Subgroups — #30772270

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×#16863221	C₂	1	2	yes	maximal
3	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#22560373	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14523977	C₂²	1	2	yes	—
9	C₂×#14523975	—	2	1	no	—
10	#16863262	—	2	1	no	—
11	#16863223	—	2	4	no	—
12	#16863221	—	2	4	no	—
13	C₄×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
14	C₂×#107×C₆₇	C₂²	1	2	yes	—
15	C₂×#106×C₆₇	—	2	1	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#382×C₆₇	—	2	1	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
22	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
24	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
25	#14523977	—	2	2	no	—
26	#14523977	—	2	2	no	—
27	#14523975	—	4	2	no	—
28	C₆₇⋊#95	—	4	2	no	—
29	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	#107×C₆₇	—	2	2	no	—
34	#107×C₆₇	—	2	2	no	—
35	#106×C₆₇	—	4	2	no	—
36	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₆₇	Q₈	1	2	yes	—
38	#95×C₆₇	—	4	2	no	—
39	C₂²×Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	C₄⋊Dic₆₇	—	4	2	no	—
42	C₆₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
43	C₆₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	D₈	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₇	D₈	1	2	yes	—
49	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×D₄	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	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₄⋊C₄×C₆₇	—	4	2	no	—
58	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
59	C₄²×C₆₇	Q₁₆	1	2	yes	—
60	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	—	2	4	no	—
62	C₂×C₄×C₆₇	—	8	2	no	—
63	C₂×C₄×C₆₇	—	8	4	no	—
64	C₂×C₄×C₆₇	#107	1	4	yes	—
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₆₇	—	8	2	no	—
72	C₂×Dic₆₇	—	8	4	no	—
73	C₈×C₆₇	#107	1	4	yes	—
74	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
79	C₂²×C₆₇	#493	1	1	yes	characteristic
80	C₂²×C₆₇	#500	1	1	yes	characteristic
81	C₂²×C₆₇	#508	1	4	yes	—
82	C₄×C₆₇	—	2	4	no	—
83	C₄×C₆₇	—	16	2	no	—
84	C₄×C₆₇	#384	1	1	yes	characteristic
85	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
86	C₄×C₆₇	#494	1	1	yes	characteristic
87	C₄×C₆₇	#500	1	1	yes	characteristic
88	Dic₆₇	—	16	2	no	—
89	#12907	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	#1845	1	1	yes	characteristic
91	C₂×C₆₇	#2158	1	2	yes	—
92	C₂×C₆₇	#2249	1	2	yes	—
93	C₂×C₆₇	#2253	1	2	yes	—
94	C₂×#367	—	67	2	no	—
95	C₂×#366	—	67	2	no	—
96	C₂×C₄×C₁₆	—	67	1	no	—
97	#1846	—	67	1	no	—
98	#1846	D₆₇	1	1	yes	characteristic
99	C₆₇	#12907	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₁₆	—	67	1	no	—
101	C₂²×C₁₆	—	67	1	no	—
102	C₂×#107	—	67	2	no	—
103	C₂×#107	C₂×D₆₇	1	2	yes	—
104	C₂×#106	—	2	1	no	—
105	C₂×#106	—	134	1	no	—
106	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
107	#382	—	2	1	no	—
108	#382	—	134	1	no	—
109	#367	—	134	4	no	—
110	#366	—	134	4	no	—
111	C₄×C₁₆	—	67	4	no	—
112	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	134	2	no	—
116	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
117	C₂×C₁₆	—	67	2	no	—
118	C₂×C₁₆	—	67	2	no	—
119	C₂×C₁₆	—	67	2	no	—
120	C₂×C₁₆	—	67	2	no	—
121	C₂×C₁₆	—	67	2	no	—
122	C₂×C₁₆	—	67	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	134	2	no	—
126	#107	—	134	2	no	—
127	#106	—	4	2	no	—
128	#106	—	268	2	no	—
129	C₄×C₈	Dic₁₃₄	1	2	yes	—
130	C₄×C₈	C₄×D₆₇	1	2	yes	—
131	#95	—	4	2	no	—
132	#95	—	268	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	268	2	no	—
135	C₂²×C₄	#11761613	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
138	C₂×C₈	#11761613	1	2	yes	—
139	C₂×C₈	#11761618	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
141	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
142	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	268	2	no	—
149	C₄⋊C₄	—	268	2	no	—
150	C₄²	Dic₆₇⋊C₄	1	2	yes	—
151	C₄²	#11761618	1	2	yes	—
152	C₁₆	—	67	4	no	—
153	C₁₆	—	67	4	no	—
154	C₂³	C₂×#11761613	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	—	536	2	no	—
159	C₂×C₄	—	536	4	no	—
160	C₂×C₄	#14523976	1	4	yes	—
161	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
162	C₂×C₄	#14524039	1	1	yes	characteristic
163	C₂×C₄	#14524041	1	1	yes	characteristic
164	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
165	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
166	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
167	C₈	#14523976	1	4	yes	—
168	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
169	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
170	C₈	#14524042	1	1	yes	characteristic
171	C₈	#14524042	1	1	yes	characteristic
172	C₂²	C₂×#14523976	1	1	yes	characteristic
173	C₂²	#16863772	1	1	yes	characteristic
174	C₂²	#16863799	1	1	yes	characteristic
175	C₂²	#16863826	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	16	2	no	—
178	C₄	—	1072	2	no	—
179	C₄	#16863265	1	1	yes	characteristic
180	C₄	C₂×#14523976	1	1	yes	characteristic
181	C₄	#16863775	1	1	yes	characteristic
182	C₄	#16863799	1	1	yes	characteristic
183	C₂	#22560368	1	1	yes	characteristic
184	C₂	#22561991	1	2	yes	—
185	C₂	#22562468	1	2	yes	—
186	C₂	#22562485	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1846	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1846	D₆₇
hypocenter	C₆₇	#12907
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
13. C₄×C₆₇⋊C₁₆	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×#16863221	2
C₂×C₄×C₆₇⋊C₁₆	2
#22560373	2
C₆₇×#1846	2
#12907	67