Subgroups — #30772269

Order 17152.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863223	C₂	1	2	yes	maximal
2	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#22560372	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×#14523974	—	2	2	no	—
9	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
10	#16863261	—	2	2	no	—
11	#16863223	—	2	4	no	—
12	#16863222	—	2	2	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×C₄×Dic₆₇	D₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
22	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
23	#14523977	—	2	2	no	—
24	#14523977	—	2	2	no	—
25	#14523974	—	4	4	no	—
26	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
27	C₈×Dic₆₇	Q₈	1	2	yes	—
28	C₆₇⋊#95	—	4	4	no	—
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	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
44	C₄×Dic₆₇	Q₁₆	1	2	yes	—
45	C₆₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	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	C₄⋊C₄×C₆₇	—	4	2	no	—
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₁₆×C₆₇	C₄⋊C₄	1	4	yes	—
57	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	8	2	no	—
59	C₂×C₄×C₆₇	—	8	4	no	—
60	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	—	8	2	no	—
69	C₂×Dic₆₇	—	8	4	no	—
70	C₂×Dic₆₇	#107	1	4	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	#508	1	2	yes	—
80	C₂²×C₆₇	#508	1	2	yes	—
81	C₄×C₆₇	—	16	2	no	—
82	C₄×C₆₇	#384	1	1	yes	characteristic
83	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
84	C₄×C₆₇	#494	1	1	yes	characteristic
85	C₄×C₆₇	#500	1	1	yes	characteristic
86	Dic₆₇	—	2	4	no	—
87	Dic₆₇	—	16	2	no	—
88	#12907	—	67	1	no	maximal, Sylow
89	C₂×C₆₇	#1845	1	1	yes	characteristic
90	C₂×C₆₇	#2158	1	2	yes	—
91	C₂×C₆₇	#2249	1	1	yes	characteristic
92	C₂×C₆₇	#2249	1	1	yes	characteristic
93	C₂×C₆₇	#2253	1	2	yes	—
94	C₂×#367	—	67	2	no	—
95	C₂×#366	—	67	1	no	—
96	C₂×#366	D₆₇	1	1	yes	characteristic
97	C₂×C₄×C₁₆	—	67	1	no	—
98	#1846	—	67	2	no	—
99	C₆₇	#12907	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₁₆	—	67	1	no	—
101	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
102	C₂×#107	—	67	2	no	—
103	C₂×#107	C₂×D₆₇	1	2	yes	—
104	C₂×#106	—	134	2	no	—
105	C₂×C₄×C₈	—	67	1	no	—
106	#382	—	134	2	no	—
107	#367	—	134	4	no	—
108	#366	—	2	2	no	—
109	#366	—	134	2	no	—
110	C₄×C₁₆	—	67	4	no	—
111	C₂²×C₈	—	67	1	no	—
112	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	134	2	no	—
115	C₂×C₄²	—	67	1	no	—
116	C₂×C₁₆	—	67	2	no	—
117	C₂×C₁₆	—	67	2	no	—
118	C₂×C₁₆	—	67	2	no	—
119	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
120	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
121	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#107	—	134	2	no	—
125	#107	—	134	2	no	—
126	#106	—	268	4	no	—
127	C₄×C₈	—	67	2	no	—
128	C₄×C₈	—	67	2	no	—
129	#95	—	268	4	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	268	2	no	—
132	C₂²×C₄	—	67	2	no	—
133	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
134	C₂×C₈	—	67	2	no	—
135	C₂×C₈	—	67	2	no	—
136	C₂×C₈	—	67	2	no	—
137	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
138	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	268	2	no	—
146	C₄⋊C₄	—	268	2	no	—
147	C₄²	—	67	2	no	—
148	C₄²	—	67	2	no	—
149	C₁₆	—	67	4	no	—
150	C₁₆	Dic₆₇⋊C₄	1	4	yes	—
151	C₂³	D₈×D₆₇	1	1	yes	characteristic
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	4	no	—
154	C₂×C₄	—	536	2	no	—
155	C₂×C₄	—	536	4	no	—
156	C₂×C₄	—	67	4	no	—
157	C₂×C₄	—	134	2	no	—
158	C₂×C₄	—	134	2	no	—
159	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
160	C₂×C₄	#14524039	1	1	yes	characteristic
161	C₂×C₄	#14524042	1	1	yes	characteristic
162	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
163	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
164	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
165	C₈	—	67	4	no	—
166	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
167	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
168	C₈	#14524041	1	1	yes	characteristic
169	C₈	#14524042	1	1	yes	characteristic
170	C₂²	#107×D₆₇	1	1	yes	characteristic
171	C₂²	#16863771	1	1	yes	characteristic
172	C₂²	#16863800	1	1	yes	characteristic
173	C₂²	#16863825	1	2	yes	—
174	C₂²	#16863829	1	2	yes	—
175	C₄	—	16	2	no	—
176	C₄	—	1072	2	no	—
177	C₄	—	134	4	no	—
178	C₄	#16863265	1	1	yes	characteristic
179	C₄	#107×D₆₇	1	1	yes	characteristic
180	C₄	#16863774	1	1	yes	characteristic
181	C₄	#16863800	1	1	yes	characteristic
182	C₂	#22560367	1	1	yes	characteristic
183	C₂	#22561990	1	2	yes	—
184	C₂	#22562467	1	1	yes	characteristic
185	C₂	#22562471	1	1	yes	characteristic
186	C₂	#22562486	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	#12907
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
99. C₆₇	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863223	2
C₂×#16863222	2
C₂×C₁₆×Dic₆₇	2
#22560372	2
C₂×#366×C₆₇	2
#12907	67