Subgroups — #30770913

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₇	C₂	1	2	yes	maximal, complemented
3	C₈×C₂²⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558286	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	—	2	4	no	—
7	C₂²×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
12	C₄×C₂²⋊Dic₆₇	C₄	1	1	yes	characteristic
13	C₁₆×Dic₆₇	—	2	4	no	—
14	#16863150	C₄	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²⋊C₁₆×C₆₇	C₄	1	4	yes	complemented
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊Dic₆₇	C₈	1	2	yes	—
20	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
21	C₂×C₆₇⋊C₁₆	—	2	8	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
26	C₂²⋊(C₆₇⋊C₈)	C₈	1	4	yes	—
27	C₈×Dic₆₇	—	2	2	no	—
28	C₈×Dic₆₇	—	2	2	no	—
29	C₆₇⋊#95	C₈	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	—	2	2	no	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₂×C₁₆×C₆₇	—	2	2	no	—
39	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
40	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
44	C₂²⋊Dic₆₇	C₁₆	1	8	yes	complemented
45	C₄×Dic₆₇	—	2	2	no	—
46	C₄×Dic₆₇	—	2	2	no	—
47	C₆₇⋊C₁₆	—	2	2	no	—
48	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
49	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	—	2	2	no	—
52	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	C₄²	1	2	yes	—
61	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₆₇	#41	1	2	yes	—
65	C₁₆×C₆₇	—	2	4	no	—
66	C₂³×C₆₇	—	2	2	no	—
67	C₂³×C₆₇	—	2	2	no	—
68	C₂³×C₆₇	C₄×C₈	1	2	yes	—
69	C₂³×C₆₇	C₂×C₁₆	1	4	yes	—
70	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#131	1	2	yes	—
80	C₂×Dic₆₇	—	2	4	no	—
81	C₂×Dic₆₇	—	2	4	no	—
82	C₂×Dic₆₇	C₂×C₁₆	1	8	yes	complemented
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₆₇	#117	1	1	yes	characteristic
86	C₈×C₆₇	C₄×D₄	1	2	yes	—
87	C₆₇⋊C₈	—	2	4	no	—
88	C₂²×C₆₇	—	2	2	no	—
89	C₂²×C₆₇	—	2	8	no	—
90	C₂²×C₆₇	—	2	2	no	—
91	C₂²×C₆₇	C₄×C₁₆	1	4	yes	complemented
92	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
93	C₂²×C₆₇	C₈×D₄	1	2	yes	—
94	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
95	C₂²×C₆₇	#504	1	2	yes	—
96	C₄×C₆₇	—	2	2	no	—
97	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₆₇	#405	1	1	yes	characteristic
99	C₄×C₆₇	C₈×D₄	1	2	yes	—
100	Dic₆₇	—	2	8	no	—
101	C₁₆×C₂²⋊C₄	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	—	2	4	no	—
103	C₂×C₆₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₆₇	C₂×C₄×C₁₆	1	1	yes	characteristic
105	C₂×C₆₇	#2108	1	1	yes	characteristic
106	C₂×C₆₇	D₄×C₁₆	1	4	yes	—
107	C₂³×C₁₆	—	67	1	no	—
108	C₂×C₂²⋊C₁₆	D₆₇	1	1	yes	characteristic
109	C₂×C₄×C₁₆	—	67	2	no	—
110	C₈×C₂²⋊C₄	—	67	1	no	—
111	#1374	—	67	2	no	—
112	C₆₇	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
114	C₂²×C₁₆	—	67	1	no	—
115	C₂²×C₁₆	—	67	4	no	—
116	C₂²×C₁₆	—	67	1	no	—
117	C₂²×C₁₆	—	134	4	no	—
118	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
119	C₂×C₂²⋊C₈	—	67	1	no	—
120	C₂×C₄×C₈	—	67	2	no	—
121	C₄×C₂²⋊C₄	—	67	1	no	—
122	C₂²⋊C₁₆	Dic₆₇	1	4	yes	complemented
123	C₄×C₁₆	—	134	4	no	—
124	#336	—	67	2	no	—
125	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	67	2	no	—
