Subgroups — #30864694

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 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	#22560962	C₂	1	4	yes	maximal
5	#22560381	C₂	1	1	yes	maximal, characteristic
6	#22560093	C₂	1	4	yes	maximal
7	C₄×#97×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×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
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	#97×C₆₇	—	2	4	no	—
45	#97×C₆₇	Q₈	1	8	yes	complemented
46	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	4	yes	—
50	C₂×Dic₁₃₄	—	2	4	no	—
51	#11761612	—	2	8	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₄⋊Dic₆₇	—	2	4	no	—
54	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
55	Dic₆₇⋊C₄	—	2	8	no	—
56	C₂²⋊Dic₆₇	—	2	4	no	—
57	C₄×Dic₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	#41	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
67	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
68	C₂×C₈×C₆₇	#41	1	4	yes	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	4	4	no	—
71	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×D₄	1	2	yes	—
74	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
75	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
85	C₂×Dic₆₇	—	2	8	no	—
86	Dic₁₃₄	—	4	8	no	—
87	C₈×C₆₇	—	2	2	no	—
88	C₈×C₆₇	C₄×Q₈	1	4	yes	—
89	C₈×C₆₇	C₄×Q₈	1	4	yes	—
90	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
91	C₆₇⋊C₈	—	4	4	no	—
92	C₂²×C₆₇	—	2	2	no	—
93	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
94	C₂²×C₆₇	#533	1	2	yes	—
95	C₄×C₆₇	—	2	2	no	—
96	C₄×C₆₇	—	4	4	no	—
97	C₄×C₆₇	#444	1	4	yes	—
98	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
99	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
100	C₄×C₆₇	#532	1	1	yes	characteristic
101	Dic₆₇	—	4	8	no	—
102	#21734	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	—	2	2	no	—
104	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#444	1	2	yes	—
106	#3160	—	67	1	no	—
107	#2960	—	67	2	no	—
108	#1994	—	67	2	no	—
109	#1946	—	67	4	no	—
110	#1848	—	67	1	no	—
111	#1773	—	67	4	no	—
112	C₄×#97	D₆₇	1	1	yes	characteristic
113	C₆₇	#21734	1	1	yes	characteristic, complemented, Sylow
114	#533	—	67	2	no	—
115	#500	—	67	2	no	—
116	#499	—	67	4	no	—
117	#498	—	67	2	no	—
118	#446	—	134	4	no	—
119	C₈×Q₈	—	134	4	no	—
120	C₂×#108	—	67	4	no	—
121	#427	—	67	4	no	—
122	#424	—	67	4	no	—
123	#423	—	67	2	no	—
124	#405	—	67	4	no	—
125	C₂×#97	C₂×D₆₇	1	2	yes	—
126	C₂×#97	C₂×D₆₇	1	2	yes	—
127	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₄³	C₂×D₆₇	1	1	yes	characteristic
130	#339	—	67	8	no	—
131	C₂×M₁₆	—	67	4	no	—
132	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
134	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	67	4	no	—
136	#125	—	67	4	no	—
137	C₂²⋊Q₈	—	134	4	no	—
138	C₄×Q₈	—	134	4	no	—
139	#117	—	67	4	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
142	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
143	#108	—	134	8	no	—
144	#107	—	67	8	no	—
145	#106	—	67	8	no	—
146	C₄⋊C₈	—	134	8	no	—
147	C₄⋊C₈	—	134	4	no	—
148	#97	—	2	4	no	—
149	#97	—	134	4	no	—
150	#97	Dic₁₃₄	1	8	yes	complemented
151	C₄×C₈	—	134	4	no	—
152	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
153	C₄×C₈	C₄×D₆₇	1	2	yes	—
154	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
155	C₄×C₈	D₂₆₈	1	4	yes	—
156	C₂×Q₈	—	134	4	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	#11761628	1	2	yes	—
159	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
160	M₁₆	—	134	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	134	4	no	—
163	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
164	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
165	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
166	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
167	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
168	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
169	C₂×C₈	#11761628	1	4	yes	—
170	C₄⋊C₄	—	67	8	no	—
171	C₄⋊C₄	—	134	8	no	—
172	C₄⋊C₄	—	134	4	no	—
173	C₂²⋊C₄	—	134	4	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	4	no	—
176	C₄²	—	134	4	no	—
177	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
179	C₄²	C₂×D₂₆₈	1	2	yes	—
180	C₄²	C₂×Dic₁₃₄	1	4	yes	—
181	C₂³	C₂×#11761628	1	1	yes	characteristic
182	Q₈	—	268	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	134	8	no	—
187	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
188	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
189	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
191	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
192	C₂×C₄	C₂×#11761628	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	268	4	no	—
195	C₈	C₄×Dic₁₃₄	1	4	yes	—
196	C₈	C₄×Dic₁₃₄	1	4	yes	—
197	C₈	#14524043	1	4	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
200	C₂²	#16863953	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	4	4	no	—
203	C₄	—	268	8	no	—
204	C₄	#16863519	1	4	yes	—
205	C₄	C₂×C₄×Dic₁₃₄	1	2	yes	—
206	C₄	C₂×#14524043	1	1	yes	characteristic
207	C₄	#16863946	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₄×#14524043	1	1	yes	characteristic
210	C₂	C₂×#16863519	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#97×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#97	D₆₇
hypocenter	C₆₇	#21734
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#97

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
45. #97×C₆₇	Q₈	semidirect
113. C₆₇	#21734	semidirect
150. #97	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
#22560962	2
#22560381	2
#22560093	2
C₄×#97×C₆₇	2
#21734	67