Subgroups — #30764435

Order 17152.

958 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	3	yes	maximal
2	C₂×#16863151	C₂	1	3	yes	maximal
3	C₄×#95×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₆₇⋊C₈	C₂²	1	3	yes	—
5	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	12	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
8	#16863151	—	2	6	no	—
9	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
10	C₂×#95×C₆₇	C₄	1	3	yes	—
11	C₂×#95×C₆₇	C₄	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₈	—	2	3	no	—
13	C₂²×C₆₇⋊C₈	—	2	3	no	—
14	C₂²×C₆₇⋊C₈	—	2	6	no	—
15	C₂²×C₆₇⋊C₈	—	2	6	no	—
16	C₂²×C₆₇⋊C₈	—	2	3	no	—
17	C₂²×C₆₇⋊C₈	—	2	3	no	—
18	C₂²×C₆₇⋊C₈	D₄	1	6	yes	—
19	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
20	C₄⋊(C₆₇⋊C₈)	—	2	24	no	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	—	2	6	no	—
25	C₂×C₄²×C₆₇	—	2	6	no	—
26	C₂×C₄²×C₆₇	D₄	1	6	yes	—
27	C₂×C₄²×C₆₇	Q₈	1	6	yes	—
28	#95×C₆₇	—	2	6	no	—
29	#95×C₆₇	C₈	1	4	yes	complemented
30	C₂×C₆₇⋊C₈	—	2	12	no	—
31	C₂×C₆₇⋊C₈	—	2	6	no	—
32	C₂×C₆₇⋊C₈	—	2	6	no	—
33	C₂×C₆₇⋊C₈	—	4	3	no	—
34	C₂×C₆₇⋊C₈	—	4	6	no	—
35	C₂×C₆₇⋊C₈	—	4	3	no	—
36	C₂×C₆₇⋊C₈	—	4	3	no	—
37	C₂×C₆₇⋊C₈	—	4	1	no	—
38	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	—	2	6	no	—
40	C₂²×C₄×C₆₇	—	2	6	no	—
41	C₂²×C₄×C₆₇	—	2	6	no	—
42	C₂²×C₄×C₆₇	—	2	6	no	—
43	C₂²×C₄×C₆₇	—	2	3	no	—
44	C₂²×C₄×C₆₇	—	2	3	no	—
45	C₂²×C₄×C₆₇	C₄⋊C₄	1	6	yes	—
46	C₂²×C₄×C₆₇	C₄⋊C₄	1	6	yes	—
47	C₂²×C₄×C₆₇	C₂×C₈	1	6	yes	—
48	C₂²×C₄×C₆₇	M₁₆	1	6	yes	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
53	C₂²×C₄×C₆₇	#41	1	3	yes	—
54	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
55	C₄²×C₆₇	—	2	6	no	—
56	C₄²×C₆₇	—	2	12	no	—
57	C₄²×C₆₇	—	2	6	no	—
58	C₂³×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
59	C₂³×C₆₇	C₄×D₄	1	3	yes	—
60	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
61	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
62	C₂³×C₆₇	C₂×M₁₆	1	3	yes	—
63	C₂³×C₆₇	#131	1	3	yes	—
64	C₂³×C₆₇	#131	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	2	6	no	—
66	C₂×C₄×C₆₇	—	2	6	no	—
67	C₂×C₄×C₆₇	—	2	12	no	—
68	C₂×C₄×C₆₇	—	2	12	no	—
69	C₂×C₄×C₆₇	—	2	6	no	—
70	C₂×C₄×C₆₇	—	2	6	no	—
71	C₂×C₄×C₆₇	—	2	6	no	—
72	C₂×C₄×C₆₇	—	4	6	no	—
73	C₂×C₄×C₆₇	—	4	6	no	—
74	C₂×C₄×C₆₇	C₄⋊C₈	1	12	yes	complemented
75	C₂×C₄×C₆₇	#120	1	3	yes	—
76	C₂×C₄×C₆₇	#121	1	3	yes	—
77	C₂×C₄×C₆₇	C₂²⋊Q₈	1	6	yes	—
78	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