129	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
130	C₂²×C₈	C₄×D₆₇	1	2	yes	—
131	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
132	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	67	2	no	—
134	C₂×C₄²	—	67	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	67	2	no	—
138	C₂×C₁₆	—	67	4	no	—
139	C₂×C₁₆	—	67	2	no	—
140	C₂×C₁₆	—	134	8	no	—
141	C₂×C₁₆	—	134	2	no	—
142	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
143	C₂²⋊C₈	—	67	4	no	—
144	C₄×C₈	—	134	2	no	—
145	C₄×C₈	—	134	2	no	—
146	#95	—	67	2	no	—
147	C₂⁴	C₈×D₆₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	67	4	no	—
151	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
152	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
153	C₂²×C₄	C₈×D₆₇	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	67	4	no	—
160	C₂×C₈	—	134	2	no	—
161	C₂×C₈	—	134	2	no	—
162	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
164	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
165	C₂×C₈	D₄×D₆₇	1	2	yes	—
166	C₂×C₈	#11761627	1	2	yes	—
167	C₂²⋊C₄	—	67	8	no	—
168	C₄²	—	134	2	no	—
169	C₄²	—	134	2	no	—
170	C₁₆	—	2	4	no	—
171	C₁₆	—	67	4	no	—
172	C₁₆	—	134	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₈×Dic₆₇	1	2	yes	—
176	C₂³	C₁₆×D₆₇	1	4	yes	—
177	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	67	8	no	—
183	C₂×C₄	—	134	4	no	—
184	C₂×C₄	—	134	4	no	—
185	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
186	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
187	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
188	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
189	C₂×C₄	#14524051	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	—	134	4	no	—
192	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
193	C₈	#14523999	1	1	yes	characteristic
194	C₈	D₄×Dic₆₇	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₁₆×Dic₆₇	1	4	yes	complemented
199	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
200	C₂²	(C₂²⋊D₆₇)⋊C₈	1	2	yes	—
201	C₂²	C₂×C₁₆×D₆₇	1	2	yes	—
202	C₂²	#16863814	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	134	8	no	—
205	C₄	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
206	C₄	#16863354	1	1	yes	characteristic
207	C₄	(C₂²⋊D₆₇)⋊C₈	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	#22559983	1	1	yes	characteristic
210	C₂	C₂×C₁₆×Dic₆₇	1	1	yes	characteristic
211	C₂	#22561808	1	1	yes	characteristic
212	C₂	(C₂²⋊D₆₇)⋊C₁₆	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₄×C₁₆
Fitting	C₂×C₂²⋊C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₆₇
hypocenter	C₆₇	C₁₆×C₂²⋊C₄
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×Dic₆₇	C₂	semidirect
4. #22558286	C₂	semidirect
7. C₂²×C₆₇⋊C₁₆	C₄	semidirect
17. C₂²⋊C₁₆×C₆₇	C₄	semidirect
44. C₂²⋊Dic₆₇	C₁₆	semidirect
48. C₆₇⋊C₁₆	C₂²⋊C₄	semidirect
82. C₂×Dic₆₇	C₂×C₁₆	semidirect
91. C₂²×C₆₇	C₄×C₁₆	semidirect
112. C₆₇	C₁₆×C₂²⋊C₄	semidirect
122. C₂²⋊C₁₆	Dic₆₇	semidirect
198. C₂²	C₁₆×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₆₇⋊C₁₆	2
C₂×C₁₆×Dic₆₇	2
C₈×C₂²⋊Dic₆₇	2
#22558286	2
C₂×C₂²⋊C₁₆×C₆₇	2
C₁₆×C₂²⋊C₄	67