Subgroups — #30749595

Order 17152.

382 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	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22560326	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863149	C₂	1	2	yes	maximal
4	C₂×#16863147	C₂	1	2	yes	maximal
5	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863151	—	2	2	no	—
11	#16863149	—	2	4	no	—
12	#16863147	—	2	4	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₄³×C₆₇	C₂²	1	1	yes	characteristic
15	#336×C₆₇	C₄	1	2	yes	—
16	#336×C₆₇	C₄	1	2	yes	—
17	C₂²×C₆₇⋊C₈	—	2	1	no	—
18	C₂²×C₆₇⋊C₈	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	—	2	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
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	D₄	1	4	yes	—
32	C₄×C₈×C₆₇	Q₈	1	4	yes	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	4	1	no	—
35	C₂×C₆₇⋊C₈	—	4	1	no	—
36	C₂×C₆₇⋊C₈	—	4	4	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₆₇	—	2	4	no	—
45	C₂×C₈×C₆₇	—	2	4	no	—
46	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
47	C₄²×C₆₇	—	2	4	no	—
48	C₄²×C₆₇	—	2	2	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	—	2	2	no	—
52	C₄²×C₆₇	C₂×D₄	1	2	yes	—
53	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
54	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
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	—	2	2	no	—
61	C₂×C₄×C₆₇	#108	1	4	yes	—
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#131	1	2	yes	—
67	C₂×C₄×C₆₇	#131	1	2	yes	—
68	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₆₇	#137	1	2	yes	—
70	C₈×C₆₇	—	2	8	no	—
71	C₆₇⋊C₈	—	8	2	no	—
72	C₆₇⋊C₈	—	8	1	no	—
73	C₆₇⋊C₈	—	8	1	no	—
74	C₂²×C₆₇	#384	1	1	yes	characteristic
75	C₂²×C₆₇	#424	1	1	yes	characteristic
76	C₂²×C₆₇	#424	1	1	yes	characteristic
77	C₂²×C₆₇	#428	1	2	yes	—
78	C₂²×C₆₇	C₂×#108	1	2	yes	—
79	C₄×C₆₇	—	2	4	no	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	—	4	2	no	—
82	C₄×C₆₇	#424	1	4	yes	—
83	C₄×C₆₇	#496	1	1	yes	characteristic
84	C₄×C₆₇	#497	1	1	yes	characteristic
85	C₄×C₆₇	#501	1	2	yes	—
86	#8812	—	67	1	no	maximal, Sylow
87	C₂×C₆₇	#1573	1	2	yes	—
88	C₂×C₆₇	#1574	1	2	yes	—
89	C₂×C₆₇	#1843	1	1	yes	characteristic
90	C₂×C₆₇	#1847	1	1	yes	characteristic
91	C₂×C₆₇	#1848	1	1	yes	characteristic
92	#1842	—	67	1	no	—
93	#1831	—	67	1	no	—
94	#1744	D₆₇	1	1	yes	characteristic
95	C₂×#335	—	67	2	no	—
96	C₂×#334	—	67	2	no	—
97	C₆₇	#8812	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊C₈	—	67	1	no	—
99	C₂×C₄⋊C₈	—	67	2	no	—
100	C₂×C₄⋊C₈	—	67	1	no	—
101	C₂×C₄⋊C₈	—	134	2	no	—
102	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
103	C₄³	C₂×D₆₇	1	1	yes	characteristic
104	#336	—	134	2	no	—
105	#336	Dic₆₇	1	2	yes	—
106	#336	Dic₆₇	1	2	yes	—
107	#335	—	134	4	no	—
108	#334	—	134	4	no	—
109	C₂²×C₈	—	134	1	no	—
110	C₂²×C₈	—	134	1	no	—
111	C₂²×C₈	—	134	2	no	—
112	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
113	C₂×C₄²	Dic₁₃₄	1	2	yes	—
114	C₂×C₄²	D₂₆₈	1	2	yes	—
115	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
117	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	C₄⋊C₈	—	268	4	no	—
119	C₄⋊C₈	—	134	2	no	—
120	C₄⋊C₈	—	134	2	no	—
121	C₄⋊C₈	—	134	2	no	—
122	C₄⋊C₈	—	134	2	no	—
123	C₄×C₈	Dic₁₃₄	1	4	yes	—
124	C₄×C₈	D₂₆₈	1	4	yes	—
125	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
126	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
127	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
129	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	268	2	no	—
133	C₂×C₈	—	268	1	no	—
134	C₂×C₈	—	268	1	no	—
135	C₂×C₈	—	268	4	no	—
136	C₂×C₈	—	268	2	no	—
137	C₂×C₈	—	268	2	no	—
138	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×D₂₆₈	1	2	yes	—
145	C₄²	C₂×Dic₁₃₄	1	2	yes	—
146	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#14523979	1	4	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
155	C₂×C₄	#14524040	1	1	yes	characteristic
156	C₂×C₄	#14524043	1	2	yes	—
157	C₂×C₄	#14524043	1	1	yes	characteristic
158	C₂×C₄	#14524050	1	2	yes	—
159	C₂×C₄	#14524050	1	2	yes	—
160	C₂×C₄	#14524071	1	2	yes	—
161	C₂×C₄	#14524076	1	2	yes	—
162	C₈	—	2	8	no	—
163	C₈	—	536	2	no	—
164	C₈	—	536	1	no	—
165	C₈	—	536	1	no	—
166	C₂²	#16863266	1	1	yes	characteristic
167	C₂²	#16863423	1	1	yes	characteristic
168	C₂²	#16863423	1	1	yes	characteristic
169	C₂²	#16863438	1	2	yes	—
170	C₂²	C₂×#14523979	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#16863423	1	4	yes	—
175	C₄	#16863785	1	1	yes	characteristic
176	C₄	#16863790	1	1	yes	characteristic
177	C₄	#16863805	1	2	yes	—
178	C₂	#22558994	1	2	yes	—
179	C₂	#22558998	1	2	yes	—
180	C₂	#22560363	1	1	yes	characteristic
181	C₂	#22560377	1	1	yes	characteristic
182	C₂	#22560381	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1744	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#8812
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
97. C₆₇	#8812	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
#22560326	2
C₂×#16863149	2
C₂×#16863147	2
C₆₇×#1744	2
#8812	67