Subgroups — #30754604

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₆₇	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	—	2	2	no	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	Q₈⋊(C₆₇⋊C₈)	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
12	#391×C₆₇	C₄	1	2	yes	—
13	#389×C₆₇	—	2	2	no	—
14	#384×C₆₇	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
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)	—	4	4	no	—
21	C₄×C₆₇⋊C₈	—	2	8	no	—
22	C₄×C₆₇⋊C₈	—	2	2	no	—
23	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
24	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	C₄⋊Q₈×C₆₇	C₈	1	4	yes	complemented
27	C₄×Q₈×C₆₇	—	2	4	no	—
28	C₄×Q₈×C₆₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	—	2	2	no	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	2	yes	—
37	C₂×C₄²×C₆₇	D₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	#95×C₆₇	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	2	8	no	—
41	C₂×C₆₇⋊C₈	—	2	4	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	4	no	—
44	C₂×Q₈×C₆₇	—	2	2	no	—
45	C₂×Q₈×C₆₇	—	2	4	no	—
46	C₂×Q₈×C₆₇	—	2	4	no	—
47	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	2	2	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	C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₄⋊C₄×C₆₇	—	4	4	no	—
58	C₄⋊C₄×C₆₇	—	4	4	no	—
59	C₄²×C₆₇	—	2	4	no	—
60	C₄²×C₆₇	—	2	2	no	—
61	C₄²×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	—	4	2	no	—
63	C₄²×C₆₇	C₂×C₈	1	2	yes	—
64	C₄²×C₆₇	M₁₆	1	2	yes	—
65	C₄²×C₆₇	SD₁₆	1	4	yes	—
66	C₄²×C₆₇	Q₁₆	1	4	yes	—
67	C₄²×C₆₇	C₂×D₄	1	2	yes	—
68	C₄²×C₆₇	#41	1	2	yes	—
69	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	Q₈×C₆₇	—	2	8	no	—
71	Q₈×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
81	C₂×C₄×C₆₇	#103	1	4	yes	—
82	C₂×C₄×C₆₇	#103	1	4	yes	—
83	C₂×C₄×C₆₇	#104	1	4	yes	—
84	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₆₇	#121	1	2	yes	—
86	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#131	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
93	C₆₇⋊C₈	—	2	8	no	—
94	C₆₇⋊C₈	—	8	2	no	—
95	C₂²×C₆₇	#390	1	1	yes	characteristic
96	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₆₇	#408	1	1	yes	characteristic
98	C₂²×C₆₇	C₂×#103	1	2	yes	—
99	C₂²×C₆₇	C₂×#104	1	2	yes	—
100	C₄×C₆₇	—	2	2	no	—
101	C₄×C₆₇	—	2	4	no	—
102	C₄×C₆₇	—	4	4	no	—
103	C₄×C₆₇	—	8	2	no	—
104	C₄×C₆₇	Q₈⋊C₈	1	8	yes	complemented
105	C₄×C₆₇	C₈×D₄	1	2	yes	—
106	C₄×C₆₇	#436	1	2	yes	—
107	C₄×C₆₇	#487	1	2	yes	—
108	C₄×C₆₇	#492	1	1	yes	characteristic
109	C₄×C₆₇	#494	1	1	yes	characteristic
110	#9637	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	C₂×Q₈⋊C₈	1	2	yes	—
112	C₂×C₆₇	#1493	1	2	yes	—
113	C₂×C₆₇	#1954	1	1	yes	characteristic
114	C₂×C₆₇	#1956	1	1	yes	characteristic
115	C₂×C₆₇	#1962	1	1	yes	characteristic
116	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
117	#1842	—	67	1	no	—
118	C₄²×C₈	—	67	1	no	—
119	C₂×Q₈⋊C₈	—	67	4	no	—
120	C₆₇	#9637	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
122	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
123	C₂×C₄⋊C₈	—	67	2	no	—
124	C₂×C₄⋊C₈	—	134	2	no	—
125	C₂×C₄×C₈	—	67	2	no	—
126	C₂×C₄×C₈	—	134	2	no	—
127	#391	Dic₆₇	1	2	yes	—
128	#389	—	2	2	no	—
129	#384	Dic₆₇	1	1	yes	characteristic
130	C₄×C₄⋊C₄	—	2	2	no	—
131	C₄³	C₂×D₆₇	1	1	yes	characteristic
132	Q₈⋊C₈	—	134	8	no	—
133	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
134	C₂²×C₈	—	134	2	no	—
135	C₂²×C₈	—	134	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	C₆₇⋊C₈	1	4	yes	complemented
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	2	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
149	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
150	C₄⋊C₈	—	268	4	no	—
151	C₄⋊C₈	—	134	4	no	—
152	C₄×C₈	—	67	4	no	—
153	C₄×C₈	—	134	8	no	—
154	C₄×C₈	—	134	2	no	—
155	#95	—	2	4	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
163	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
164	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂×C₈	—	268	2	no	—
168	C₂×C₈	—	268	4	no	—
169	C₂×C₈	—	134	8	no	—
170	C₂×C₈	—	134	4	no	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
180	C₄²	#11761612	1	2	yes	—
181	C₄²	#11761615	1	4	yes	—
182	C₄²	#11761618	1	4	yes	—
183	C₄²	D₄×D₆₇	1	2	yes	—
184	C₄²	#11761627	1	2	yes	—
185	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
186	Q₈	—	2	8	no	—
187	Q₈	—	4	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
197	C₂×C₄	#14523967	1	4	yes	—
198	C₂×C₄	#14523967	1	4	yes	—
199	C₂×C₄	#14523970	1	4	yes	—
200	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
201	C₂×C₄	#14524014	1	2	yes	—
202	C₂×C₄	#14524030	1	1	yes	characteristic
203	C₂×C₄	#14524039	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
205	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
206	C₂×C₄	#14524050	1	2	yes	—
207	C₂×C₄	C₂×#11761615	1	2	yes	—
208	C₂×C₄	C₂×#11761618	1	2	yes	—
209	C₈	—	536	2	no	—
210	C₈	—	134	8	no	—
211	C₂²	#16863299	1	1	yes	characteristic
212	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
213	C₂²	#16863363	1	1	yes	characteristic
214	C₂²	C₂×#14523967	1	2	yes	—
215	C₂²	C₂×#14523970	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	2	no	—
220	C₄	Q₈⋊(C₆₇⋊C₈)	1	8	yes	complemented
221	C₄	D₄×C₆₇⋊C₈	1	2	yes	—
222	C₄	#16863474	1	2	yes	—
223	C₄	#16863746	1	2	yes	—
224	C₄	#16863770	1	1	yes	characteristic
225	C₄	#16863775	1	1	yes	characteristic
226	C₂	C₂×Q₈⋊(C₆₇⋊C₈)	1	2	yes	—
227	C₂	#22558664	1	2	yes	—
228	C₂	#22561004	1	1	yes	characteristic
229	C₂	#22561012	1	1	yes	characteristic
230	C₂	#22561038	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#9637
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
26. C₄⋊Q₈×C₆₇	C₈	semidirect
104. C₄×C₆₇	Q₈⋊C₈	semidirect
120. C₆₇	#9637	semidirect
137. C₄⋊Q₈	C₆₇⋊C₈	semidirect
220. C₄	Q₈⋊(C₆₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₄²×C₆₇⋊C₈	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
C₄×C₄⋊Q₈×C₆₇	2
#9637	67