Subgroups — #22569296

Order 8576.

364 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#16863526	C₂	1	2	yes	maximal, complemented
3	#16863525	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₃₄	C₂	1	2	yes	maximal, complemented
5	#16863453	C₂	1	1	yes	maximal, characteristic
6	#16863450	C₂	1	2	yes	maximal
7	#16863446	C₂	1	2	yes	maximal
8	#431×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₃₄	C₄	1	1	yes	characteristic
10	C₄×Dic₁₃₄	C₄	1	2	yes	—
11	C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
12	C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
13	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₃₄	C₂²	1	2	yes	—
15	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
17	C₂×Dic₆₇⋊C₄	C₄	1	2	yes	—
18	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
19	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
20	Dic₆₇⋊C₈	C₂²	1	4	yes	—
21	Dic₆₇⋊C₈	C₂²	1	4	yes	—
22	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
23	#14523954	C₂²	1	2	yes	—
24	#14523953	C₂²	1	4	yes	—
25	C₈×Dic₆₇	C₂²	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	C₂×Dic₁₃₄	—	2	4	no	—
32	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
36	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
38	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
41	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
42	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
43	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
44	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
45	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
46	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₆₇	C₂³	1	2	yes	—
48	C₄×Dic₆₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂×C₈×C₆₇	C₂³	1	2	yes	—
54	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₆₇	—	2	4	no	—
69	C₂×Dic₆₇	—	2	4	no	—
70	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
71	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
73	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
74	Dic₁₃₄	—	2	2	no	—
75	Dic₁₃₄	—	2	4	no	—
76	Dic₁₃₄	M₁₆	1	4	yes	—
77	C₈×C₆₇	—	2	2	no	—
78	C₈×C₆₇	—	2	2	no	—
79	C₆₇⋊C₈	—	2	4	no	—
80	C₂²×C₆₇	—	2	1	no	—
81	C₂²×C₆₇	—	2	1	no	—
82	C₂²×C₆₇	#131	1	2	yes	—
83	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₆₇	—	2	1	no	—
85	C₄×C₆₇	—	2	1	no	—
86	C₄×C₆₇	—	2	1	no	—
87	C₄×C₆₇	—	2	1	no	—
88	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
89	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
90	Dic₆₇	—	2	2	no	—
91	Dic₆₇	—	2	2	no	—
92	Dic₆₇	—	2	2	no	—
93	Dic₆₇	C₂×M₁₆	1	4	yes	—
94	C₂×C₆₇	—	2	2	no	—
95	C₂×C₆₇	#519	1	1	yes	characteristic
96	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
97	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
98	#2984	—	67	1	no	maximal, Sylow
99	C₆₇	#2984	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄×Q₈	—	67	1	no	—
101	#446	—	67	4	no	—
102	#446	—	67	2	no	—
103	C₈×Q₈	—	67	2	no	—
104	#432	—	67	1	no	—
105	#432	—	67	2	no	—
106	#431	—	67	2	no	—
107	#431	D₆₇	1	1	yes	characteristic
108	C₂²×Q₈	—	67	1	no	—
109	C₄×Q₈	—	67	1	no	—
110	C₄×Q₈	—	67	2	no	—
111	C₄×Q₈	—	67	1	no	—
112	C₄×Q₈	—	67	1	no	—
113	C₄×Q₈	—	67	2	no	—
114	C₄×Q₈	—	67	1	no	—
115	C₂×C₄⋊C₄	—	67	2	no	—
116	C₂×C₄⋊C₄	—	67	1	no	—
117	C₂×C₄²	—	67	2	no	—
118	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
119	C₄⋊C₈	—	67	4	no	—
120	C₄⋊C₈	—	67	4	no	—
121	C₄⋊C₈	—	67	2	no	—
122	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
123	C₂²⋊C₈	—	67	2	no	—
124	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
125	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
126	#97	—	67	4	no	—
127	#97	—	67	2	no	—
128	C₄×C₈	—	67	4	no	—
129	C₄×C₈	C₂×D₆₇	1	2	yes	—
130	C₂×Q₈	—	67	1	no	—
131	C₂×Q₈	—	67	2	no	—
132	C₂×Q₈	—	67	1	no	—
133	C₂×Q₈	—	134	4	no	—
134	C₂²×C₄	—	67	2	no	—
135	C₂²×C₄	—	67	2	no	—
136	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
139	C₂×C₈	—	67	4	no	—
140	C₂×C₈	C₂²×D₆₇	1	2	yes	—
141	C₂×C₈	C₂²×D₆₇	1	2	yes	—
142	C₄⋊C₄	—	67	2	no	—
143	C₄⋊C₄	—	67	2	no	—
144	C₄⋊C₄	—	67	2	no	—
145	C₄⋊C₄	—	67	2	no	—
146	C₄⋊C₄	—	67	2	no	—
147	C₄⋊C₄	—	67	1	no	—
148	C₄⋊C₄	—	67	1	no	—
149	C₄²	—	67	2	no	—
150	C₄²	—	67	2	no	—
151	C₄²	—	67	2	no	—
152	C₄²	—	67	2	no	—
153	C₄²	C₄×D₆₇	1	1	yes	characteristic
154	C₄²	C₄×D₆₇	1	1	yes	characteristic
155	C₄²	C₂²×D₆₇	1	1	yes	characteristic
156	C₄²	C₂²×D₆₇	1	1	yes	characteristic
157	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
158	Q₈	—	67	4	no	—
159	Q₈	—	134	4	no	—
160	Q₈	—	134	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	67	2	no	—
165	C₂×C₄	—	67	2	no	—
166	C₂×C₄	—	67	2	no	—
167	C₂×C₄	—	67	2	no	—
168	C₂×C₄	—	134	4	no	—
169	C₂×C₄	—	134	4	no	—
170	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	134	4	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	#14524049	1	2	yes	—
182	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	67	4	no	—
188	C₄	—	134	2	no	—
189	C₄	—	134	2	no	—
190	C₄	—	134	2	no	—
191	C₄	M₁₆×D₆₇	1	2	yes	—
192	C₄	#14524090	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#16863873	1	1	yes	characteristic
195	C₂	C₂×M₁₆×D₆₇	1	1	yes	characteristic
196	C₂	C₂×#14524049	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#431×C₆₇	C₂
hypercenter	#431	D₆₇
hypocenter	C₆₇	#2984
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16863526	C₂	semidirect
3. #16863525	C₂	semidirect
4. C₈×Dic₁₃₄	C₂	semidirect
99. C₆₇	#2984	semidirect