Subgroups — #30767444

Order 17152.

246 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	#22562008	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22558279	C₂	1	2	yes	maximal, complemented
4	#22558273	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
10	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
12	C₄×C₆₇⋊C₁₆	—	2	1	no	—
13	#431×C₆₇	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#322×C₆₇	C₄	1	2	yes	—
17	C₈²×C₆₇	C₂²	1	2	yes	—
18	C₂×C₆₇⋊C₁₆	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	1	no	—
20	C₂×C₆₇⋊C₁₆	—	2	1	no	—
21	C₂×C₆₇⋊C₁₆	—	4	2	no	—
22	C₂×M₁₆×C₆₇	C₈	1	1	yes	characteristic
23	C₂×M₁₆×C₆₇	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	1	no	—
28	C₂²⋊C₈×C₆₇	—	2	1	no	—
29	#97×C₆₇	C₈	1	2	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₆₇⋊C₁₆	—	4	1	no	—
37	C₆₇⋊C₁₆	—	4	2	no	—
38	C₆₇⋊C₁₆	—	4	1	no	—
39	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
42	M₁₆×C₆₇	—	2	2	no	—
43	M₁₆×C₆₇	C₁₆	1	4	yes	complemented
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	—	4	1	no	—
48	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
57	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	—	2	1	no	—
61	C₂×C₄×C₆₇	—	4	1	no	—
62	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₆₇	—	2	1	no	—
71	C₈×C₆₇	—	2	1	no	—
72	C₈×C₆₇	—	2	1	no	—
73	C₈×C₆₇	—	2	2	no	—
74	C₈×C₆₇	—	4	2	no	—
75	C₈×C₆₇	C₂×C₁₆	1	4	yes	complemented
76	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₈×C₆₇	#137	1	1	yes	characteristic
78	C₂²×C₆₇	—	2	1	no	—
79	C₂²×C₆₇	—	2	1	no	—
80	C₂²×C₆₇	C₄⋊C₁₆	1	2	yes	—
81	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₄×C₆₇	—	2	1	no	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	2	1	no	—
85	C₄×C₆₇	C₄⋊C₁₆	1	2	yes	—
86	C₄×C₆₇	#428	1	1	yes	characteristic
87	C₄×C₆₇	#430	1	1	yes	characteristic
88	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
89	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
90	#12015	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	2	2	no	—
92	C₂×C₆₇	#1565	1	1	yes	characteristic
93	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
94	C₂×C₆₇	#2153	1	1	yes	characteristic
95	#2162	—	67	1	no	—
96	C₂×C₄⋊C₁₆	—	67	1	no	—
97	C₈×M₁₆	D₆₇	1	1	yes	characteristic
98	#1370	—	67	2	no	—
99	#1367	—	67	2	no	—
100	C₆₇	#12015	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	134	1	no	—
102	#431	Dic₆₇	1	1	yes	characteristic
103	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
105	C₄⋊C₁₆	—	67	1	no	—
106	C₄⋊C₁₆	—	67	2	no	—
107	C₄⋊C₁₆	—	67	1	no	—
108	C₄⋊C₁₆	—	134	1	no	—
109	C₂²⋊C₁₆	—	134	1	no	—
110	C₄×C₁₆	—	134	1	no	—
111	#322	Dic₆₇	1	2	yes	—
112	C₈²	C₂×D₆₇	1	2	yes	—
113	C₂×M₁₆	C₆₇⋊C₈	1	1	yes	characteristic
114	C₂×M₁₆	C₆₇⋊C₈	1	1	yes	characteristic
115	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
116	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
118	C₂×C₁₆	—	268	2	no	—
119	C₂×C₁₆	—	134	1	no	—
120	C₂×C₁₆	—	134	1	no	—
121	C₂×C₁₆	—	134	2	no	—
122	C₄⋊C₈	—	2	1	no	—
123	C₂²⋊C₈	—	2	1	no	—
124	#97	C₆₇⋊C₈	1	2	yes	—
125	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
126	C₄×C₈	D₂₆₈	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
128	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
129	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
130	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
131	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
133	C₂²×C₄	#11761612	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	C₆₇⋊C₁₆	1	4	yes	complemented
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
142	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
143	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
144	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
146	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
147	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
148	C₄²	#11761612	1	1	yes	characteristic
149	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
150	C₁₆	—	268	2	no	—
151	C₁₆	—	268	1	no	—
152	C₁₆	—	268	1	no	—
153	C₂³	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
158	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
159	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
160	C₂×C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
161	C₂×C₄	#14523983	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
164	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	2	no	—
170	C₈	C₂×C₆₇⋊C₁₆	1	4	yes	complemented
171	C₈	#14524071	1	1	yes	characteristic
172	C₈	#14524076	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊(C₆₇⋊C₁₆)	1	2	yes	—
176	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₄⋊(C₆₇⋊C₁₆)	1	2	yes	—
181	C₄	#16863438	1	1	yes	characteristic
182	C₄	#16863445	1	1	yes	characteristic
183	C₄	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
184	C₄	C₂×#14523983	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#22558961	1	1	yes	characteristic
187	C₂	C₂×C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
188	C₂	#22561975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#12015
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22558279	C₂	semidirect
4. #22558273	C₂	semidirect
43. M₁₆×C₆₇	C₁₆	semidirect
75. C₈×C₆₇	C₂×C₁₆	semidirect
100. C₆₇	#12015	semidirect
135. M₁₆	C₆₇⋊C₁₆	semidirect
170. C₈	C₂×C₆₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22562008	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22558279	2
#22558273	2
C₈×M₁₆×C₆₇	2
#12015	67