Subgroups — #30772266

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863224	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863223	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#22560367	C₂	1	2	yes	maximal
5	C₂×#367×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14523977	C₂²	1	2	yes	—
8	C₂×#14523976	—	2	2	no	—
9	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
10	#16863265	—	2	2	no	—
11	#16863224	—	2	2	no	—
12	#16863223	—	2	4	no	—
13	C₁₆×Dic₆₇	C₄	1	4	yes	complemented
14	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	#367×C₆₇	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×Dic₆₇⋊C₄	—	4	4	no	—
20	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
24	#14523977	—	2	2	no	—
25	#14523977	—	2	2	no	—
26	#14523976	—	4	4	no	—
27	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
28	C₈×Dic₆₇	Q₈	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
33	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
34	#107×C₆₇	—	2	2	no	—
35	#107×C₆₇	—	2	2	no	—
36	C₂²×Dic₆₇	—	4	2	no	—
37	C₂²×Dic₆₇	D₈	1	2	yes	—
38	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
39	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
40	C₂×C₆₇⋊C₈	Q₁₆	1	2	yes	—
41	C₄⋊Dic₆₇	—	4	2	no	—
42	C₄⋊Dic₆₇	—	4	2	no	—
43	Dic₆₇⋊C₄	—	8	8	no	—
44	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
45	C₄×Dic₆₇	Q₁₆	1	2	yes	—
46	C₆₇⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
47	C₂²×C₄×C₆₇	—	4	2	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	8	2	no	—
60	C₂×C₄×C₆₇	—	8	4	no	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₆₇	—	8	2	no	—
68	C₂×Dic₆₇	—	8	4	no	—
69	C₂×Dic₆₇	#107	1	4	yes	—
70	C₂×Dic₆₇	SD₃₂	1	8	yes	—
71	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₆₇⋊C₈	#107	1	4	yes	—
76	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
77	C₂²×C₆₇	#493	1	1	yes	characteristic
78	C₂²×C₆₇	#500	1	1	yes	characteristic
79	C₂²×C₆₇	C₂×SD₃₂	1	4	yes	—
80	C₄×C₆₇	—	16	2	no	—
81	C₄×C₆₇	#384	1	1	yes	characteristic
82	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
83	C₄×C₆₇	#494	1	1	yes	characteristic
84	C₄×C₆₇	#500	1	1	yes	characteristic
85	Dic₆₇	—	16	2	no	—
86	Dic₆₇	#367	1	8	yes	complemented
87	#12906	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	#1845	1	1	yes	characteristic
89	C₂×C₆₇	C₂×#367	1	2	yes	—
90	C₂×C₆₇	#2248	1	2	yes	—
91	C₂×C₆₇	#2254	1	2	yes	—
92	C₂×#367	—	67	1	no	—
93	C₂×#367	—	67	2	no	—
94	C₂×#367	D₆₇	1	1	yes	characteristic
95	C₂×C₄×C₁₆	—	67	1	no	—
96	#1845	—	67	2	no	—
97	C₆₇	#12906	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₁₆	—	67	1	no	—
99	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
100	C₂×#107	—	67	2	no	—
101	C₂×#107	—	134	2	no	—
102	C₂×#107	C₂×D₆₇	1	2	yes	—
103	C₂×C₄×C₈	—	67	1	no	—
104	#384	—	134	2	no	—
105	#367	—	2	2	no	—
106	#367	—	134	4	no	—
107	#367	—	134	2	no	—
108	C₄×C₁₆	—	67	4	no	—
109	C₂²×C₈	—	67	1	no	—
110	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	268	4	no	—
113	C₂×C₄⋊C₄	—	134	2	no	—
114	C₂×C₄²	—	67	1	no	—
115	C₂×C₁₆	—	67	2	no	—
116	C₂×C₁₆	—	67	2	no	—
117	C₂×C₁₆	—	67	2	no	—
118	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
119	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
121	#107	—	2	2	no	—
122	#107	—	2	2	no	—
123	#107	—	268	4	no	—
124	#107	—	134	2	no	—
125	#107	—	134	2	no	—
126	C₄×C₈	—	67	2	no	—
127	C₄×C₈	—	67	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	268	2	no	—
130	C₂²×C₄	—	67	2	no	—
131	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
132	C₂×C₈	—	67	2	no	—
133	C₂×C₈	—	67	2	no	—
134	C₂×C₈	—	67	2	no	—
135	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
136	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	268	2	no	—
144	C₄⋊C₄	—	268	2	no	—
145	C₄⋊C₄	—	536	8	no	—
146	C₄²	—	67	2	no	—
147	C₄²	—	67	2	no	—
148	C₁₆	—	67	4	no	—
149	C₁₆	Dic₆₇⋊C₄	1	4	yes	complemented
150	C₂³	D₈×D₆₇	1	1	yes	characteristic
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	536	2	no	—
154	C₂×C₄	—	536	4	no	—
155	C₂×C₄	—	67	8	no	—
156	C₂×C₄	—	67	4	no	—
157	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
158	C₂×C₄	#14524039	1	1	yes	characteristic
159	C₂×C₄	#14524042	1	1	yes	characteristic
160	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
161	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
162	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
163	C₈	—	67	4	no	—
164	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
165	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
166	C₈	#14524041	1	1	yes	characteristic
167	C₈	#14524042	1	1	yes	characteristic
168	C₂²	#107×D₆₇	1	1	yes	characteristic
169	C₂²	#16863771	1	1	yes	characteristic
170	C₂²	#16863800	1	1	yes	characteristic
171	C₂²	SD₃₂×D₆₇	1	4	yes	—
172	C₄	—	16	2	no	—
173	C₄	—	1072	2	no	—
174	C₄	—	67	8	no	—
175	C₄	#16863265	1	1	yes	characteristic
176	C₄	#107×D₆₇	1	1	yes	characteristic
177	C₄	#16863774	1	1	yes	characteristic
178	C₄	#16863800	1	1	yes	characteristic
179	C₂	#22560367	1	1	yes	characteristic
180	C₂	#367×D₆₇	1	2	yes	—
181	C₂	#22562463	1	2	yes	—
182	C₂	#22562490	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#367×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#367	D₆₇
hypocenter	C₆₇	#12906
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
13. C₁₆×Dic₆₇	C₄	semidirect
46. C₆₇⋊C₁₆	C₄⋊C₄	semidirect
57. C₁₆×C₆₇	C₄⋊C₄	semidirect
86. Dic₆₇	#367	semidirect
97. C₆₇	#12906	semidirect
149. C₁₆	Dic₆₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863224	2
C₂×#16863223	2
C₂×C₁₆×Dic₆₇	2
#22560367	2
C₂×#367×C₆₇	2
#12906	67