Subgroups — #30749631

Order 17152.

354 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	#22560380	C₂	1	2	yes	maximal
2	#22558965	C₂	1	1	yes	maximal, characteristic
3	#22558964	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523978	—	2	4	no	—
7	#16863420	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#16863147	—	2	2	no	—
11	#16863146	—	2	4	no	—
12	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
16	#423×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#334×C₆₇	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
21	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
22	#14523978	—	4	8	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₇	—	2	2	no	—
35	C₄⋊C₈×C₆₇	—	2	2	no	—
36	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₇	D₄	1	2	yes	—
39	#11761612	—	4	4	no	—
40	C₂×C₆₇⋊C₈	—	2	4	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₆₇⋊C₈	SD₁₆	1	8	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	M₁₆×C₆₇	—	4	4	no	—
51	C₂×C₈×C₆₇	—	2	4	no	—
52	C₂×C₈×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
69	C₈×C₆₇	—	2	2	no	—
70	C₈×C₆₇	—	8	2	no	—
71	C₈×C₆₇	#108	1	4	yes	complemented
72	C₆₇⋊C₈	—	2	2	no	—
73	C₆₇⋊C₈	—	8	2	no	—
74	C₆₇⋊C₈	#106	1	4	yes	—
75	C₆₇⋊C₈	#106	1	4	yes	—
76	C₆₇⋊C₈	#108	1	4	yes	complemented
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	—	2	1	no	—
79	C₂²×C₆₇	#384	1	1	yes	characteristic
80	C₂²×C₆₇	#498	1	2	yes	—
81	C₄×C₆₇	—	2	1	no	—
82	C₄×C₆₇	—	2	1	no	—
83	C₄×C₆₇	C₂×#106	1	2	yes	—
84	C₄×C₆₇	C₂×#108	1	2	yes	—
85	C₄×C₆₇	C₂×#108	1	2	yes	—
86	C₄×C₆₇	#492	1	2	yes	—
87	#8822	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	2	no	—
89	C₂×C₆₇	#1844	1	1	yes	characteristic
90	C₂×C₆₇	#1848	1	1	yes	characteristic
91	C₂×C₆₇	#1848	1	1	yes	characteristic
92	#1848	—	67	2	no	—
93	#1566	—	67	2	no	—
94	#1566	—	67	1	no	—
95	#1566	D₆₇	1	1	yes	characteristic
96	C₂×C₈²	—	67	1	no	—
97	C₆₇	#8822	1	1	yes	characteristic, complemented, Sylow
98	C₂×#108	—	134	4	no	—
99	#423	—	67	2	no	—
100	#423	C₂×D₆₇	1	2	yes	—
101	C₂×C₄×C₈	—	67	1	no	—
102	C₂×C₄×C₈	—	67	1	no	—
103	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
104	#334	—	2	2	no	—
105	#334	—	134	4	no	—
106	#334	—	134	2	no	—
107	C₈²	—	67	1	no	—
108	C₈²	—	67	1	no	—
109	C₈²	—	67	1	no	—
110	C₈²	—	67	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	134	2	no	—
113	C₂²×C₈	—	67	2	no	—
114	C₂²×C₈	—	67	2	no	—
115	C₂²×C₈	Dic₁₃₄	1	2	yes	—
116	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
117	#108	—	268	8	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	134	2	no	—
121	C₄⋊C₈	—	134	2	no	—
122	C₄×C₈	—	67	1	no	—
123	C₄×C₈	—	67	1	no	—
124	C₄×C₈	—	67	1	no	—
125	C₄×C₈	—	67	1	no	—
126	C₄×C₈	—	67	2	no	—
127	C₄×C₈	—	67	2	no	—
128	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
129	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
131	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
132	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
134	M₁₆	—	4	4	no	—
135	M₁₆	—	268	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	268	2	no	—
139	C₂×C₈	—	67	2	no	—
140	C₂×C₈	—	67	2	no	—
141	C₂×C₈	—	67	2	no	—
142	C₂×C₈	—	67	8	no	—
143	C₂×C₈	—	67	2	no	—
144	C₂×C₈	—	134	4	no	—
145	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
146	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
147	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
148	C₄²	D₄×D₆₇	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
150	C₄²	Q₈×D₆₇	1	1	yes	characteristic
151	C₂³	#14524042	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
155	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#14524039	1	1	yes	characteristic
158	C₂×C₄	#14524041	1	1	yes	characteristic
159	C₂×C₄	#14524042	1	1	yes	characteristic
160	C₂×C₄	SD₁₆×D₆₇	1	4	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	8	2	no	—
163	C₈	—	536	2	no	—
164	C₈	—	67	4	no	—
165	C₈	—	67	4	no	—
166	C₈	—	67	4	no	—
167	C₈	—	134	2	no	—
168	C₈	#14523978	1	4	yes	complemented
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#16863265	1	1	yes	characteristic
172	C₂²	#16863793	1	2	yes	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#106×D₆₇	1	2	yes	—
176	C₄	#108×D₆₇	1	2	yes	—
177	C₄	C₂×#14523978	1	2	yes	—
178	C₄	#16863767	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#22560364	1	1	yes	characteristic
181	C₂	#22560379	1	1	yes	characteristic
182	C₂	#22560380	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#8822
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
71. C₈×C₆₇	#108	semidirect
76. C₆₇⋊C₈	#108	semidirect
97. C₆₇	#8822	semidirect
168. C₈	#14523978	semidirect

Maximal subgroups

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

subgroup	index
#22560380	2
#22558965	2
#22558964	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1566	2
#8822	67