Subgroups — #30746259

Order 17152.

446 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	#22559954	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇⋊#1273	C₂	1	2	yes	maximal
4	C₆₇⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
10	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
11	#16863150	C₄	1	4	yes	—
12	#16863150	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₆₇⋊C₈	—	2	1	no	—
22	C₄×C₆₇⋊C₈	—	2	2	no	—
23	C₄×C₆₇⋊C₈	—	2	1	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₈×Dic₆₇	—	2	4	no	—
26	C₈×Dic₆₇	C₈	1	8	yes	complemented
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	—	2	2	no	—
33	C₂²×Dic₆₇	C₄²	1	1	yes	characteristic
34	C₂²×Dic₆₇	C₄²	1	1	yes	characteristic
35	C₂²×Dic₆₇	C₄²	1	2	yes	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	2	no	—
38	C₂×C₆₇⋊C₈	—	2	4	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	2	4	no	—
41	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
42	C₄×Dic₆₇	—	2	4	no	—
43	C₄×Dic₆₇	—	2	4	no	—
44	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
45	C₄×Dic₆₇	M₁₆	1	4	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	—	2	4	no	—
52	C₂×C₈×C₆₇	—	2	4	no	—
53	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
54	C₄²×C₆₇	#41	1	2	yes	—
55	C₄²×C₆₇	#41	1	1	yes	characteristic
56	C₄²×C₆₇	#41	1	1	yes	characteristic
57	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#117	1	2	yes	—
60	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#117	1	2	yes	—
62	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₆₇	#131	1	4	yes	—
65	C₂×C₄×C₆₇	#131	1	4	yes	—
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	—	2	2	no	—
69	C₂×Dic₆₇	—	2	2	no	—
70	C₂×Dic₆₇	C₄×C₈	1	2	yes	—
71	C₂×Dic₆₇	C₄×C₈	1	2	yes	—
72	C₂×Dic₆₇	#97	1	4	yes	—
73	C₈×C₆₇	—	2	4	no	—
74	C₈×C₆₇	—	4	4	no	—
75	C₆₇⋊C₈	—	2	4	no	—
76	C₆₇⋊C₈	—	4	4	no	—
77	C₂²×C₆₇	#376	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
79	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
80	C₂²×C₆₇	#405	1	2	yes	—
81	C₂²×C₆₇	#405	1	2	yes	—
82	C₄×C₆₇	#431	1	2	yes	—
83	C₄×C₆₇	#431	1	2	yes	—
84	C₄×C₆₇	#433	1	4	yes	—
85	C₄×C₆₇	#433	1	4	yes	—
86	Dic₆₇	—	2	2	no	—
87	Dic₆₇	—	2	2	no	—
88	Dic₆₇	—	2	4	no	—
89	#8133	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	#1450	1	1	yes	characteristic
91	C₂×C₆₇	#1450	1	1	yes	characteristic
92	C₂×C₆₇	#1452	1	2	yes	—
93	C₂×C₆₇	#1744	1	1	yes	characteristic
94	C₂×C₆₇	#1746	1	2	yes	—
95	#1744	—	67	2	no	—
96	C₄²×C₈	—	67	1	no	—
97	#1273	—	67	2	no	—
98	#1273	—	67	1	no	—
99	#1273	D₆₇	1	1	yes	characteristic
100	C₆₇	#8133	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×C₈	—	67	2	no	—
102	C₂×C₄×C₈	—	67	2	no	—
103	C₂×C₄×C₈	—	67	1	no	—
104	C₂×C₄×C₈	—	67	2	no	—
105	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
107	C₄³	—	67	1	no	—
108	#336	—	67	4	no	—
109	#336	—	67	4	no	—
110	C₂²×C₈	—	67	2	no	—
111	C₂²×C₈	—	67	4	no	—
112	C₂²×C₈	C₄×D₆₇	1	4	yes	—
113	C₂²×C₈	C₄×D₆₇	1	2	yes	—
114	C₂×C₄²	—	67	1	no	—
115	C₂×C₄²	—	67	1	no	—
116	C₂×C₄²	—	67	2	no	—
117	C₂×C₄²	—	67	2	no	—
118	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	67	8	no	—
123	C₄×C₈	—	134	4	no	—
124	C₄×C₈	—	134	2	no	—
125	C₄×C₈	—	134	1	no	—
126	C₄×C₈	—	134	2	no	—
127	C₄×C₈	—	134	1	no	—
128	C₂²×C₄	—	67	1	no	—
129	C₂²×C₄	—	67	1	no	—
130	C₂²×C₄	—	67	2	no	—
131	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	67	4	no	—
139	C₂×C₈	—	134	4	no	—
140	C₂×C₈	—	134	4	no	—
141	C₂×C₈	—	134	4	no	—
142	C₂×C₈	—	134	2	no	—
143	C₂×C₈	—	134	2	no	—
144	C₂×C₈	C₈×D₆₇	1	4	yes	—
145	C₄²	—	67	4	no	—
146	C₄²	—	67	4	no	—
147	C₄²	—	134	4	no	—
148	C₄²	—	134	4	no	—
149	C₄²	#11761627	1	2	yes	—
150	C₄²	#11761627	1	1	yes	characteristic
151	C₄²	#11761629	1	1	yes	characteristic
152	C₂³	C₄²×D₆₇	1	1	yes	characteristic
153	C₂×C₄	—	67	2	no	—
154	C₂×C₄	—	67	2	no	—
155	C₂×C₄	—	67	4	no	—
156	C₂×C₄	—	134	2	no	—
157	C₂×C₄	—	134	2	no	—
158	C₂×C₄	—	134	2	no	—
159	C₂×C₄	—	134	2	no	—
160	C₂×C₄	#14523999	1	1	yes	characteristic
161	C₂×C₄	#14523999	1	2	yes	—
162	C₂×C₄	#14524001	1	1	yes	characteristic
163	C₂×C₄	#14524001	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
165	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
166	C₂×C₄	#14524049	1	4	yes	—
167	C₂×C₄	#14524051	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	4	4	no	—
170	C₈	—	268	4	no	—
171	C₈	—	134	4	no	—
172	C₂²	#16863241	1	1	yes	characteristic
173	C₂²	C₄×C₈×D₆₇	1	1	yes	characteristic
174	C₂²	#97×D₆₇	1	1	yes	characteristic
175	C₂²	#16863350	1	2	yes	—
176	C₂²	#16863353	1	2	yes	—
177	C₄	—	134	2	no	—
178	C₄	—	134	2	no	—
179	C₄	—	134	4	no	—
180	C₄	#16863446	1	2	yes	—
181	C₄	#16863448	1	2	yes	—
182	C₄	#16863454	1	4	yes	—
183	C₄	#16863456	1	4	yes	—
184	C₂	#22558491	1	1	yes	characteristic
185	C₂	#22558494	1	1	yes	characteristic
186	C₂	#22558500	1	2	yes	—
187	C₂	#22559953	1	1	yes	characteristic
188	C₂	#22559959	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₆₇×#1273	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1273	D₆₇
hypocenter	C₆₇	#8133
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
26. C₈×Dic₆₇	C₈	semidirect
100. C₆₇	#8133	semidirect

Maximal subgroups

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

subgroup	index
#22559954	2
C₄×C₈×Dic₆₇	2
C₆₇⋊#1273	2
C₆₇×#1273	2
#8133	67