Subgroups — #22565240

Order 8576.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14524034	C₂	1	2	yes	maximal
2	#16863328	C₂	1	4	yes	maximal
3	#16863295	C₂	1	2	yes	maximal
4	#16863294	C₂	1	2	yes	maximal
5	#16863266	C₂	1	2	yes	maximal
6	#16863266	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#14524034	—	2	8	no	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
17	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
18	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
19	C₆₇⋊#95	C₂²	1	2	yes	—
20	C₆₇⋊#95	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#95×C₆₇	C₂²	1	2	yes	—
26	C₂²×Dic₆₇	C₂³	1	2	yes	—
27	C₂²×Dic₆₇	C₂³	1	2	yes	—
28	C₂²×Dic₆₇	C₂³	1	4	yes	—
29	C₄⋊Dic₆₇	—	2	8	no	—
30	C₄⋊Dic₆₇	—	2	4	no	—
31	C₄⋊Dic₆₇	—	2	2	no	—
32	C₄⋊Dic₆₇	—	2	2	no	—
33	C₄⋊Dic₆₇	—	2	4	no	—
34	C₄⋊Dic₆₇	—	4	4	no	—
35	C₄⋊Dic₆₇	—	4	2	no	—
36	Dic₆₇⋊C₄	—	2	8	no	—
37	C₄×Dic₆₇	—	2	4	no	—
38	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₆₇	D₄	1	4	yes	—
44	C₄⋊C₄×C₆₇	Q₈	1	4	yes	—
45	C₄²×C₆₇	—	2	4	no	—
46	C₄²×C₆₇	Q₈	1	4	yes	—
47	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	1	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
55	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
58	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₆₇	#41	1	2	yes	—
62	C₂×C₄×C₆₇	#41	1	2	yes	—
63	C₂×Dic₆₇	—	2	2	no	—
64	C₂×Dic₆₇	—	2	2	no	—
65	C₂×Dic₆₇	—	2	4	no	—
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	—	2	4	no	—
69	C₂×Dic₆₇	—	2	2	no	—
70	C₂×Dic₆₇	—	2	4	no	—
71	C₂×Dic₆₇	—	2	2	no	—
72	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
76	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
77	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	—	2	4	no	—
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	#125	1	4	yes	complemented
83	C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
84	Dic₆₇	—	4	2	no	—
85	Dic₆₇	—	4	2	no	—
86	Dic₆₇	—	4	4	no	—
87	C₂×C₆₇	C₂×#125	1	1	yes	characteristic
88	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₆₇	#544	1	1	yes	characteristic
90	C₂×C₆₇	#547	1	1	yes	characteristic
91	C₂×C₆₇	#550	1	1	yes	characteristic
92	C₂×C₆₇	#554	1	1	yes	characteristic
93	C₂×C₆₇	#557	1	1	yes	characteristic
94	#2552	—	67	1	no	maximal, Sylow
95	C₆₇	#2552	1	1	yes	characteristic, complemented, Sylow
96	C₂×#125	—	67	2	no	—
97	#398	—	67	4	no	—
98	#389	—	67	2	no	—
99	#389	—	67	2	no	—
100	#384	—	67	2	no	—
101	#384	—	67	1	no	—
102	C₄×C₄⋊C₄	—	67	1	no	—
103	C₄×C₄⋊C₄	D₆₇	1	1	yes	characteristic
104	#125	—	134	8	no	—
105	C₂×C₄⋊C₄	—	67	4	no	—
106	C₂×C₄⋊C₄	—	67	2	no	—
107	C₂×C₄⋊C₄	—	67	4	no	—
108	C₂×C₄⋊C₄	—	67	2	no	—
109	C₂×C₄⋊C₄	—	67	4	no	—
110	C₂×C₄⋊C₄	—	67	1	no	—
111	C₂×C₄⋊C₄	—	67	1	no	—
112	C₂×C₄⋊C₄	—	67	2	no	—
113	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂×C₄²	—	67	2	no	—
116	C₂×C₄²	C₂×D₆₇	1	2	yes	—
117	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
118	#95	—	67	2	no	—
119	#95	—	67	4	no	—
120	#95	C₂×D₆₇	1	2	yes	—
121	C₂²×C₄	—	67	2	no	—
122	C₂²×C₄	—	67	2	no	—
123	C₂²×C₄	—	67	4	no	—
124	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
125	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
127	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
129	C₄⋊C₄	—	268	4	no	—
130	C₄⋊C₄	—	268	2	no	—
131	C₄⋊C₄	—	134	8	no	—
132	C₄⋊C₄	—	134	8	no	—
133	C₄⋊C₄	—	134	4	no	—
134	C₄⋊C₄	—	134	2	no	—
135	C₄⋊C₄	—	134	2	no	—
136	C₄⋊C₄	—	134	4	no	—
137	C₄⋊C₄	Dic₁₃₄	1	4	yes	—
138	C₄⋊C₄	D₂₆₈	1	4	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	134	4	no	—
141	C₄²	Dic₁₃₄	1	4	yes	—
142	C₂³	C₂³×D₆₇	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	134	2	no	—
150	C₂×C₄	—	134	2	no	—
151	C₂×C₄	—	134	4	no	—
152	C₂×C₄	—	134	2	no	—
153	C₂×C₄	—	134	2	no	—
154	C₂×C₄	—	134	4	no	—
155	C₂×C₄	—	134	2	no	—
156	C₂×C₄	—	134	4	no	—
157	C₂×C₄	—	134	2	no	—
158	C₂×C₄	C₂×D₂₆₈	1	4	yes	—
159	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
160	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
162	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
163	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
164	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
165	C₂×C₄	#11761627	1	2	yes	—
166	C₂×C₄	#11761629	1	2	yes	—
167	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
168	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
169	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
170	C₂²	C₂×#11761627	1	1	yes	characteristic
171	C₂²	C₂×#11761629	1	1	yes	characteristic
172	C₂²	#14524089	1	1	yes	characteristic
173	C₂²	#14524090	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	4	no	—
176	C₄	—	2	4	no	—
177	C₄	—	268	2	no	—
178	C₄	—	268	2	no	—
179	C₄	—	268	4	no	—
180	C₄	#14524034	1	4	yes	complemented
181	C₄	#14524043	1	4	yes	—
182	C₂	C₂×#14524034	1	1	yes	characteristic
183	C₂	C₂×#14524043	1	1	yes	characteristic
184	C₂	#16864024	1	1	yes	characteristic
185	C₂	#16864050	1	1	yes	characteristic
186	C₂	#16864068	1	1	yes	characteristic
187	C₂	#16864103	1	1	yes	characteristic
188	C₂	#16864125	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₄×C₄⋊C₄×C₆₇	C₂
hypercenter	C₄×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#2552
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
82. C₄×C₆₇	#125	semidirect
95. C₆₇	#2552	semidirect
180. C₄	#14524034	semidirect