Subgroups — #30774355

Order 17152.

386 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
3	#22558287	C₂	1	4	yes	maximal
4	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
8	#16863172	—	2	2	no	—
9	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
10	C₈×Q₈×C₆₇	—	2	2	no	—
11	C₈×Q₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₆₇⋊C₁₆	—	2	2	no	—
17	C₂×C₆₇⋊C₁₆	—	2	2	no	—
18	C₂×C₆₇⋊C₁₆	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	4	no	—
20	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
21	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₄×Q₈×C₆₇	—	2	2	no	—
26	C₄×Q₈×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	—	2	4	no	—
32	C₄⋊C₈×C₆₇	—	2	2	no	—
33	C₄×C₈×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	D₄	1	2	yes	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₆₇⋊C₁₆	—	2	4	no	—
38	C₆₇⋊C₁₆	—	4	2	no	—
39	C₂×Q₈×C₆₇	—	2	2	no	—
40	C₂×Q₈×C₆₇	—	2	4	no	—
41	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	—	2	4	no	—
49	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₆₇	#41	1	2	yes	—
54	C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₄²×C₆₇	—	2	2	no	—
57	C₄²×C₆₇	—	2	2	no	—
58	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₆₇	M₁₆	1	2	yes	—
60	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
61	Q₈×C₆₇	—	2	8	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
71	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#131	1	2	yes	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	4	2	no	—
77	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₆₇	#124	1	1	yes	characteristic
79	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
82	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
83	C₂²×C₆₇	#504	1	2	yes	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	2	4	no	—
86	C₄×C₆₇	—	2	2	no	—
87	C₄×C₆₇	—	4	2	no	—
88	C₄×C₆₇	#391	1	1	yes	characteristic
89	C₄×C₆₇	#408	1	1	yes	characteristic
90	C₄×C₆₇	#446	1	2	yes	—
91	#13264	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	#1984	1	1	yes	characteristic
93	C₂×C₆₇	#2112	1	1	yes	characteristic
94	C₂×C₆₇	#2113	1	1	yes	characteristic
95	C₂×C₆₇	#2183	1	4	yes	—
96	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	—	67	1	no	—
98	C₂×#346	—	67	1	no	—
99	#1374	—	67	4	no	—
100	C₆₇	#13264	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
102	C₂²×C₁₆	—	67	2	no	—
103	C₂²×C₁₆	—	67	2	no	—
104	C₈×Q₈	—	2	2	no	—
105	C₈×Q₈	—	2	2	no	—
106	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
107	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	C₄⋊C₁₆	—	134	2	no	—
111	#346	—	134	2	no	—
112	C₂²×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
114	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
116	C₄×Q₈	—	2	2	no	—
117	C₄×Q₈	—	2	2	no	—
118	C₂×C₄⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₆₇⋊C₈	1	2	yes	—
120	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
121	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
122	C₂×C₁₆	—	67	4	no	—
123	C₂×C₁₆	—	134	2	no	—
124	C₂×C₁₆	—	134	4	no	—
125	C₂×C₁₆	—	134	2	no	—
126	C₂×C₁₆	—	134	2	no	—
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
132	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
136	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
145	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₆₇	1	2	yes	—
147	C₂×C₈	#11761629	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
153	C₄²	#11761612	1	2	yes	—
154	C₁₆	—	268	2	no	—
155	C₁₆	—	134	4	no	—
156	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
157	Q₈	—	2	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
165	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
166	C₂×C₄	#14523983	1	4	yes	—
167	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
169	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
170	C₂×C₄	#14524050	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	4	2	no	—
173	C₈	#14524019	1	2	yes	—
174	C₈	#14524032	1	1	yes	characteristic
175	C₈	#14524041	1	1	yes	characteristic
176	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
177	C₂²	Q₈×C₆₇⋊C₈	1	2	yes	—
178	C₂²	C₂×#14523983	1	2	yes	—
179	C₂²	#16863813	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	2	no	—
184	C₄	#16863303	1	1	yes	characteristic
185	C₄	#16863363	1	1	yes	characteristic
186	C₄	#16863527	1	2	yes	—
187	C₂	#22561146	1	1	yes	characteristic
188	C₂	#22561820	1	1	yes	characteristic
189	C₂	#22561823	1	1	yes	characteristic
190	C₂	#22562118	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂²×C₈
Fitting	C₂×C₈×Q₈×C₆₇	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	#13264
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
20. C₂×C₆₇⋊C₁₆	Q₈	semidirect
100. C₆₇	#13264	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863172	2
#22558287	2
C₂×C₈×Q₈×C₆₇	2
#13264	67