Subgroups — #30767101

Order 17152.

302 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561965	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558261	C₂	1	4	yes	maximal, complemented
4	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	—	2	1	no	—
6	#16863449	—	2	1	no	—
7	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
9	#16863149	—	2	1	no	—
10	#16863147	C₂²	1	2	yes	—
11	C₂×#125×C₆₇	—	2	1	no	—
12	#499×C₆₇	—	2	1	no	—
13	#499×C₆₇	—	2	1	no	—
14	#499×C₆₇	C₄	1	2	yes	—
15	#499×C₆₇	C₂²	1	2	yes	—
16	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
17	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#106×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#14523983	—	4	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
24	C₄×C₆₇⋊C₈	—	4	1	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	#125×C₆₇	—	2	2	no	—
28	#125×C₆₇	—	2	2	no	—
29	#125×C₆₇	—	4	1	no	—
30	#125×C₆₇	—	4	1	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
34	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	#107×C₆₇	—	2	1	no	—
37	#107×C₆₇	—	2	1	no	—
38	#107×C₆₇	C₂×C₄	1	2	yes	—
39	#107×C₆₇	C₂×C₄	1	2	yes	—
40	#106×C₆₇	—	2	2	no	—
41	#106×C₆₇	—	2	2	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₆₇⋊C₁₆	—	4	2	no	—
48	C₂²×C₄×C₆₇	—	4	1	no	—
49	C₂²×C₄×C₆₇	—	4	1	no	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
57	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₄⋊C₄×C₆₇	—	4	1	no	—
63	C₄⋊C₄×C₆₇	—	4	1	no	—
64	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₆₇	D₈	1	1	yes	characteristic
70	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
71	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₆₇	#102	1	1	yes	characteristic
73	C₂×C₄×C₆₇	—	4	1	no	—
74	C₂×C₄×C₆₇	—	4	1	no	—
75	C₂×C₄×C₆₇	—	4	2	no	—
76	C₂×C₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	—	8	2	no	—
78	C₂×C₄×C₆₇	—	8	2	no	—
79	C₂×C₄×C₆₇	#101	1	2	yes	—
80	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#103	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₆₇	—	2	1	no	—
88	C₈×C₆₇	—	2	1	no	—
89	C₈×C₆₇	#103	1	2	yes	—
90	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
92	C₆₇⋊C₈	—	8	2	no	—
93	C₂²×C₆₇	—	2	1	no	—
94	C₂²×C₆₇	—	2	1	no	—
95	C₂²×C₆₇	#332	1	2	yes	—
96	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
97	C₄×C₆₇	—	2	1	no	—
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	8	2	no	—
102	C₄×C₆₇	—	8	2	no	—
103	C₄×C₆₇	#332	1	2	yes	—
104	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
105	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
106	#11935	—	67	1	no	maximal, Sylow
107	C₂×C₆₇	—	2	2	no	—
108	C₂×C₆₇	C₂×#332	1	1	yes	characteristic
109	C₂×C₆₇	#2143	1	1	yes	characteristic
110	C₂×C₆₇	#2145	1	1	yes	characteristic
111	C₂×#499	D₆₇	1	1	yes	characteristic
112	#2150	—	67	1	no	—
113	#1564	—	67	1	no	—
114	#1363	—	67	4	no	—
115	C₆₇	#11935	1	1	yes	characteristic, complemented, Sylow
116	C₂×#125	—	2	1	no	—
117	C₂×M₃₂	—	134	1	no	—
118	#499	—	2	1	no	—
119	#499	—	2	1	no	—
120	#499	Dic₆₇	1	2	yes	—
121	#499	C₂×D₆₇	1	2	yes	—
122	#431	—	134	1	no	—
123	C₂×#107	Dic₆₇	1	1	yes	characteristic
124	C₂×#107	Dic₆₇	1	1	yes	characteristic
125	C₂×#106	—	2	1	no	—
126	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
127	C₄⋊C₁₆	—	67	2	no	—
128	C₄⋊C₁₆	—	134	1	no	—
129	#335	—	134	1	no	—
130	#334	—	67	2	no	—
131	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
133	#125	—	2	2	no	—
134	#125	—	2	2	no	—
135	#125	—	4	1	no	—
136	#125	—	4	1	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
142	M₃₂	—	268	2	no	—
143	C₂×C₁₆	—	134	2	no	—
144	#107	—	2	1	no	—
145	#107	—	2	1	no	—
146	#107	C₂×Dic₆₇	1	2	yes	—
147	#107	C₂×Dic₆₇	1	2	yes	—
148	#106	—	2	2	no	—
149	#106	—	2	2	no	—
150	C₄⋊C₈	—	134	2	no	—
151	C₂²⋊C₈	—	268	1	no	—
152	C₄×C₈	—	268	1	no	—
153	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
157	C₂²×C₄	—	4	1	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
160	C₂²×C₄	#11761613	1	1	yes	characteristic
161	C₂²×C₄	#11761616	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	268	2	no	—
164	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
166	C₂×C₈	#11761615	1	2	yes	—
167	C₂×C₈	#11761618	1	2	yes	—
168	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
179	C₄²	#11761613	1	1	yes	characteristic
180	C₄²	#11761616	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
182	C₁₆	—	268	2	no	—
183	C₂³	#14523964	1	1	yes	characteristic
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	#14523962	1	2	yes	—
191	C₂×C₄	#14523964	1	1	yes	characteristic
192	C₂×C₄	#14523964	1	1	yes	characteristic
193	C₂×C₄	#14523964	1	1	yes	characteristic
194	C₂×C₄	#14523967	1	2	yes	—
195	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
197	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	536	2	no	—
201	C₈	#14523967	1	2	yes	—
202	C₈	C₂×#11761615	1	1	yes	characteristic
203	C₈	C₂×#11761618	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	#16863142	1	2	yes	—
207	C₂²	C₂×#14523964	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	2	no	—
213	C₄	—	8	2	no	—
214	C₄	#16863142	1	2	yes	—
215	C₄	C₂×#14523962	1	1	yes	characteristic
216	C₄	C₂×#14523967	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	C₂×#16863142	1	1	yes	characteristic
219	C₂	#22561937	1	1	yes	characteristic
220	C₂	#22561945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	#11935
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
3. #22558261	C₂	semidirect
115. C₆₇	#11935	semidirect

Maximal subgroups

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

subgroup	index
#22561965	2
#22558957	2
#22558261	2
C₂×#499×C₆₇	2
#11935	67