Subgroups — #30772242

Order 17152.

414 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	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₆₇⋊C₁₆	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
9	C₄²×Dic₆₇	C₄	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
11	Dic₆₇⋊C₁₆	—	2	8	no	—
12	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₆₇	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₈	1	2	yes	—
17	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₆₇⋊C₁₆	—	2	4	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₄×C₆₇⋊C₈	—	2	2	no	—
23	C₈×Dic₆₇	—	2	4	no	—
24	C₈×Dic₆₇	D₄	1	4	yes	—
25	C₈×Dic₆₇	Q₈	1	4	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	—	2	4	no	—
31	C₄×C₈×C₆₇	D₄	1	2	yes	—
32	C₄×C₈×C₆₇	Q₈	1	2	yes	—
33	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
34	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₄×Dic₆₇	—	2	2	no	—
39	C₄×Dic₆₇	—	2	4	no	—
40	C₄×Dic₆₇	C₁₆	1	4	yes	complemented
41	C₄×Dic₆₇	C₄⋊C₄	1	4	yes	—
42	C₄×Dic₆₇	M₁₆	1	4	yes	—
43	C₆₇⋊C₁₆	—	4	2	no	—
44	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₄²×C₆₇	M₁₆	1	2	yes	—
55	C₁₆×C₆₇	—	4	2	no	—
56	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
69	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
70	C₂×Dic₆₇	C₂×C₁₆	1	4	yes	—
71	C₂×Dic₆₇	M₃₂	1	4	yes	—
72	C₈×C₆₇	—	2	2	no	—
73	C₈×C₆₇	#127	1	1	yes	characteristic
74	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
76	C₆₇⋊C₈	—	2	4	no	—
77	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
78	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
80	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
81	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	C₄⋊C₁₆	1	4	yes	complemented
84	C₄×C₆₇	#384	1	1	yes	characteristic
85	C₄×C₆₇	#423	1	2	yes	—
86	C₄×C₆₇	#423	1	1	yes	characteristic
87	Dic₆₇	—	2	4	no	—
88	Dic₆₇	C₄⋊C₁₆	1	8	yes	complemented
89	#12898	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	#1842	1	1	yes	characteristic
91	C₂×C₆₇	C₂×C₄⋊C₁₆	1	2	yes	—
92	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
93	C₂×C₆₇	#2150	1	2	yes	—
94	C₂×C₆₇	#2150	1	1	yes	characteristic
95	C₂×C₄⋊C₁₆	—	67	4	no	—
96	C₂×C₄⋊C₁₆	—	67	1	no	—
97	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
98	C₄²×C₈	—	67	1	no	—
99	C₆₇	#12898	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₁₆	—	67	2	no	—
101	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
102	C₂×C₄×C₈	—	67	2	no	—
103	C₂×C₄×C₈	—	67	2	no	—
104	C₂×C₄×C₈	—	67	1	no	—
105	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
106	C₄³	—	67	1	no	—
107	C₄⋊C₁₆	—	2	2	no	—
108	C₄⋊C₁₆	—	134	8	no	—
109	C₄⋊C₁₆	—	134	2	no	—
110	C₂²×C₈	—	67	2	no	—
111	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
113	C₂×C₄²	—	67	1	no	—
114	C₂×C₄²	—	67	2	no	—
115	C₂×C₄²	—	67	2	no	—
116	C₂×C₄²	—	67	1	no	—
117	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
118	C₂×C₁₆	—	2	4	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	134	4	no	—
121	C₂×C₁₆	—	134	2	no	—
122	C₄×C₈	—	67	4	no	—
123	C₄×C₈	—	67	4	no	—
124	C₄×C₈	—	134	4	no	—
125	C₄×C₈	—	134	2	no	—
126	C₄×C₈	Dic₁₃₄	1	2	yes	—
127	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
128	C₂²×C₄	—	67	2	no	—
129	C₂²×C₄	—	67	2	no	—
130	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
131	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	67	4	no	—
135	C₂×C₈	—	134	2	no	—
136	C₂×C₈	—	134	2	no	—
137	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
138	C₂×C₈	D₄×D₆₇	1	2	yes	—
139	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₆₇	1	2	yes	—
141	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
142	C₄²	—	67	4	no	—
143	C₄²	—	67	4	no	—
144	C₄²	—	67	4	no	—
145	C₄²	—	134	4	no	—
146	C₄²	—	134	2	no	—
147	C₄²	Dic₆₇⋊C₄	1	2	yes	—
148	C₄²	#11761611	1	2	yes	—
149	C₁₆	—	4	2	no	—
150	C₁₆	—	268	2	no	—
151	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	67	4	no	—
154	C₂×C₄	—	67	4	no	—
155	C₂×C₄	—	67	4	no	—
156	C₂×C₄	—	67	4	no	—
157	C₂×C₄	—	134	2	no	—
158	C₂×C₄	—	134	2	no	—
159	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
160	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
161	C₂×C₄	C₁₆×D₆₇	1	2	yes	—
162	C₂×C₄	#14523982	1	2	yes	—
163	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
164	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
165	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
166	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	134	4	no	—
169	C₈	#14524039	1	1	yes	characteristic
170	C₈	#14524041	1	1	yes	characteristic
171	C₈	#14524042	1	2	yes	—
172	C₂²	C₄⋊C₈×D₆₇	1	2	yes	—
173	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
174	C₂²	C₂×C₁₆×D₆₇	1	1	yes	characteristic
175	C₂²	M₃₂×D₆₇	1	2	yes	—
176	C₂²	C₂×#14523982	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	67	8	no	—
179	C₄	—	134	4	no	—
180	C₄	Dic₆₇⋊C₁₆	1	4	yes	complemented
181	C₄	#16863265	1	1	yes	characteristic
182	C₄	#16863418	1	2	yes	—
183	C₄	#16863419	1	1	yes	characteristic
184	C₂	#22560360	1	1	yes	characteristic
185	C₂	C₄⋊C₁₆×D₆₇	1	2	yes	—
186	C₂	C₂×Dic₆₇⋊C₁₆	1	1	yes	characteristic
187	C₂	#22561963	1	2	yes	—
188	C₂	#22561964	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#12898
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
40. C₄×Dic₆₇	C₁₆	semidirect
83. C₄×C₆₇	C₄⋊C₁₆	semidirect
88. Dic₆₇	C₄⋊C₁₆	semidirect
99. C₆₇	#12898	semidirect
180. C₄	Dic₆₇⋊C₁₆	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×Dic₆₇⋊C₁₆	2
C₄×C₈×Dic₆₇	2
C₂×C₄⋊C₁₆×C₆₇	2
#12898	67