Subgroups — #30864678

Order 17152.

702 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	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	2	yes	maximal
3	#22561186	C₂	1	2	yes	maximal
4	#22560957	C₂	1	4	yes	maximal
5	#22560381	C₂	1	1	yes	maximal, characteristic
6	#22560088	C₂	1	4	yes	maximal
7	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₂²	1	2	yes	—
9	#16863801	C₄	1	2	yes	complemented
10	#16863798	C₄	1	4	yes	complemented
11	#16863794	C₄	1	2	yes	complemented
12	#16863526	—	2	4	no	—
13	C₈×Dic₁₃₄	—	2	4	no	—
14	C₂×#14523979	C₂²	1	4	yes	—
15	#16863434	C₂²	1	4	yes	—
16	#16863422	C₂²	1	4	yes	—
17	#16863420	C₂²	1	2	yes	—
18	#16863351	C₂²	1	4	yes	—
19	#16863156	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄³×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	C₂³	1	4	yes	—
26	#14524043	C₂×C₄	1	4	yes	—
27	#14524035	C₂×C₄	1	4	yes	—
28	#14524022	—	2	4	no	—
29	C₄×Dic₁₃₄	—	2	4	no	—
30	#14524002	C₂³	1	4	yes	—
31	#14523979	—	2	8	no	—
32	#14523977	C₂×C₄	1	8	yes	—
33	#14523975	C₂×C₄	1	8	yes	—
34	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
35	Dic₆₇⋊C₈	—	2	8	no	—
36	#14523953	—	2	4	no	—
37	C₈×Dic₆₇	—	2	4	no	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₆₇	—	2	4	no	—
45	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	D₄	1	2	yes	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
51	C₂×Dic₁₃₄	—	2	4	no	—
52	#11761612	—	2	8	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₄⋊Dic₆₇	—	2	4	no	—
55	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
56	Dic₆₇⋊C₄	—	2	8	no	—
57	C₂²⋊Dic₆₇	—	2	4	no	—
58	C₄×Dic₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	#41	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₆₇	#41	1	4	yes	—
71	C₄²×C₆₇	—	2	8	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂×D₄	1	2	yes	—
76	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
77	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	2	8	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
87	C₂×Dic₆₇	—	2	8	no	—
88	Dic₁₃₄	—	4	8	no	—
89	C₈×C₆₇	—	2	2	no	—
90	C₈×C₆₇	C₄×Q₈	1	4	yes	—
91	C₈×C₆₇	C₄×Q₈	1	4	yes	—
92	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
93	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
94	C₆₇⋊C₈	—	4	4	no	—
95	C₂²×C₆₇	—	2	2	no	—
96	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²×C₆₇	#533	1	2	yes	—
98	C₄×C₆₇	—	2	8	no	—
99	C₄×C₆₇	—	2	2	no	—
100	C₄×C₆₇	#443	1	4	yes	—
101	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
102	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₆₇	#532	1	1	yes	characteristic
104	Dic₆₇	—	4	8	no	—
105	#21732	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	—	2	2	no	—
107	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#443	1	2	yes	—
109	#3160	—	67	1	no	—
110	#2960	—	67	2	no	—
111	#1994	—	67	2	no	—
112	#1945	—	67	4	no	—
113	#1848	—	67	1	no	—
114	#1772	—	67	4	no	—
115	C₄²×C₈	D₆₇	1	1	yes	characteristic
116	C₆₇	#21732	1	1	yes	characteristic, complemented, Sylow
117	#533	—	67	2	no	—
118	#500	—	67	2	no	—
119	#499	—	67	4	no	—
120	#498	—	67	2	no	—
121	#446	—	134	4	no	—
122	C₈×Q₈	—	134	4	no	—
123	C₂×#108	—	67	4	no	—
124	#427	—	67	4	no	—
125	#424	—	67	4	no	—
126	#423	—	67	2	no	—
127	#405	—	67	4	no	—
128	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	C₄³	C₂×D₆₇	1	1	yes	characteristic
133	#339	—	67	8	no	—
134	C₂×M₁₆	—	67	4	no	—
135	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
137	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	67	4	no	—
139	#125	—	67	4	no	—
140	C₂²⋊Q₈	—	134	4	no	—
141	C₄×Q₈	—	134	4	no	—
142	#117	—	67	4	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
145	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
146	#108	—	134	8	no	—
147	#107	—	67	8	no	—
148	#106	—	67	8	no	—
149	C₄⋊C₈	—	134	8	no	—
150	C₄⋊C₈	—	134	4	no	—
151	#97	—	134	4	no	—
152	C₄×C₈	—	2	4	no	—
153	C₄×C₈	—	134	4	no	—
154	C₄×C₈	Dic₁₃₄	1	8	yes	complemented
155	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
156	C₄×C₈	C₄×D₆₇	1	2	yes	—
157	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
158	C₄×C₈	D₂₆₈	1	2	yes	—
159	C₄×C₈	D₂₆₈	1	2	yes	—
160	C₂×Q₈	—	134	4	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	#11761628	1	2	yes	—
163	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
164	M₁₆	—	134	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	134	4	no	—
167	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
168	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
169	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
170	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
171	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
172	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
173	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
174	C₂×C₈	#11761628	1	4	yes	—
175	C₄⋊C₄	—	67	8	no	—
176	C₄⋊C₄	—	134	8	no	—
177	C₄⋊C₄	—	134	4	no	—
178	C₂²⋊C₄	—	134	4	no	—
179	C₄²	—	2	8	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	134	4	no	—
182	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
183	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
184	C₄²	C₂×D₂₆₈	1	2	yes	—
185	C₄²	C₂×Dic₁₃₄	1	4	yes	—
186	C₂³	C₂×#11761628	1	1	yes	characteristic
187	Q₈	—	268	8	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	134	8	no	—
192	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
193	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
194	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
197	C₂×C₄	C₂×#11761628	1	2	yes	—
198	C₈	—	2	2	no	—
199	C₈	—	268	4	no	—
200	C₈	C₄×Dic₁₃₄	1	4	yes	—
201	C₈	C₄×Dic₁₃₄	1	4	yes	—
202	C₈	#14524043	1	2	yes	—
203	C₈	#14524043	1	2	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
206	C₂²	#16863953	1	2	yes	—
207	C₄	—	2	8	no	—
208	C₄	—	2	2	no	—
209	C₄	—	268	8	no	—
210	C₄	#16863514	1	4	yes	—
211	C₄	C₂×C₄×Dic₁₃₄	1	2	yes	—
212	C₄	C₂×#14524043	1	1	yes	characteristic
213	C₄	#16863946	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₄×#14524043	1	1	yes	characteristic
216	C₂	C₂×#16863514	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#21732
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
9. #16863801	C₄	semidirect
10. #16863798	C₄	semidirect
11. #16863794	C₄	semidirect
50. C₄×C₈×C₆₇	Q₈	semidirect
116. C₆₇	#21732	semidirect
154. C₄×C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22569143	2
#22561186	2
#22560957	2
#22560381	2
#22560088	2
C₄²×C₈×C₆₇	2
#21732	67