Subgroups — #30749599

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560369	C₂	1	1	yes	maximal, characteristic
2	#22560365	C₂	1	2	yes	maximal
3	#22560336	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	2	no	—
7	C₂×#14523974	—	2	4	no	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#16863266	C₂²	1	2	yes	—
11	#16863158	—	2	4	no	—
12	C₈×C₆₇⋊C₈	C₄	1	4	yes	complemented
13	C₂×#107×C₆₇	—	2	2	no	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#384×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
17	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	#14523977	—	4	4	no	—
22	#14523974	—	4	8	no	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
26	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₆₇	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	#107×C₆₇	—	4	4	no	—
33	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₇	Q₈	1	2	yes	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	2	no	—
38	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
39	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
40	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
41	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	—
42	C₄⋊Dic₆₇	—	4	4	no	—
43	C₄⋊Dic₆₇	—	4	4	no	—
44	C₄⋊Dic₆₇	—	4	4	no	—
45	C₂²×C₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
50	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
51	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
52	C₄⋊C₄×C₆₇	—	4	4	no	—
53	C₄⋊C₄×C₆₇	—	4	4	no	—
54	C₄⋊C₄×C₆₇	—	4	4	no	—
55	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₆₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	8	2	no	—
61	C₂×C₄×C₆₇	—	8	2	no	—
62	C₂×C₄×C₆₇	—	8	2	no	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₆₇	#135	1	2	yes	—
74	C₂×C₄×C₆₇	#135	1	2	yes	—
75	C₂×Dic₆₇	—	8	2	no	—
76	C₂×Dic₆₇	—	8	2	no	—
77	C₂×Dic₆₇	—	8	2	no	—
78	C₈×C₆₇	#106	1	8	yes	complemented
79	C₆₇⋊C₈	—	2	4	no	—
80	C₆₇⋊C₈	#107	1	8	yes	complemented
81	C₂²×C₆₇	#384	1	1	yes	characteristic
82	C₂²×C₆₇	#492	1	1	yes	characteristic
83	C₂²×C₆₇	#492	1	1	yes	characteristic
84	C₂²×C₆₇	#493	1	1	yes	characteristic
85	C₂²×C₆₇	#494	1	1	yes	characteristic
86	C₂²×C₆₇	#495	1	1	yes	characteristic
87	C₂²×C₆₇	#495	1	1	yes	characteristic
88	C₄×C₆₇	—	16	2	no	—
89	C₄×C₆₇	C₂×#106	1	2	yes	—
90	C₄×C₆₇	C₂×#107	1	2	yes	—
91	C₄×C₆₇	#427	1	2	yes	—
92	C₄×C₆₇	#498	1	2	yes	—
93	C₄×C₆₇	#499	1	2	yes	—
94	C₄×C₆₇	#500	1	2	yes	—
95	Dic₆₇	—	16	2	no	—
96	#8814	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	#1706	1	1	yes	characteristic
98	C₂×C₆₇	#1706	1	1	yes	characteristic
99	C₂×C₆₇	#1707	1	1	yes	characteristic
100	C₂×C₆₇	#1707	1	1	yes	characteristic
101	C₂×C₆₇	#1844	1	1	yes	characteristic
102	C₂×C₆₇	#1845	1	1	yes	characteristic
103	C₂×C₆₇	#1846	1	1	yes	characteristic
104	#1845	—	67	1	no	—
105	#1845	D₆₇	1	1	yes	characteristic
106	#1844	—	67	2	no	—
107	#1834	—	67	2	no	—
108	C₂×C₈²	—	67	1	no	—
109	C₆₇	#8814	1	1	yes	characteristic, complemented, Sylow
110	C₂×#107	—	2	2	no	—
111	C₂×#107	—	134	2	no	—
112	C₂×#106	—	134	4	no	—
113	C₂×C₄×C₈	—	67	1	no	—
114	C₂×C₄×C₈	—	67	1	no	—
115	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
116	#384	—	67	2	no	—
