Subgroups — #30772263

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863221	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#22560367	C₂	1	2	yes	maximal
5	C₂×#366×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	#16863222	—	2	2	no	—
12	#16863221	—	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	#366×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	D₁₆	1	4	yes	—
71	C₂×Dic₆₇	Q₃₂	1	4	yes	—
72	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₆₇⋊C₈	#107	1	4	yes	—
77	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
78	C₂²×C₆₇	#493	1	1	yes	characteristic
79	C₂²×C₆₇	#500	1	1	yes	characteristic
80	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
81	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
82	C₄×C₆₇	—	16	2	no	—
83	C₄×C₆₇	#384	1	1	yes	characteristic
84	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
85	C₄×C₆₇	#494	1	1	yes	characteristic
86	C₄×C₆₇	#500	1	1	yes	characteristic
87	Dic₆₇	—	16	2	no	—
88	Dic₆₇	#366	1	8	yes	complemented
89	#12905	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	#1845	1	1	yes	characteristic
91	C₂×C₆₇	C₂×#366	1	2	yes	—
92	C₂×C₆₇	#2246	1	1	yes	characteristic
93	C₂×C₆₇	#2247	1	1	yes	characteristic
94	C₂×C₆₇	#2252	1	2	yes	—
95	C₂×#366	—	67	1	no	—
96	C₂×#366	—	67	2	no	—
97	C₂×#366	D₆₇	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	67	1	no	—
99	#1845	—	67	2	no	—
100	C₆₇	#12905	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	67	1	no	—
102	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
103	C₂×#107	—	67	2	no	—
104	C₂×#107	—	134	2	no	—
105	C₂×#107	C₂×D₆₇	1	2	yes	—
106	C₂×C₄×C₈	—	67	1	no	—
107	#384	—	134	2	no	—
108	#366	—	2	2	no	—
109	#366	—	134	4	no	—
110	#366	—	134	2	no	—
111	C₄×C₁₆	—	67	4	no	—
112	C₂²×C₈	—	67	1	no	—
113	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	268	4	no	—
116	C₂×C₄⋊C₄	—	134	2	no	—
117	C₂×C₄²	—	67	1	no	—
118	C₂×C₁₆	—	67	2	no	—
119	C₂×C₁₆	—	67	2	no	—
120	C₂×C₁₆	—	67	2	no	—
121	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
122	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	268	4	no	—
127	#107	—	134	2	no	—
128	#107	—	134	2	no	—
129	C₄×C₈	—	67	2	no	—
130	C₄×C₈	—	67	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	268	2	no	—
133	C₂²×C₄	—	67	2	no	—
134	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
135	C₂×C₈	—	67	2	no	—
136	C₂×C₈	—	67	2	no	—
137	C₂×C₈	—	67	2	no	—
138	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
139	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	268	2	no	—
147	C₄⋊C₄	—	268	2	no	—
148	C₄⋊C₄	—	536	8	no	—
149	C₄²	—	67	2	no	—
150	C₄²	—	67	2	no	—
151	C₁₆	—	67	4	no	—
152	C₁₆	Dic₆₇⋊C₄	1	4	yes	complemented
153	C₂³	D₈×D₆₇	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	536	2	no	—
157	C₂×C₄	—	536	4	no	—
158	C₂×C₄	—	67	4	no	—
159	C₂×C₄	—	67	4	no	—
160	C₂×C₄	—	67	4	no	—
161	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
162	C₂×C₄	#14524039	1	1	yes	characteristic
163	C₂×C₄	#14524042	1	1	yes	characteristic
164	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
165	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
167	C₈	—	67	4	no	—
168	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
169	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
170	C₈	#14524041	1	1	yes	characteristic
171	C₈	#14524042	1	1	yes	characteristic
172	C₂²	#107×D₆₇	1	1	yes	characteristic
173	C₂²	#16863771	1	1	yes	characteristic
174	C₂²	#16863800	1	1	yes	characteristic
175	C₂²	D₁₆×D₆₇	1	2	yes	—
176	C₂²	Q₃₂×D₆₇	1	2	yes	—
177	C₄	—	16	2	no	—
178	C₄	—	1072	2	no	—
179	C₄	—	67	8	no	—
180	C₄	#16863265	1	1	yes	characteristic
181	C₄	#107×D₆₇	1	1	yes	characteristic
182	C₄	#16863774	1	1	yes	characteristic
183	C₄	#16863800	1	1	yes	characteristic
184	C₂	#22560367	1	1	yes	characteristic
185	C₂	#366×D₆₇	1	2	yes	—
186	C₂	#22562457	1	1	yes	characteristic
187	C₂	#22562460	1	1	yes	characteristic
188	C₂	#22562483	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#366×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#366	D₆₇
hypocenter	C₆₇	#12905
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

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
88. Dic₆₇	#366	semidirect
100. C₆₇	#12905	semidirect
152. 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×#16863222	2
C₂×#16863221	2
C₂×C₁₆×Dic₆₇	2
#22560367	2
C₂×#366×C₆₇	2
#12905	67