79	C₂×C₄×C₆₇	#123	1	3	yes	—
80	C₂×C₄×C₆₇	#125	1	3	yes	—
81	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₄⋊Q₈	1	3	yes	—
83	C₂×C₄×C₆₇	C₄⋊Q₈	1	3	yes	—
84	C₆₇⋊C₈	—	4	6	no	—
85	C₆₇⋊C₈	—	4	2	no	—
86	C₂²×C₆₇	#384	1	1	yes	characteristic
87	C₂²×C₆₇	#385	1	3	yes	—
88	C₂²×C₆₇	#389	1	3	yes	—
89	C₂²×C₆₇	C₂×C₄⋊C₈	1	3	yes	—
90	C₂²×C₆₇	#423	1	3	yes	—
91	C₂²×C₆₇	#424	1	3	yes	—
92	C₂²×C₆₇	#424	1	3	yes	—
93	C₂²×C₆₇	C₈×D₄	1	3	yes	—
94	C₂²×C₆₇	#435	1	6	yes	—
95	C₂²×C₆₇	#436	1	3	yes	—
96	C₂²×C₆₇	C₈×Q₈	1	1	yes	characteristic
97	C₂²×C₆₇	#446	1	3	yes	—
98	C₄×C₆₇	—	2	12	no	—
99	C₄×C₆₇	—	4	6	no	—
100	C₄×C₆₇	#392	1	1	yes	characteristic
101	C₄×C₆₇	#395	1	3	yes	—
102	C₄×C₆₇	#398	1	3	yes	—
103	C₄×C₆₇	#399	1	1	yes	characteristic
104	#11322	—	67	1	no	maximal, Sylow
105	C₂×C₆₇	#1444	1	1	yes	characteristic
106	C₂×C₆₇	#1842	1	1	yes	characteristic
107	C₂×C₆₇	#1843	1	1	yes	characteristic
108	C₂×C₆₇	#1850	1	3	yes	—
109	C₂×C₆₇	#1851	1	3	yes	—
110	C₂×C₆₇	#1939	1	3	yes	—
111	C₂×C₆₇	#1940	1	3	yes	—
112	C₂²×C₄⋊C₈	—	67	3	no	—
113	C₂×#336	—	67	3	no	—
114	C₄×#95	D₆₇	1	1	yes	characteristic
115	C₆₇	#11322	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₈	—	67	3	no	—
117	C₂³×C₈	—	67	1	no	—
118	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
119	C₂×C₄⋊C₈	—	134	12	no	—
120	C₂×C₄⋊C₈	—	134	6	no	—
121	C₂×#95	Dic₆₇	1	3	yes	—
122	C₂×#95	Dic₆₇	1	1	yes	characteristic
123	#336	—	134	6	no	—
124	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
125	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
126	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
127	C₂²×C₈	—	67	6	no	—
128	C₂²×C₈	—	67	2	no	—
129	C₂²×C₈	—	134	6	no	—
130	C₂²×C₈	—	134	3	no	—
131	C₂²×C₈	—	134	3	no	—
132	C₂²×C₈	—	134	6	no	—
133	C₂²×C₈	—	134	3	no	—
134	C₂²×C₈	—	134	3	no	—
135	C₂×C₄²	—	2	6	no	—
136	C₂×C₄²	—	2	6	no	—
137	C₂×C₄²	Dic₁₃₄	1	6	yes	—
138	C₂×C₄²	D₂₆₈	1	6	yes	—
139	C₄⋊C₈	—	134	24	no	—
140	#95	—	2	6	no	—
141	#95	C₆₇⋊C₈	1	4	yes	complemented
142	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
143	C₂²×C₄	—	2	6	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	—	2	6	no	—
146	C₂²×C₄	—	2	6	no	—
147	C₂²×C₄	—	2	3	no	—
148	C₂²×C₄	—	2	3	no	—
149	C₂²×C₄	C₄⋊Dic₆₇	1	6	yes	—
150	C₂²×C₄	C₄⋊Dic₆₇	1	6	yes	—
151	C₂²×C₄	C₂×C₆₇⋊C₈	1	6	yes	—
152	C₂²×C₄	#11761612	1	6	yes	—
153	C₂²×C₄	D₄×D₆₇	1	3	yes	—
154	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
