Subgroups — #22565579

Order 8576.

484 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#16863329	C₂	1	4	yes	maximal
3	#16863303	C₂	1	2	yes	maximal
4	#16863302	C₂	1	2	yes	maximal
5	#16863294	C₂	1	2	yes	maximal
6	#16863292	C₂	1	2	yes	maximal
7	#16863266	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₃₄	—	2	4	no	—
11	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
15	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
18	C₆₇⋊#95	C₂²	1	4	yes	—
19	C₆₇⋊#95	C₂²	1	4	yes	—
20	C₆₇⋊#95	C₂²	1	4	yes	—
21	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₃₄	—	2	2	no	—
28	C₂×Dic₁₃₄	—	2	2	no	—
29	C₂×Dic₁₃₄	Q₈	1	4	yes	complemented
30	C₂²×Dic₆₇	C₂³	1	4	yes	—
31	C₂²×Dic₆₇	C₂³	1	2	yes	—
32	C₂²×Dic₆₇	C₂³	1	2	yes	—
33	C₄⋊Dic₆₇	—	2	2	no	—
34	C₄⋊Dic₆₇	—	2	4	no	—
35	C₄⋊Dic₆₇	—	2	4	no	—
36	C₄⋊Dic₆₇	—	4	2	no	—
37	Dic₆₇⋊C₄	—	2	4	no	—
38	Dic₆₇⋊C₄	—	4	4	no	—
39	C₄×Dic₆₇	—	2	4	no	—
40	C₂×Q₈×C₆₇	—	2	2	no	—
41	C₂×Q₈×C₆₇	—	2	2	no	—
42	C₂×Q₈×C₆₇	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₆₇	—	2	4	no	—
49	C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	2	4	no	—
51	C₄²×C₆₇	D₄	1	4	yes	—
52	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
53	Q₈×C₆₇	—	2	8	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	1	no	—
57	C₂×C₄×C₆₇	—	2	1	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	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₆₇	#41	1	2	yes	—
68	C₂×C₄×C₆₇	#41	1	2	yes	—
69	C₂×Dic₆₇	—	2	2	no	—
70	C₂×Dic₆₇	—	2	2	no	—
71	C₂×Dic₆₇	—	2	2	no	—
72	C₂×Dic₆₇	—	2	4	no	—
73	C₂×Dic₆₇	—	2	4	no	—
74	C₂×Dic₆₇	—	2	2	no	—
75	C₂×Dic₆₇	—	2	4	no	—
76	C₂×Dic₆₇	—	2	2	no	—
77	C₂×Dic₆₇	C₂×Q₈	1	4	yes	—
78	Dic₁₃₄	—	2	8	no	—
79	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
83	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
84	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
85	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
86	C₄×C₆₇	—	2	4	no	—
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	—	4	2	no	—
89	C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
90	C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
91	Dic₆₇	—	2	4	no	—
92	Dic₆₇	—	4	2	no	—
93	Dic₆₇	—	4	4	no	—
94	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₆₇	#537	1	1	yes	characteristic
97	C₂×C₆₇	#557	1	1	yes	characteristic
98	C₂×C₆₇	#557	1	1	yes	characteristic
99	C₂×C₆₇	Q₈²	1	1	yes	characteristic
100	C₂×C₆₇	#559	1	1	yes	characteristic
101	#2582	—	67	1	no	maximal, Sylow
102	C₆₇	#2582	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄×Q₈	—	67	1	no	—
104	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
105	#398	—	67	4	no	—
106	#391	—	67	2	no	—
107	#391	—	67	2	no	—
108	#389	—	67	2	no	—
109	#389	—	67	2	no	—
110	#384	—	67	1	no	—
111	C₂²×Q₈	—	67	1	no	—
112	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
113	C₄×Q₈	—	2	4	no	—
114	C₄×Q₈	—	134	4	no	—
115	C₂×C₄⋊C₄	—	67	4	no	—
116	C₂×C₄⋊C₄	—	67	2	no	—
117	C₂×C₄⋊C₄	—	67	2	no	—
118	C₂×C₄⋊C₄	—	67	1	no	—
119	C₂×C₄⋊C₄	—	67	2	no	—
120	C₂×C₄⋊C₄	—	67	2	no	—
121	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
123	C₂×C₄²	—	67	2	no	—
124	C₂×C₄²	C₂×D₆₇	1	2	yes	—
125	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
126	#95	—	67	4	no	—
127	#95	—	67	4	no	—
128	#95	—	67	4	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	67	4	no	—
132	C₂×Q₈	—	134	2	no	—
133	C₂×Q₈	—	134	2	no	—
134	C₂×Q₈	Dic₁₃₄	1	4	yes	complemented
135	C₂²×C₄	—	67	2	no	—
136	C₂²×C₄	—	67	4	no	—
137	C₂²×C₄	—	67	2	no	—
138	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
139	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
141	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	268	4	no	—
146	C₄⋊C₄	—	268	2	no	—
147	C₄⋊C₄	—	134	4	no	—
148	C₄⋊C₄	—	134	2	no	—
149	C₄⋊C₄	—	134	4	no	—
150	C₄⋊C₄	—	134	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	134	4	no	—
153	C₄²	C₂²⋊D₆₇	1	4	yes	—
154	C₂³	C₂³×D₆₇	1	1	yes	characteristic
155	Q₈	—	2	8	no	—
156	Q₈	—	134	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	67	4	no	—
165	C₂×C₄	—	134	2	no	—
166	C₂×C₄	—	134	4	no	—
167	C₂×C₄	—	134	4	no	—
168	C₂×C₄	—	134	4	no	—
169	C₂×C₄	—	134	2	no	—
170	C₂×C₄	—	134	2	no	—
171	C₂×C₄	—	134	2	no	—
172	C₂×C₄	—	134	2	no	—
173	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
175	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
176	C₂×C₄	Q₈×D₆₇	1	2	yes	—
177	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
178	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
179	C₂×C₄	#11761628	1	2	yes	—
180	C₂×C₄	#11761629	1	2	yes	—
181	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂²	C₂×Q₈×D₆₇	1	1	yes	characteristic
183	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
184	C₂²	C₂×#11761628	1	1	yes	characteristic
185	C₂²	C₂×#11761629	1	1	yes	characteristic
186	C₂²	#14524089	1	1	yes	characteristic
187	C₂²	#14524090	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	268	4	no	—
192	C₄	—	268	2	no	—
193	C₄	—	134	4	no	—
194	C₄	#14524019	1	4	yes	—
195	C₄	#14524022	1	4	yes	—
196	C₂	C₂×#14524019	1	1	yes	characteristic
197	C₂	C₂×#14524022	1	1	yes	characteristic
198	C₂	#16863974	1	1	yes	characteristic
199	C₂	#16864124	1	1	yes	characteristic
200	C₂	#16864125	1	1	yes	characteristic
201	C₂	Q₈×Dic₁₃₄	1	1	yes	characteristic
202	C₂	#16864131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×C₄×Q₈×C₆₇	C₂
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	#2582
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
29. C₂×Dic₁₃₄	Q₈	semidirect
42. C₂×Q₈×C₆₇	Q₈	semidirect
102. C₆₇	#2582	semidirect
134. C₂×Q₈	Dic₁₃₄	semidirect