Subgroups — #30749580

Order 17152.

418 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	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22560326	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863149	C₂	1	2	yes	maximal
4	C₂×#16863147	C₂	1	2	yes	maximal
5	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863151	—	2	2	no	—
11	#16863149	—	2	4	no	—
12	#16863147	—	2	4	no	—
13	C₂×#97×C₆₇	C₄	1	2	yes	—
14	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
15	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
17	C₄³×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	D₄	1	2	yes	—
31	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	#97×C₆₇	—	2	4	no	—
34	#97×C₆₇	—	2	2	no	—
35	#97×C₆₇	C₈	1	4	yes	complemented
36	C₄×C₈×C₆₇	D₄	1	4	yes	—
37	C₄×C₈×C₆₇	Q₈	1	4	yes	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	1	no	—
40	C₂×C₆₇⋊C₈	—	4	1	no	—
41	C₂×C₆₇⋊C₈	—	4	4	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
55	C₄²×C₆₇	—	2	2	no	—
56	C₄²×C₆₇	—	2	2	no	—
57	C₄²×C₆₇	—	2	2	no	—
58	C₄²×C₆₇	—	4	2	no	—
59	C₄²×C₆₇	C₂×C₈	1	2	yes	—
60	C₄²×C₆₇	M₁₆	1	2	yes	—
61	C₄²×C₆₇	C₂×D₄	1	2	yes	—
62	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
63	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₆₇	#137	1	2	yes	—
80	C₈×C₆₇	—	2	4	no	—
81	C₈×C₆₇	C₄⋊C₈	1	8	yes	complemented
82	C₆₇⋊C₈	—	8	2	no	—
83	C₆₇⋊C₈	—	8	1	no	—
84	C₆₇⋊C₈	—	8	1	no	—
85	C₂²×C₆₇	#384	1	1	yes	characteristic
86	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₆₇	#423	1	1	yes	characteristic
88	C₂²×C₆₇	#428	1	2	yes	—
89	C₂²×C₆₇	#430	1	2	yes	—
90	C₄×C₆₇	—	2	4	no	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	4	2	no	—
93	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
94	C₄×C₆₇	#423	1	2	yes	—
95	C₄×C₆₇	#496	1	1	yes	characteristic
96	C₄×C₆₇	#497	1	1	yes	characteristic
97	C₄×C₆₇	#501	1	2	yes	—
98	#8808	—	67	1	no	maximal, Sylow
99	C₂×C₆₇	#1565	1	2	yes	—
100	C₂×C₆₇	#1569	1	2	yes	—
101	C₂×C₆₇	#1842	1	1	yes	characteristic
102	C₂×C₆₇	#1847	1	1	yes	characteristic
103	C₂×C₆₇	#1849	1	1	yes	characteristic
104	#1842	—	67	1	no	—
105	#1831	—	67	1	no	—
106	C₄×#97	D₆₇	1	1	yes	characteristic
107	C₂×#335	—	67	2	no	—
108	C₂×#334	—	67	2	no	—
109	C₆₇	#8808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊C₈	—	67	1	no	—
111	C₂×C₄⋊C₈	—	67	2	no	—
112	C₂×C₄⋊C₈	—	67	1	no	—
113	C₂×C₄⋊C₈	—	134	2	no	—
114	C₂×#97	Dic₆₇	1	2	yes	—
115	C₂×#97	Dic₆₇	1	1	yes	characteristic
116	C₂×#97	Dic₆₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
118	C₄³	C₂×D₆₇	1	1	yes	characteristic
119	#336	—	134	2	no	—
120	#335	—	134	4	no	—
121	#334	—	134	4	no	—
122	C₂²×C₈	—	134	1	no	—
123	C₂²×C₈	—	134	1	no	—
124	C₂²×C₈	—	134	2	no	—
125	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
126	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
127	C₂×C₄²	Dic₁₃₄	1	2	yes	—
128	C₂×C₄²	D₂₆₈	1	2	yes	—
129	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
131	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
132	C₄⋊C₈	—	268	4	no	—
133	C₄⋊C₈	—	134	2	no	—
134	C₄⋊C₈	—	134	2	no	—
135	C₄⋊C₈	—	134	2	no	—
136	C₄⋊C₈	—	134	2	no	—
137	#97	—	2	4	no	—
138	#97	—	2	2	no	—
139	#97	C₆₇⋊C₈	1	4	yes	complemented
140	C₄×C₈	Dic₁₃₄	1	4	yes	—
141	C₄×C₈	D₂₆₈	1	4	yes	—
142	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
143	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	268	2	no	—
150	C₂×C₈	—	268	1	no	—
151	C₂×C₈	—	268	1	no	—
152	C₂×C₈	—	268	4	no	—
153	C₂×C₈	—	268	2	no	—
154	C₂×C₈	—	268	2	no	—
155	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
156	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
157	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
158	C₂×C₈	#11761612	1	4	yes	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
164	C₄²	#11761612	1	2	yes	—
165	C₄²	C₂×D₂₆₈	1	2	yes	—
166	C₄²	C₂×Dic₁₃₄	1	2	yes	—
167	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
176	C₂×C₄	#14524040	1	1	yes	characteristic
177	C₂×C₄	#14524043	1	2	yes	—
178	C₂×C₄	#14524043	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×#11761612	1	2	yes	—
181	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
182	C₂×C₄	#14524071	1	2	yes	—
183	C₂×C₄	#14524076	1	2	yes	—
184	C₈	—	2	4	no	—
185	C₈	—	536	2	no	—
186	C₈	—	536	1	no	—
187	C₈	—	536	1	no	—
188	C₈	C₄⋊(C₆₇⋊C₈)	1	8	yes	complemented
189	C₂²	#16863266	1	1	yes	characteristic
190	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
191	C₂²	#16863420	1	1	yes	characteristic
192	C₂²	#16863438	1	2	yes	—
193	C₂²	#16863445	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
198	C₄	#16863420	1	2	yes	—
199	C₄	#16863785	1	1	yes	characteristic
200	C₄	#16863790	1	1	yes	characteristic
201	C₄	#16863805	1	2	yes	—
202	C₂	#22558961	1	2	yes	—
203	C₂	#22558976	1	2	yes	—
204	C₂	#22560361	1	1	yes	characteristic
205	C₂	#22560377	1	1	yes	characteristic
206	C₂	#22560384	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#97×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#97	D₆₇
hypocenter	C₆₇	#8808
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
35. #97×C₆₇	C₈	semidirect
81. C₈×C₆₇	C₄⋊C₈	semidirect
109. C₆₇	#8808	semidirect
139. #97	C₆₇⋊C₈	semidirect
188. C₈	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
#22560361	2
#22560326	2
C₂×#16863149	2
C₂×#16863147	2
C₄×#97×C₆₇	2
#8808	67