Subgroups — #30754864

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568607	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863143	C₂	1	4	yes	maximal
4	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524043	—	2	2	no	—
6	C₂×#14524043	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
9	#16863266	—	2	1	no	—
10	#16863266	—	2	1	no	—
11	#16863266	C₄	1	4	yes	—
12	#16863151	—	2	1	no	—
13	#16863143	—	2	8	no	—
14	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₃₄	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	#14524043	—	2	4	no	—
20	#14524043	—	2	4	no	—
21	#14524043	—	4	4	no	—
22	#14524043	—	4	4	no	—
23	C₂×C₄⋊Dic₆₇	—	2	4	no	—
24	C₂×C₄⋊Dic₆₇	—	2	4	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
29	C₄×C₆₇⋊C₈	—	4	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₆₇	—	2	4	no	—
37	C₄⋊C₈×C₆₇	—	4	4	no	—
38	C₂×Dic₁₃₄	—	4	2	no	—
39	C₂×Dic₁₃₄	—	4	2	no	—
40	C₂×Dic₁₃₄	—	4	4	no	—
41	C₂×Dic₁₃₄	—	4	4	no	—
42	C₂²×Dic₆₇	—	2	4	no	—
43	C₂×C₆₇⋊C₈	—	2	4	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₄⋊Dic₆₇	—	2	8	no	—
47	C₄⋊Dic₆₇	—	4	8	no	—
48	C₄⋊Dic₆₇	—	4	8	no	—
49	C₄⋊Dic₆₇	—	4	4	no	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	—	4	2	no	—
56	C₂×C₈×C₆₇	—	4	4	no	—
57	C₄²×C₆₇	—	2	4	no	—
58	C₄²×C₆₇	—	2	2	no	—
59	C₄²×C₆₇	D₈	1	4	yes	—
60	C₄²×C₆₇	SD₁₆	1	4	yes	—
61	C₄²×C₆₇	SD₁₆	1	2	yes	—
62	C₄²×C₆₇	Q₁₆	1	2	yes	—
63	C₄²×C₆₇	C₂×D₄	1	2	yes	—
64	C₄²×C₆₇	#41	1	2	yes	—
65	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#101	1	4	yes	—
70	C₂×C₄×C₆₇	#102	1	4	yes	—
71	C₂×C₄×C₆₇	#102	1	4	yes	—
72	C₂×C₄×C₆₇	#103	1	4	yes	—
73	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₆₇	#121	1	2	yes	—
75	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#137	1	2	yes	—
82	C₂×Dic₆₇	—	4	4	no	—
83	C₂×Dic₆₇	—	4	8	no	—
84	Dic₁₃₄	—	8	4	no	—
85	Dic₁₃₄	—	8	2	no	—
86	Dic₁₃₄	—	8	2	no	—
87	C₈×C₆₇	—	8	2	no	—
88	C₆₇⋊C₈	—	4	4	no	—
89	C₂²×C₆₇	#390	1	1	yes	characteristic
90	C₂²×C₆₇	C₂×#101	1	2	yes	—
91	C₂²×C₆₇	C₂×#102	1	2	yes	—
92	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
93	C₂²×C₆₇	#419	1	1	yes	characteristic
94	C₄×C₆₇	—	2	4	no	—
95	C₄×C₆₇	—	2	2	no	—
96	C₄×C₆₇	#332	1	8	yes	complemented
97	C₄×C₆₇	#461	1	2	yes	—
98	C₄×C₆₇	#462	1	2	yes	—
99	C₄×C₆₇	#486	1	2	yes	—
100	C₄×C₆₇	#492	1	1	yes	characteristic
101	C₄×C₆₇	#493	1	1	yes	characteristic
102	Dic₆₇	—	8	4	no	—
103	#9692	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	C₂×#332	1	2	yes	—
105	C₂×C₆₇	#1546	1	2	yes	—
106	C₂×C₆₇	#1957	1	1	yes	characteristic
107	C₂×C₆₇	#1966	1	1	yes	characteristic
108	C₂×C₆₇	#1981	1	1	yes	characteristic
109	#2868	—	67	1	no	—
110	#1842	D₆₇	1	1	yes	characteristic
111	C₄×C₄⋊C₈	—	67	1	no	—
112	C₂×#332	—	67	4	no	—
113	C₆₇	#9692	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄⋊Q₈	—	67	2	no	—
115	C₂×C₄⋊Q₈	—	134	2	no	—
116	C₂×C₄⋊C₈	—	2	2	no	—
117	C₂×C₄⋊C₈	—	67	2	no	—
118	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
119	C₂×C₄×C₈	—	134	1	no	—
120	#384	—	67	4	no	—
121	#384	—	134	1	no	—
122	#384	—	134	1	no	—
