	1	2	yes	—
19	C₂×#11761612	C₂×C₄	1	4	yes	—
20	#14523979	—	2	4	no	—
21	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
23	#14523954	—	2	2	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₆₇	—	2	1	no	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₆₇	—	2	1	no	—
32	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
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	2	yes	—
38	C₄×C₈×C₆₇	D₄	1	2	yes	—
39	#11761612	—	2	4	no	—
40	#11761612	—	2	4	no	—
41	C₂×C₆₇⋊C₈	—	2	4	no	—
42	C₂×C₆₇⋊C₈	—	2	4	no	—
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	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	—	4	2	no	—
49	C₂×C₈×C₆₇	—	4	2	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	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
56	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₇	#41	1	1	yes	characteristic
60	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₆₇	#131	1	2	yes	—
71	C₈×C₆₇	—	2	1	no	—
72	C₈×C₆₇	—	2	1	no	—
73	C₈×C₆₇	—	2	4	no	—
74	C₈×C₆₇	—	4	2	no	—
75	C₈×C₆₇	#97	1	2	yes	—
76	C₈×C₆₇	#97	1	2	yes	—
77	C₆₇⋊C₈	—	4	4	no	—
78	C₆₇⋊C₈	—	4	4	no	—
79	C₂²×C₆₇	—	2	1	no	—
80	C₂²×C₆₇	—	2	1	no	—
81	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₆₇	#446	1	2	yes	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	2	1	no	—
85	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
86	C₄×C₆₇	#405	1	1	yes	characteristic
87	C₄×C₆₇	#430	1	2	yes	—
88	C₄×C₆₇	#436	1	2	yes	—
89	C₄×C₆₇	#444	1	2	yes	—
90	#8295	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	2	2	no	—
92	C₂×C₆₇	#1770	1	1	yes	characteristic
93	C₂×C₆₇	#1773	1	1	yes	characteristic
94	C₂×C₆₇	#1778	1	1	yes	characteristic
95	C₄×#108	—	67	1	no	—
96	#1564	—	67	1	no	—
97	#1451	—	67	2	no	—
98	C₂×#322	D₆₇	1	1	yes	characteristic
99	#1278	—	67	2	no	—
100	C₆₇	#8295	1	1	yes	characteristic, complemented, Sylow
101	#431	—	67	2	no	—
102	C₂×#108	—	67	2	no	—
103	C₄×M₁₆	—	67	2	no	—
104	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
107	#341	—	67	2	no	—
108	#335	—	67	2	no	—
109	#334	—	67	2	no	—
110	#322	—	134	2	no	—
111	#322	—	134	2	no	—
112	#322	Dic₆₇	1	2	yes	—
113	#322	Dic₆₇	1	2	yes	—
114	C₂×M₁₆	—	67	4	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	Dic₁₃₄	1	2	yes	—
117	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
120	#108	—	134	4	no	—
121	C₄⋊C₈	—	67	4	no	—
122	C₂²⋊C₈	—	134	2	no	—
123	#97	—	134	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	134	2	no	—
126	C₄×C₈	—	134	2	no	—
127	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
128	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
129	C₄×C₈	D₂₆₈	1	2	yes	—
130	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
131	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
133	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
136	C₂²×C₄	#11761628	1	1	yes	characteristic
137	M₁₆	—	134	4	no	—
138	M₁₆	—	134	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	134	4	no	—
144	C₂×C₈	—	134	4	no	—
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	C₄×Dic₆₇	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
150	C₂×C₈	#11761611	1	4	yes	—
151	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
152	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
153	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
154	C₄²	#11761628	1	1	yes	characteristic
155	C₂³	C₄×Dic₁₃₄	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
160	C₂×C₄	#14524002	1	1	yes	characteristic
161	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
162	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
163	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
164	C₂×C₄	C₂×#11761611	1	2	yes	—
165	C₂×C₄	#14524049	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	4	no	—
169	C₈	—	4	2	no	—
170	C₈	—	268	4	no	—
171	C₈	—	268	4	no	—
172	C₈	#14523953	1	2	yes	—
173	C₈	#14523953	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
177	C₂²	#16863526	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×#14523953	1	1	yes	characteristic
181	C₄	#16863351	1	1	yes	characteristic
182	C₄	#16863445	1	2	yes	—
183	C₄	#16863475	1	2	yes	—
184	C₄	#16863519	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	#22560076	1	1	yes	characteristic
187	C₂	#22560093	1	1	yes	characteristic
188	C₂	#22560115	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	C₄×C₄⋊C₄
prime radical	C₂³×C₆₇	C₄×Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#8295
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
100. C₆₇	#8295	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523979	2
#22558957	2
#22558498	2
#22558046	2
C₂×#322×C₆₇	2
#8295	67