Subgroups — #30754875

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863142	C₂	1	4	yes	maximal
4	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
9	#16863151	—	2	1	no	—
10	#16863142	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	#384×C₆₇	—	2	2	no	—
15	#384×C₆₇	C₄	1	4	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
22	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
23	C₄×C₆₇⋊C₈	—	4	2	no	—
24	C₂²×Q₈×C₆₇	—	2	2	no	—
25	C₄⋊Q₈×C₆₇	—	2	4	no	—
26	C₄⋊Q₈×C₆₇	—	2	4	no	—
27	C₄⋊Q₈×C₆₇	—	4	4	no	—
28	C₄⋊Q₈×C₆₇	—	4	2	no	—
29	C₄⋊Q₈×C₆₇	—	4	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	4	no	—
44	C₂×Q₈×C₆₇	—	4	4	no	—
45	C₂×Q₈×C₆₇	—	4	2	no	—
46	C₂×Q₈×C₆₇	—	4	2	no	—
47	C₂×Q₈×C₆₇	—	4	4	no	—
48	C₂²×C₄×C₆₇	—	2	4	no	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₄⋊C₄×C₆₇	—	4	8	no	—
56	C₄⋊C₄×C₆₇	—	4	8	no	—
57	C₄⋊C₄×C₆₇	—	4	2	no	—
58	C₄⋊C₄×C₆₇	—	4	2	no	—
59	C₄²×C₆₇	—	2	4	no	—
60	C₄²×C₆₇	—	2	2	no	—
61	C₄²×C₆₇	D₈	1	2	yes	—
62	C₄²×C₆₇	SD₁₆	1	4	yes	—
63	C₄²×C₆₇	SD₁₆	1	2	yes	—
64	C₄²×C₆₇	Q₁₆	1	4	yes	—
65	C₄²×C₆₇	C₂×D₄	1	2	yes	—
66	C₄²×C₆₇	#41	1	2	yes	—
67	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	Q₈×C₆₇	—	8	4	no	—
69	Q₈×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	4	4	no	—
74	C₂×C₄×C₆₇	—	4	4	no	—
75	C₂×C₄×C₆₇	—	4	4	no	—
76	C₂×C₄×C₆₇	#101	1	4	yes	—
77	C₂×C₄×C₆₇	#102	1	4	yes	—
78	C₂×C₄×C₆₇	#103	1	4	yes	—
79	C₂×C₄×C₆₇	#103	1	4	yes	—
80	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₆₇	#121	1	2	yes	—
82	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
89	C₆₇⋊C₈	—	4	4	no	—
90	C₆₇⋊C₈	—	8	2	no	—
91	C₂²×C₆₇	#390	1	1	yes	characteristic
92	C₂²×C₆₇	C₂×#101	1	2	yes	—
93	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
94	C₂²×C₆₇	C₂×#103	1	2	yes	—
95	C₂²×C₆₇	#418	1	1	yes	characteristic
96	C₄×C₆₇	—	2	4	no	—
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	—	8	4	no	—
99	C₄×C₆₇	#332	1	8	yes	complemented
100	C₄×C₆₇	#459	1	2	yes	—
101	C₄×C₆₇	#460	1	2	yes	—
102	C₄×C₆₇	#487	1	2	yes	—
103	C₄×C₆₇	#492	1	1	yes	characteristic
104	C₄×C₆₇	#494	1	1	yes	characteristic
105	#9694	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	C₂×#332	1	2	yes	—
107	C₂×C₆₇	#1549	1	2	yes	—
108	C₂×C₆₇	#1956	1	1	yes	characteristic
109	C₂×C₆₇	#1966	1	1	yes	characteristic
110	C₂×C₆₇	#1982	1	1	yes	characteristic
111	#2868	D₆₇	1	1	yes	characteristic
112	#1842	—	67	1	no	—
113	C₄×C₄⋊C₈	—	67	1	no	—
114	C₂×#332	—	67	4	no	—
115	C₆₇	#9694	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×C₄⋊Q₈	—	2	1	no	—
118	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
119	C₂×C₄⋊C₈	—	67	2	no	—
120	C₂×C₄⋊C₈	—	67	2	no	—
121	C₂×C₄⋊C₈	—	134	2	no	—
122	C₂×C₄×C₈	—	134	1	no	—
123	#384	—	2	2	no	—
124	#384	Dic₆₇	1	4	yes	—
125	C₄³	C₂×D₆₇	1	1	yes	characteristic
126	#336	—	134	1	no	—
127	#332	—	134	8	no	—
128	C₂²×Q₈	—	2	2	no	—
129	C₂²×C₈	—	134	2	no	—
130	C₂²×C₈	—	134	2	no	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	2	4	no	—
133	C₄⋊Q₈	—	4	4	no	—
134	C₄⋊Q₈	—	4	2	no	—
135	C₄⋊Q₈	—	4	2	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
140	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
141	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
144	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
145	C₄⋊C₈	—	268	4	no	—
146	C₄⋊C₈	—	67	4	no	—
147	C₄⋊C₈	—	134	4	no	—
148	C₄⋊C₈	—	134	2	no	—
149	C₄×C₈	—	268	2	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
156	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
160	C₂×C₈	—	268	2	no	—
161	C₂×C₈	—	268	2	no	—
162	C₂×C₈	—	268	2	no	—
163	C₂×C₈	—	268	4	no	—
164	C₂×C₈	—	134	4	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	2	2	no	—
172	C₄²	#11761613	1	2	yes	—
173	C₄²	#11761615	1	4	yes	—
174	C₄²	#11761616	1	2	yes	—
175	C₄²	#11761618	1	4	yes	—
176	C₄²	D₄×D₆₇	1	2	yes	—
177	C₄²	#11761627	1	2	yes	—
178	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	#14523962	1	4	yes	—
188	C₂×C₄	#14523964	1	4	yes	—
189	C₂×C₄	#14523967	1	4	yes	—
190	C₂×C₄	#14523967	1	4	yes	—
191	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
192	C₂×C₄	#14524014	1	2	yes	—
193	C₂×C₄	#14524030	1	1	yes	characteristic
194	C₂×C₄	#14524039	1	1	yes	characteristic
195	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
196	C₂×C₄	C₂×#11761615	1	2	yes	—
197	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
198	C₂×C₄	C₂×#11761618	1	2	yes	—
199	C₂×C₄	#14524069	1	2	yes	—
200	C₈	—	268	4	no	—
201	C₈	—	536	2	no	—
202	C₂²	#16863299	1	1	yes	characteristic
203	C₂²	C₂×#14523962	1	2	yes	—
204	C₂²	C₂×#14523964	1	1	yes	characteristic
205	C₂²	C₂×#14523967	1	2	yes	—
206	C₂²	#16863401	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	8	4	no	—
210	C₄	#16863142	1	8	yes	complemented
211	C₄	#16863588	1	2	yes	—
212	C₄	#16863593	1	2	yes	—
213	C₄	#16863746	1	2	yes	—
214	C₄	#16863770	1	1	yes	characteristic
215	C₄	#16863775	1	1	yes	characteristic
216	C₂	C₂×#16863142	1	2	yes	—
217	C₂	#22558894	1	2	yes	—
218	C₂	#22561012	1	1	yes	characteristic
219	C₂	#22561054	1	1	yes	characteristic
220	C₂	#22561137	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2868	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2868	D₆₇
hypocenter	C₆₇	#9694
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
99. C₄×C₆₇	#332	semidirect
115. C₆₇	#9694	semidirect
210. C₄	#16863142	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863142	2
C₆₇×#2868	2
#9694	67