Subgroups — #30770917

Order 17152.

486 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×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863172	C₂	1	2	yes	maximal, complemented
4	#22558287	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊C₄×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	2	yes	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	complemented
12	#16863172	—	2	4	no	—
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
16	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
17	#336×C₆₇	C₄	1	2	yes	—
18	C₂×C₆₇⋊C₁₆	—	2	8	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
24	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²⋊C₈×C₆₇	C₈	1	4	yes	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	#95×C₆₇	C₈	1	2	yes	—
40	C₆₇⋊C₁₆	—	2	2	no	—
41	C₆₇⋊C₁₆	—	2	4	no	—
42	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
43	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	4	no	—
57	C₂×C₈×C₆₇	C₄²	1	2	yes	—
58	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₂²⋊C₄×C₆₇	—	2	4	no	—
64	C₄²×C₆₇	—	2	2	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₂³×C₆₇	—	2	2	no	—
67	C₂³×C₆₇	—	2	2	no	—
68	C₂³×C₆₇	C₄×C₈	1	2	yes	—
69	C₂³×C₆₇	M₃₂	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	—	2	4	no	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#131	1	2	yes	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	—	2	4	no	—
85	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₆₇	#117	1	1	yes	characteristic
87	C₈×C₆₇	C₄×D₄	1	2	yes	—
88	C₂²×C₆₇	—	2	2	no	—
89	C₂²×C₆₇	—	2	8	no	—
90	C₂²×C₆₇	—	2	2	no	—
91	C₂²×C₆₇	#346	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	C₂×M₃₂	1	2	yes	—
95	C₂²×C₆₇	#504	1	2	yes	—
96	C₄×C₆₇	—	2	2	no	—
97	C₄×C₆₇	—	4	4	no	—
98	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₆₇	#405	1	1	yes	characteristic
100	C₄×C₆₇	C₈×D₄	1	2	yes	—
101	#12611	—	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×#346	1	1	yes	characteristic
105	C₂×C₆₇	#2108	1	1	yes	characteristic
106	C₂×C₆₇	#2168	1	4	yes	—
107	C₂³×C₁₆	—	67	1	no	—
108	C₂×C₂²⋊C₁₆	—	67	1	no	—
109	C₂×#346	—	67	2	no	—
110	C₈×C₂²⋊C₄	D₆₇	1	1	yes	characteristic
111	#1374	—	67	2	no	—
112	C₆₇	#12611	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	2	no	—
117	C₂²×C₁₆	—	67	1	no	—
118	C₂²×C₁₆	—	134	4	no	—
119	C₂×C₂²⋊C₈	Dic₆₇	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
121	C₄×C₂²⋊C₄	Dic₆₇	1	1	yes	characteristic
122	C₂²⋊C₁₆	—	67	4	no	—
123	#346	—	134	4	no	—
124	#336	Dic₆₇	1	2	yes	—
125	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	C₄×D₆₇	1	2	yes	—
129	C₂²×C₈	C₄×D₆₇	1	2	yes	—
130	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
132	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₆₇⋊C₈	1	2	yes	—
134	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
135	C₂×C₁₆	—	67	2	no	—
136	C₂×C₁₆	—	67	4	no	—
137	C₂×C₁₆	—	67	4	no	—
138	C₂×C₁₆	—	67	2	no	—
139	C₂×C₁₆	—	134	8	no	—
140	C₂×C₁₆	—	134	2	no	—
141	C₂×C₁₆	—	134	2	no	—
142	C₂×C₁₆	—	134	2	no	—
143	C₂²⋊C₈	C₆₇⋊C₈	1	4	yes	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	#95	C₆₇⋊C₈	1	2	yes	—
147	C₂⁴	C₂×C₆₇⋊C₈	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
151	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
152	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
154	C₂²×C₄	C₂²×Dic₆₇	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	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
162	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
163	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
165	C₂×C₈	D₄×D₆₇	1	2	yes	—
166	C₂×C₈	#11761627	1	2	yes	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₁₆	—	67	4	no	—
171	C₁₆	—	134	2	no	—
172	C₁₆	—	134	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₄×C₆₇⋊C₈	1	2	yes	—
176	C₂³	#14523983	1	4	yes	—
177	C₂³	C₂²×C₆₇⋊C₈	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	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
186	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
188	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
189	C₂×C₄	#14524050	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	—	2	4	no	—
192	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
193	C₈	#14523999	1	1	yes	characteristic
194	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	#16863172	1	4	yes	complemented
199	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
200	C₂²	D₄×C₆₇⋊C₈	1	2	yes	—
201	C₂²	C₂×#14523983	1	2	yes	—
202	C₂²	#16863813	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
206	C₄	#16863352	1	1	yes	characteristic
207	C₄	D₄×C₆₇⋊C₈	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	C₂²⋊C₄×C₆₇⋊C₈	1	1	yes	characteristic
210	C₂	C₂×#16863172	1	1	yes	characteristic
211	C₂	#22561806	1	1	yes	characteristic
212	C₂	#22562035	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₈×C₂²⋊C₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₈×C₂²⋊C₄	D₆₇
hypocenter	C₆₇	#12611
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#16863172	C₂	semidirect
4. #22558287	C₂	semidirect
7. C₂²×C₆₇⋊C₁₆	C₄	semidirect
11. C₂²⋊(C₆₇⋊C₁₆)	C₄	semidirect
42. C₆₇⋊C₁₆	C₂²⋊C₄	semidirect
91. C₂²×C₆₇	#346	semidirect
112. C₆₇	#12611	semidirect
198. C₂²	#16863172	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×C₂²⋊(C₆₇⋊C₁₆)	2
C₂×#16863172	2
#22558287	2
C₈×C₂²⋊C₄×C₆₇	2
#12611	67