Subgroups — #30773837

Order 17152.

338 conjugacy classes of subgroups.

The subgroup lattice has 212 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×C₄⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal
3	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
4	#22558287	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
10	C₄⋊(C₆₇⋊C₁₆)	—	2	4	no	—
11	#16863172	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
16	#336×C₆₇	C₄	1	2	yes	—
17	C₂×C₆₇⋊C₁₆	—	2	1	no	—
18	C₂×C₆₇⋊C₁₆	—	2	1	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	C₂×C₆₇⋊C₁₆	—	2	1	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
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	Q₈	1	2	yes	—
37	#95×C₆₇	C₈	1	1	yes	characteristic
38	#95×C₆₇	C₈	1	1	yes	characteristic
39	C₆₇⋊C₁₆	—	2	2	no	—
40	C₆₇⋊C₁₆	—	2	2	no	—
41	C₆₇⋊C₁₆	—	4	2	no	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	—	2	1	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
58	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₄²×C₆₇	—	2	2	no	—
61	C₄²×C₆₇	—	2	2	no	—
62	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₄²×C₆₇	M₁₆	1	2	yes	—
64	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
73	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
81	C₈×C₆₇	—	2	2	no	—
82	C₈×C₆₇	—	4	2	no	—
83	C₈×C₆₇	#121	1	1	yes	characteristic
84	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
85	C₈×C₆₇	#125	1	1	yes	characteristic
86	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
87	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
89	C₂²×C₆₇	C₈×Q₈	1	1	yes	characteristic
90	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
91	C₂²×C₆₇	#504	1	2	yes	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	—	2	4	no	—
95	C₄×C₆₇	—	4	2	no	—
96	C₄×C₆₇	#389	1	1	yes	characteristic
97	C₄×C₆₇	#424	1	1	yes	characteristic
98	C₄×C₆₇	#436	1	1	yes	characteristic
99	C₄×C₆₇	#446	1	1	yes	characteristic
100	#13160	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	#1939	1	1	yes	characteristic
102	C₂×C₆₇	#2150	1	1	yes	characteristic
103	C₂×C₆₇	#2151	1	1	yes	characteristic
104	C₂×C₆₇	#2168	1	2	yes	—
105	C₂×C₆₇	#2183	1	2	yes	—
106	C₂×C₄⋊C₁₆	—	67	2	no	—
107	C₂×C₄⋊C₁₆	—	67	1	no	—
108	C₂×#346	—	67	1	no	—
109	C₈×C₄⋊C₄	D₆₇	1	1	yes	characteristic
110	#1374	—	67	2	no	—
111	C₆₇	#13160	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₁₆	—	67	1	no	—
113	C₂²×C₁₆	—	67	1	no	—
114	C₂²×C₁₆	—	67	2	no	—
115	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
119	C₄⋊C₁₆	—	134	4	no	—
120	C₄⋊C₁₆	—	134	2	no	—
121	#346	—	134	2	no	—
122	#336	Dic₆₇	1	2	yes	—
123	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
124	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₆₇⋊C₈	1	2	yes	—
127	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
128	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
129	C₂×C₁₆	—	67	2	no	—
130	C₂×C₁₆	—	67	2	no	—
131	C₂×C₁₆	—	134	2	no	—
132	C₂×C₁₆	—	134	2	no	—
133	C₂×C₁₆	—	134	2	no	—
134	C₂×C₁₆	—	134	1	no	—
135	C₂×C₁₆	—	134	1	no	—
136	C₂×C₁₆	—	134	1	no	—
137	C₂×C₁₆	—	134	1	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	Dic₁₃₄	1	2	yes	—
142	C₄×C₈	D₂₆₈	1	2	yes	—
143	#95	C₆₇⋊C₈	1	1	yes	characteristic
144	#95	C₆₇⋊C₈	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
146	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
156	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
158	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
160	C₂×C₈	#11761627	1	1	yes	characteristic
161	C₂×C₈	#11761629	1	1	yes	characteristic
162	C₄⋊C₄	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	C₄⋊Dic₆₇	1	2	yes	—
166	C₄²	#11761612	1	2	yes	—
167	C₁₆	—	268	2	no	—
168	C₁₆	—	134	2	no	—
169	C₁₆	—	134	2	no	—
170	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
179	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
180	C₂×C₄	#14523983	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
185	C₂×C₄	#14524050	1	1	yes	characteristic
186	C₂×C₄	#14524050	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#14524018	1	1	yes	characteristic
190	C₈	#14524023	1	1	yes	characteristic
191	C₈	#14524034	1	1	yes	characteristic
192	C₈	#14524042	1	1	yes	characteristic
193	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
194	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
195	C₂²	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
196	C₂²	C₂×#14523983	1	2	yes	—
197	C₂²	#16863813	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	4	2	no	—
202	C₄	#16863295	1	1	yes	characteristic
203	C₄	#16863423	1	1	yes	characteristic
204	C₄	#16863474	1	1	yes	characteristic
205	C₄	#16863527	1	1	yes	characteristic
206	C₂	#22560928	1	1	yes	characteristic
207	C₂	#22561965	1	1	yes	characteristic
208	C₂	#22561968	1	1	yes	characteristic
209	C₂	#22562035	1	2	yes	—
210	C₂	#22562118	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂²×C₈
Fitting	C₈×C₄⋊C₄×C₆₇	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#13160
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
111. C₆₇	#13160	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
C₈×C₄⋊C₄×C₆₇	2
#13160	67