123	C₄³	C₂×D₆₇	1	1	yes	characteristic
124	#336	—	134	1	no	—
125	#332	—	134	8	no	—
126	C₂²×Q₈	—	134	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	134	2	no	—
129	C₄⋊Q₈	—	268	4	no	—
130	C₄⋊Q₈	—	268	4	no	—
131	C₄⋊Q₈	—	134	4	no	—
132	C₄⋊Q₈	—	134	4	no	—
133	C₂×C₄⋊C₄	—	67	4	no	—
134	C₂×C₄⋊C₄	—	134	4	no	—
135	C₂×C₄⋊C₄	—	134	4	no	—
136	C₂×C₄⋊C₄	—	134	2	no	—
137	C₂×C₄²	C₄×D₆₇	1	2	yes	—
138	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
139	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
141	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	4	4	no	—
144	C₄⋊C₈	—	67	4	no	—
145	C₄⋊C₈	—	134	2	no	—
146	C₄×C₈	—	268	2	no	—
147	C₂×Q₈	—	268	2	no	—
148	C₂×Q₈	—	268	2	no	—
149	C₂×Q₈	—	268	4	no	—
150	C₂×Q₈	—	268	4	no	—
151	C₂²×C₄	—	134	4	no	—
152	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
153	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	4	no	—
159	C₂×C₈	—	268	2	no	—
160	C₂×C₈	—	268	2	no	—
161	C₂×C₈	—	134	4	no	—
162	C₄⋊C₄	—	268	8	no	—
163	C₄⋊C₄	—	268	8	no	—
164	C₄⋊C₄	—	268	4	no	—
165	C₄⋊C₄	—	134	8	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	#11761613	1	4	yes	—
169	C₄²	#11761615	1	4	yes	—
170	C₄²	#11761617	1	2	yes	—
171	C₄²	Dic₂₆₈	1	2	yes	—
172	C₄²	D₄×D₆₇	1	2	yes	—
173	C₄²	#11761629	1	2	yes	—
174	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
175	Q₈	—	536	4	no	—
176	Q₈	—	536	2	no	—
177	Q₈	—	536	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	268	4	no	—
182	C₂×C₄	—	268	8	no	—
183	C₂×C₄	#14523961	1	4	yes	—
184	C₂×C₄	#14523963	1	4	yes	—
185	C₂×C₄	#14523963	1	4	yes	—
186	C₂×C₄	#14523968	1	4	yes	—
187	C₂×C₄	#14524005	1	2	yes	—
188	C₂×C₄	#14524018	1	2	yes	—
189	C₂×C₄	#14524032	1	1	yes	characteristic
190	C₂×C₄	#14524039	1	1	yes	characteristic
191	C₂×C₄	C₂×#11761613	1	2	yes	—
192	C₂×C₄	C₂×#11761615	1	2	yes	—
193	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
194	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
195	C₂×C₄	#14524076	1	2	yes	—
196	C₈	—	8	2	no	—
197	C₈	—	268	4	no	—
198	C₂²	#16863297	1	1	yes	characteristic
199	C₂²	C₂×#14523961	1	2	yes	—
200	C₂²	C₂×#14523963	1	2	yes	—
201	C₂²	C₂×#14523968	1	1	yes	characteristic
202	C₂²	#16863406	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	536	4	no	—
206	C₄	#16863143	1	8	yes	complemented
207	C₄	#16863602	1	2	yes	—
208	C₄	#16863607	1	2	yes	—
209	C₄	#16863744	1	2	yes	—
210	C₄	#16863770	1	1	yes	characteristic
211	C₄	#16863772	1	1	yes	characteristic
212	C₂	C₂×#16863143	1	2	yes	—
213	C₂	#22558882	1	2	yes	—
214	C₂	#22561014	1	1	yes	characteristic
215	C₂	#22561052	1	1	yes	characteristic
216	C₂	#22561131	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1842	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#9692
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
96. C₄×C₆₇	#332	semidirect
113. C₆₇	#9692	semidirect
206. C₄	#16863143	semidirect

Maximal subgroups

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

subgroup	index
#22568607	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863143	2
C₆₇×#1842	2
#9692	67