117	#384	C₂×D₆₇	1	2	yes	—
118	#340	—	134	4	no	—
119	C₈²	—	67	4	no	—
120	C₂²×C₈	—	67	2	no	—
121	C₂²×C₈	—	67	2	no	—
122	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	134	2	no	—
127	C₂×C₄⋊C₄	—	134	2	no	—
128	C₂×C₄⋊C₄	—	134	2	no	—
129	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	#107	—	4	4	no	—
131	#107	—	268	4	no	—
132	#106	—	268	8	no	—
133	C₄×C₈	—	67	2	no	—
134	C₄×C₈	—	67	2	no	—
135	C₄×C₈	—	67	2	no	—
136	C₄×C₈	—	67	2	no	—
137	C₄×C₈	Dic₁₃₄	1	2	yes	—
138	C₄×C₈	C₄×D₆₇	1	2	yes	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	268	2	no	—
141	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
142	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂×C₈	—	67	4	no	—
145	C₂×C₈	—	67	4	no	—
146	C₂×C₈	—	67	4	no	—
147	C₂×C₈	—	67	4	no	—
148	C₂×C₈	—	134	2	no	—
149	C₂×C₈	—	134	2	no	—
150	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
151	C₂×C₈	#11761615	1	4	yes	—
152	C₂×C₈	#11761616	1	4	yes	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	268	4	no	—
157	C₄⋊C₄	—	268	4	no	—
158	C₄⋊C₄	—	268	4	no	—
159	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
160	C₄²	Q₈×D₆₇	1	1	yes	characteristic
161	C₄²	Q₈×D₆₇	1	1	yes	characteristic
162	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
163	C₂³	#14524039	1	1	yes	characteristic
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	536	2	no	—
168	C₂×C₄	—	536	2	no	—
169	C₂×C₄	—	536	2	no	—
170	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
171	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#14524041	1	1	yes	characteristic
174	C₂×C₄	#14524042	1	1	yes	characteristic
175	C₂×C₄	#14524042	1	1	yes	characteristic
176	C₂×C₄	D₈×D₆₇	1	2	yes	—
177	C₂×C₄	C₂×#11761615	1	2	yes	—
178	C₂×C₄	C₂×#11761616	1	2	yes	—
179	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
180	C₂×C₄	#14524062	1	2	yes	—
181	C₂×C₄	#14524066	1	2	yes	—
182	C₈	—	67	8	no	—
183	C₈	—	134	4	no	—
184	C₈	#14523974	1	8	yes	complemented
185	C₂²	#16863265	1	1	yes	characteristic
186	C₂²	#16863768	1	1	yes	characteristic
187	C₂²	#16863770	1	1	yes	characteristic
188	C₂²	#16863771	1	1	yes	characteristic
189	C₂²	#16863774	1	1	yes	characteristic
190	C₂²	#16863779	1	1	yes	characteristic
191	C₂²	#16863781	1	1	yes	characteristic
192	C₄	—	16	2	no	—
193	C₄	—	1072	2	no	—
194	C₄	C₂×#14523974	1	2	yes	—
195	C₄	#107×D₆₇	1	2	yes	—
196	C₄	#16863433	1	2	yes	—
197	C₄	#16863792	1	2	yes	—
198	C₄	#16863797	1	2	yes	—
199	C₄	#16863800	1	2	yes	—
200	C₂	#22559798	1	1	yes	characteristic
201	C₂	#22559802	1	1	yes	characteristic
202	C₂	#22559803	1	1	yes	characteristic
203	C₂	#22559807	1	1	yes	characteristic
204	C₂	#22560365	1	1	yes	characteristic
205	C₂	#22560367	1	1	yes	characteristic
206	C₂	#22560372	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#8814
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
12. C₈×C₆₇⋊C₈	C₄	semidirect
78. C₈×C₆₇	#106	semidirect
80. C₆₇⋊C₈	#107	semidirect
109. C₆₇	#8814	semidirect
184. C₈	#14523974	semidirect

Maximal subgroups

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

subgroup	index
#22560369	2
#22560365	2
#22560336	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1845	2
#8814	67