Subgroups — #30749568

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863146	C₂	1	4	yes	maximal
4	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
11	#16863146	—	2	8	no	—
12	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₄³×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	—	2	2	no	—
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
20	C₄×C₆₇⋊C₈	—	2	4	no	—
21	C₄×C₆₇⋊C₈	—	2	2	no	—
22	C₄×C₆₇⋊C₈	C₈	1	4	yes	complemented
23	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
24	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
25	C₂²×C₈×C₆₇	—	2	2	no	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	—	2	4	no	—
32	C₄⋊C₈×C₆₇	—	4	4	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	4	no	—
37	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
39	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
40	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
41	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₆₇	—	4	2	no	—
47	C₂×C₈×C₆₇	—	4	4	no	—
48	C₄²×C₆₇	—	2	4	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	C₂×C₈	1	2	yes	—
51	C₄²×C₆₇	M₁₆	1	2	yes	—
52	C₄²×C₆₇	SD₁₆	1	4	yes	—
53	C₄²×C₆₇	SD₁₆	1	4	yes	—
54	C₄²×C₆₇	C₂×D₄	1	2	yes	—
55	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
61	C₂×C₄×C₆₇	#106	1	4	yes	—
62	C₂×C₄×C₆₇	#106	1	4	yes	—
63	C₂×C₄×C₆₇	#108	1	4	yes	—
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
68	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
73	C₈×C₆₇	—	8	2	no	—
74	C₆₇⋊C₈	—	2	4	no	—
75	C₆₇⋊C₈	—	8	2	no	—
76	C₆₇⋊C₈	C₄⋊C₈	1	8	yes	complemented
77	C₂²×C₆₇	#384	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₆₇	#423	1	1	yes	characteristic
80	C₂²×C₆₇	C₂×#106	1	2	yes	—
81	C₂²×C₆₇	C₂×#108	1	2	yes	—
82	C₄×C₆₇	—	2	4	no	—
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	#334	1	8	yes	complemented
85	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
86	C₄×C₆₇	#423	1	2	yes	—
87	C₄×C₆₇	#492	1	1	yes	characteristic
88	C₄×C₆₇	#492	1	1	yes	characteristic
89	C₄×C₆₇	#498	1	2	yes	—
90	#8805	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	C₂×#334	1	2	yes	—
92	C₂×C₆₇	#1566	1	2	yes	—
93	C₂×C₆₇	#1842	1	1	yes	characteristic
94	C₂×C₆₇	#1844	1	1	yes	characteristic
95	C₂×C₆₇	#1848	1	1	yes	characteristic
96	#1842	—	67	1	no	—
97	#1842	D₆₇	1	1	yes	characteristic
98	C₄²×C₈	—	67	1	no	—
99	C₂×#334	—	67	4	no	—
100	C₆₇	#8805	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊C₈	—	2	2	no	—
102	C₂×C₄⋊C₈	—	67	2	no	—
103	C₂×C₄⋊C₈	—	134	2	no	—
104	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
105	C₂×C₄×C₈	—	67	1	no	—
106	C₂×C₄×C₈	—	67	2	no	—
107	C₂×C₄×C₈	—	67	2	no	—
108	C₂×C₄×C₈	—	67	1	no	—
109	C₄³	C₂×D₆₇	1	1	yes	characteristic
110	#334	—	134	8	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	67	2	no	—
113	C₂²×C₈	—	67	2	no	—
114	C₂²×C₈	—	134	2	no	—
115	C₂×C₄²	Dic₁₃₄	1	2	yes	—
116	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
117	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
119	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
120	C₄⋊C₈	—	2	4	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄⋊C₈	—	268	4	no	—
123	C₄⋊C₈	—	134	4	no	—
124	C₄×C₈	—	67	4	no	—
125	C₄×C₈	—	67	4	no	—
126	C₄×C₈	—	67	4	no	—
127	C₄×C₈	—	134	4	no	—
128	C₄×C₈	—	134	2	no	—
129	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
130	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	268	2	no	—
137	C₂×C₈	—	268	4	no	—
138	C₂×C₈	—	67	4	no	—
139	C₂×C₈	—	67	4	no	—
140	C₂×C₈	—	67	4	no	—
141	C₂×C₈	—	67	4	no	—
142	C₂×C₈	—	134	2	no	—
143	C₂×C₈	—	134	2	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₈×D₆₇	1	2	yes	—
147	C₄²	#11761611	1	2	yes	—
148	C₄²	#11761615	1	4	yes	—
149	C₄²	#11761616	1	4	yes	—
150	C₄²	D₄×D₆₇	1	2	yes	—
151	C₄²	Q₈×D₆₇	1	2	yes	—
152	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
157	C₂×C₄	#14523974	1	4	yes	—
158	C₂×C₄	#14523974	1	4	yes	—
159	C₂×C₄	#14523978	1	4	yes	—
160	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
161	C₂×C₄	#14524039	1	1	yes	characteristic
162	C₂×C₄	#14524041	1	1	yes	characteristic
163	C₂×C₄	#14524042	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
165	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
166	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
167	C₂×C₄	C₂×#11761615	1	2	yes	—
168	C₂×C₄	C₂×#11761616	1	2	yes	—
169	C₈	—	8	2	no	—
170	C₈	—	536	2	no	—
171	C₈	—	67	8	no	—
172	C₈	—	134	4	no	—
173	C₂²	#16863265	1	1	yes	characteristic
174	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
175	C₂²	#16863419	1	1	yes	characteristic
176	C₂²	C₂×#14523974	1	2	yes	—
177	C₂²	C₂×#14523978	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	2	4	no	—
180	C₄	#16863146	1	8	yes	complemented
181	C₄	C₄⋊C₈×D₆₇	1	2	yes	—
182	C₄	#16863418	1	2	yes	—
183	C₄	#16863768	1	1	yes	characteristic
184	C₄	#16863770	1	1	yes	characteristic
185	C₄	#16863792	1	2	yes	—
186	C₂	C₂×#16863146	1	2	yes	—
187	C₂	#22558964	1	2	yes	—
188	C₂	#22560360	1	1	yes	characteristic
189	C₂	#22560365	1	1	yes	characteristic
190	C₂	#22560380	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1842	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#8805
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
22. C₄×C₆₇⋊C₈	C₈	semidirect
76. C₆₇⋊C₈	C₄⋊C₈	semidirect
84. C₄×C₆₇	#334	semidirect
100. C₆₇	#8805	semidirect
180. C₄	#16863146	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₄²×C₆₇⋊C₈	2
C₂×#16863146	2
C₆₇×#1842	2
#8805	67