155	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₁₃₄	1	3	yes	—
157	C₂²×C₄	#11761627	1	3	yes	—
158	C₂²×C₄	#11761629	1	1	yes	characteristic
159	C₂×C₈	—	268	3	no	—
160	C₂×C₈	—	268	6	no	—
161	C₂×C₈	—	268	3	no	—
162	C₂×C₈	—	268	3	no	—
163	C₂×C₈	—	268	1	no	—
164	C₂×C₈	—	134	12	no	—
165	C₂×C₈	—	134	6	no	—
166	C₂×C₈	—	134	6	no	—
167	C₄²	—	2	6	no	—
168	C₄²	—	2	12	no	—
169	C₄²	—	2	6	no	—
170	C₂³	C₂×C₄⋊Dic₆₇	1	3	yes	—
171	C₂³	D₄×Dic₆₇	1	3	yes	—
172	C₂³	Q₈×Dic₆₇	1	1	yes	characteristic
173	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
174	C₂³	C₂×#11761612	1	3	yes	—
175	C₂³	#14524050	1	1	yes	characteristic
176	C₂³	#14524050	1	3	yes	—
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	4	6	no	—
185	C₂×C₄	—	4	6	no	—
186	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	12	yes	complemented
187	C₂×C₄	#14524013	1	3	yes	—
188	C₂×C₄	#14524018	1	3	yes	—
189	C₂×C₄	#14524020	1	6	yes	—
190	C₂×C₄	#14524023	1	3	yes	—
191	C₂×C₄	#14524027	1	3	yes	—
192	C₂×C₄	#14524034	1	3	yes	—
193	C₂×C₄	#14524040	1	1	yes	characteristic
194	C₂×C₄	#14524042	1	3	yes	—
195	C₂×C₄	#14524043	1	3	yes	—
196	C₈	—	268	6	no	—
197	C₈	—	268	2	no	—
198	C₂²	#16863266	1	1	yes	characteristic
199	C₂²	#16863271	1	3	yes	—
200	C₂²	#16863295	1	3	yes	—
201	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	3	yes	—
202	C₂²	#16863420	1	3	yes	—
203	C₂²	#16863423	1	3	yes	—
204	C₂²	#16863423	1	3	yes	—
205	C₂²	D₄×C₆₇⋊C₈	1	3	yes	—
206	C₂²	#16863468	1	6	yes	—
207	C₂²	#16863474	1	3	yes	—
208	C₂²	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
209	C₂²	#16863527	1	3	yes	—
210	C₄	—	2	12	no	—
211	C₄	—	4	6	no	—
212	C₄	#16863306	1	1	yes	characteristic
213	C₄	#16863317	1	3	yes	—
214	C₄	#16863328	1	3	yes	—
215	C₄	#16863333	1	1	yes	characteristic
216	C₂	#22558474	1	1	yes	characteristic
217	C₂	#22560361	1	1	yes	characteristic
218	C₂	#22560363	1	1	yes	characteristic
219	C₂	#22560389	1	3	yes	—
220	C₂	#22560394	1	3	yes	—
221	C₂	#22560928	1	3	yes	—
222	C₂	#22560933	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₆₇
commutator	C₂³×C₆₇	C₂²×C₈
Fitting	C₄×#95×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₄×#95	D₆₇
hypocenter	C₆₇	#11322
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
29. #95×C₆₇	C₈	semidirect
74. C₂×C₄×C₆₇	C₄⋊C₈	semidirect
115. C₆₇	#11322	semidirect
141. #95	C₆₇⋊C₈	semidirect
186. C₂×C₄	C₄⋊(C₆₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863151	2
C₄×#95×C₆₇	2
#11322	67