Subgroups — #30749585

Order 17152.

402 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560360	C₂	1	2	yes	maximal
2	#22559954	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863148	C₂	1	2	yes	maximal
5	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₆₇⋊C₈	—	2	4	no	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
10	#16863150	C₄	1	2	yes	—
11	#16863150	C₄	1	2	yes	—
12	#16863149	—	2	2	no	—
13	#16863148	—	2	4	no	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#335×C₆₇	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
21	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	Dic₆₇⋊C₈	—	4	8	no	—
25	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
26	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	—	2	2	no	—
32	C₄⋊C₈×C₆₇	—	2	2	no	—
33	C₄×C₈×C₆₇	D₄	1	2	yes	—
34	C₄×C₈×C₆₇	Q₈	1	2	yes	—
35	C₂²×Dic₆₇	C₄⋊C₄	1	4	yes	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	2	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	4	no	—
40	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
41	C₄×Dic₆₇	—	2	2	no	—
42	C₄×Dic₆₇	—	2	2	no	—
43	C₄×Dic₆₇	—	4	2	no	—
44	C₄×Dic₆₇	D₈	1	4	yes	—
45	C₄×Dic₆₇	Q₁₆	1	4	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	—	4	2	no	—
52	C₂×C₈×C₆₇	—	4	4	no	—
53	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×D₄	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	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#131	1	2	yes	—
68	C₂×C₄×C₆₇	#131	1	2	yes	—
69	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
71	C₂×Dic₆₇	—	2	4	no	—
72	C₂×Dic₆₇	—	2	4	no	—
73	C₂×Dic₆₇	#107	1	8	yes	—
74	C₈×C₆₇	—	2	2	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	8	2	no	—
77	C₆₇⋊C₈	—	2	4	no	—
78	C₆₇⋊C₈	—	8	2	no	—
79	C₂²×C₆₇	#384	1	1	yes	characteristic
80	C₂²×C₆₇	#424	1	2	yes	—
81	C₂²×C₆₇	C₂×#107	1	2	yes	—
82	C₂²×C₆₇	#430	1	2	yes	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	2	1	no	—
85	C₄×C₆₇	#424	1	2	yes	—
86	C₄×C₆₇	#424	1	1	yes	characteristic
87	C₄×C₆₇	#424	1	1	yes	characteristic
88	C₄×C₆₇	#493	1	1	yes	characteristic
89	C₄×C₆₇	#494	1	1	yes	characteristic
90	C₄×C₆₇	#500	1	2	yes	—
91	Dic₆₇	—	2	8	no	—
92	#8810	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	#1571	1	4	yes	—
94	C₂×C₆₇	#1843	1	1	yes	characteristic
95	C₂×C₆₇	#1845	1	1	yes	characteristic
96	C₂×C₆₇	#1849	1	1	yes	characteristic
97	#1842	—	67	2	no	—
98	#1744	—	67	1	no	—
99	C₂×#335	—	67	2	no	—
100	C₂×#335	—	67	1	no	—
101	C₂×#335	D₆₇	1	1	yes	characteristic
102	C₆₇	#8810	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊C₈	—	67	2	no	—
104	C₂×C₄⋊C₈	—	134	4	no	—
105	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
106	C₂×C₄×C₈	—	67	1	no	—
107	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
108	C₄³	—	67	1	no	—
109	#336	—	67	2	no	—
110	#336	—	67	2	no	—
111	#335	—	2	2	no	—
112	#335	—	134	4	no	—
113	#335	—	134	2	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	67	2	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	C₂×C₄²	—	67	2	no	—
120	C₂×C₄²	—	67	2	no	—
121	C₂×C₄²	—	67	2	no	—
122	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	268	8	no	—
126	C₄⋊C₈	—	134	2	no	—
127	C₄⋊C₈	—	134	2	no	—
128	C₄×C₈	—	67	2	no	—
129	C₄×C₈	—	67	2	no	—
130	C₄×C₈	Dic₁₃₄	1	2	yes	—
131	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
132	C₂²×C₄	—	67	4	no	—
133	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	268	2	no	—
141	C₂×C₈	—	268	4	no	—
142	C₂×C₈	—	67	4	no	—
143	C₂×C₈	—	134	2	no	—
144	C₂×C₈	—	134	2	no	—
145	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
146	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
147	C₄²	—	268	2	no	—
148	C₄²	—	67	4	no	—
149	C₄²	—	67	4	no	—
150	C₄²	—	134	2	no	—
151	C₄²	—	134	2	no	—
152	C₄²	D₄×D₆₇	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₄²	Q₈×D₆₇	1	1	yes	characteristic
155	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
156	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	67	8	no	—
159	C₂×C₄	—	134	4	no	—
160	C₂×C₄	—	134	4	no	—
161	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#14524039	1	1	yes	characteristic
164	C₂×C₄	#14524041	1	1	yes	characteristic
165	C₂×C₄	#14524042	1	1	yes	characteristic
166	C₂×C₄	#14524042	1	1	yes	characteristic
167	C₂×C₄	#14524049	1	2	yes	—
168	C₂×C₄	#14524051	1	2	yes	—
169	C₂×C₄	D₈×D₆₇	1	2	yes	—
170	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	536	2	no	—
175	C₈	—	134	4	no	—
176	C₂²	#16863265	1	1	yes	characteristic
177	C₂²	#16863421	1	2	yes	—
178	C₂²	#107×D₆₇	1	2	yes	—
179	C₂²	#16863443	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	134	8	no	—
183	C₄	#16863421	1	2	yes	—
184	C₄	#16863422	1	1	yes	characteristic
185	C₄	#16863424	1	1	yes	characteristic
186	C₄	#16863771	1	1	yes	characteristic
187	C₄	#16863774	1	1	yes	characteristic
188	C₄	#16863800	1	2	yes	—
189	C₂	#22558982	1	4	yes	—
190	C₂	#22560362	1	1	yes	characteristic
191	C₂	#22560367	1	1	yes	characteristic
192	C₂	#22560382	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	#8810
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
102. C₆₇	#8810	semidirect

Maximal subgroups

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

subgroup	index
#22560360	2
#22559954	2
C₂×#16863149	2
C₂×#16863148	2
C₂×#335×C₆₇	2
#8810	67