Subgroups — #30747085

Order 17152.

398 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523975	C₂	1	1	yes	maximal, characteristic
2	#22560076	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
4	#22558049	C₂	1	2	yes	maximal
5	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523975	C₄	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14523953	—	2	2	no	—
9	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	#16863159	C₄	1	2	yes	—
11	#16863150	—	2	2	no	—
12	#16863149	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#322×C₆₇	C₄	1	4	yes	complemented
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₆₇	—	2	2	no	—
20	#14523975	—	2	4	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
23	#14523953	—	4	4	no	—
24	C₆₇⋊#95	—	2	2	no	—
25	C₂²×C₈×C₆₇	—	2	1	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₆₇	—	2	1	no	—
31	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₇	Q₈	1	2	yes	—
36	C₂²×Dic₆₇	—	2	4	no	—
37	C₂×C₆₇⋊C₈	—	2	4	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₄⋊Dic₆₇	—	2	4	no	—
41	C₄⋊Dic₆₇	—	2	4	no	—
42	C₄×Dic₆₇	—	4	2	no	—
43	C₄×Dic₆₇	—	4	2	no	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	4	2	no	—
52	C₂×C₈×C₆₇	C₄²	1	2	yes	—
53	C₂×C₈×C₆₇	C₄²	1	2	yes	—
54	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
56	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₆₇	#41	1	1	yes	characteristic
60	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄×D₄	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	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₆₇	#131	1	2	yes	—
70	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₆₇	#137	1	2	yes	—
72	C₂×Dic₆₇	—	4	4	no	—
73	C₂×Dic₆₇	—	4	4	no	—
74	C₂×Dic₆₇	—	4	4	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	2	4	no	—
77	C₈×C₆₇	—	4	2	no	—
78	C₈×C₆₇	#97	1	4	yes	complemented
79	C₆₇⋊C₈	—	4	4	no	—
80	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₆₇	#436	1	2	yes	—
82	C₂²×C₆₇	#440	1	2	yes	—
83	C₂²×C₆₇	#444	1	2	yes	—
84	C₄×C₆₇	—	2	1	no	—
85	C₄×C₆₇	—	2	1	no	—
86	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
87	C₄×C₆₇	#405	1	1	yes	characteristic
88	C₄×C₆₇	#428	1	2	yes	—
89	C₄×C₆₇	#441	1	1	yes	characteristic
90	C₄×C₆₇	#442	1	1	yes	characteristic
91	C₄×C₆₇	#446	1	2	yes	—
92	Dic₆₇	—	8	4	no	—
93	#8286	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	#1592	1	4	yes	—
95	C₂×C₆₇	#1770	1	1	yes	characteristic
96	C₂×C₆₇	#1773	1	1	yes	characteristic
97	C₂×C₆₇	#1776	1	1	yes	characteristic
98	C₄×#106	—	67	1	no	—
99	#1770	—	67	2	no	—
100	C₂×#335	—	67	1	no	—
101	C₂×#322	D₆₇	1	1	yes	characteristic
102	#1279	—	67	2	no	—
103	C₆₇	#8286	1	1	yes	characteristic, complemented, Sylow
104	C₂×#106	—	67	2	no	—
105	C₂×C₄⋊C₈	—	67	2	no	—
106	C₂×#97	—	134	2	no	—
107	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	67	2	no	—
111	#340	—	67	2	no	—
112	#336	—	134	2	no	—
113	#335	—	67	4	no	—
114	#322	Dic₆₇	1	4	yes	complemented
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	134	2	no	—
117	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
119	C₂²×C₈	D₂₆₈	1	2	yes	—
120	C₂×C₄⋊C₄	—	67	4	no	—
121	C₂×C₄²	—	134	2	no	—
122	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
123	#106	—	134	4	no	—
124	C₄⋊C₈	—	67	4	no	—
125	C₄⋊C₈	—	134	2	no	—
126	#97	—	268	4	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	Dic₁₃₄	1	2	yes	—
129	C₄×C₈	C₄×D₆₇	1	2	yes	—
130	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
131	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
132	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
133	#95	—	134	2	no	—
134	C₂²×C₄	—	134	4	no	—
135	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
137	C₂²×C₄	#11761628	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	268	2	no	—
144	C₂×C₈	—	268	2	no	—
145	C₂×C₈	—	134	4	no	—
146	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
147	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
148	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
149	C₂×C₈	#11761611	1	4	yes	—
150	C₄⋊C₄	—	134	4	no	—
151	C₄⋊C₄	—	134	4	no	—
152	C₄²	—	268	2	no	—
153	C₄²	—	268	2	no	—
154	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
155	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
156	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
157	C₄²	#11761628	1	1	yes	characteristic
158	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	268	4	no	—
161	C₂×C₄	—	268	4	no	—
162	C₂×C₄	—	268	4	no	—
163	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
165	C₂×C₄	#14524002	1	1	yes	characteristic
166	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
167	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
169	C₂×C₄	C₂×#11761611	1	2	yes	—
170	C₂×C₄	#14524049	1	2	yes	—
171	C₂×C₄	#14524071	1	2	yes	—
172	C₂×C₄	#14524076	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	4	no	—
175	C₈	—	4	2	no	—
176	C₈	—	268	4	no	—
177	C₈	#14523953	1	4	yes	complemented
178	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
179	C₂²	#16863475	1	2	yes	—
180	C₂²	#16863498	1	2	yes	—
181	C₂²	#16863519	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	536	4	no	—
185	C₄	C₂×#14523953	1	1	yes	characteristic
186	C₄	#16863351	1	1	yes	characteristic
187	C₄	#16863438	1	2	yes	—
188	C₄	#16863504	1	1	yes	characteristic
189	C₄	#16863509	1	1	yes	characteristic
190	C₄	#16863526	1	2	yes	—
191	C₂	#22559113	1	4	yes	—
192	C₂	#22560076	1	1	yes	characteristic
193	C₂	#22560093	1	1	yes	characteristic
194	C₂	#22560106	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#8286
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
12. #16863149	C₄	semidirect
16. #322×C₆₇	C₄	semidirect
78. C₈×C₆₇	#97	semidirect
103. C₆₇	#8286	semidirect
114. #322	Dic₆₇	semidirect
177. C₈	#14523953	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523975	2
#22560076	2
C₂×#16863149	2
#22558049	2
C₂×#322×C₆₇	2
#8286	67