Subgroups — C₂×#16864131 #22573797

Order 8576.

1240 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864131	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14524040	C₂	1	1	yes	maximal, characteristic
3	C₂×#14524032	C₂	1	2	yes	maximal, complemented
4	C₂×#14524031	C₂	1	2	yes	maximal, complemented
5	C₂×#14524028	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₂₆₈	C₂	1	2	yes	maximal, complemented
8	C₂×#14524005	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	C₂²×D₂₆₈	C₂²	1	2	yes	—
12	C₂²×D₂₆₈	C₂²	1	2	yes	—
13	C₂²×D₂₆₈	C₂²	1	2	yes	—
14	C₂²×C₄×D₆₇	C₂²	1	4	yes	—
15	#14524040	C₂²	1	8	yes	—
16	#14524032	C₂²	1	16	yes	complemented
17	#14524031	C₂²	1	16	yes	complemented
18	#14524028	C₂²	1	32	yes	complemented
19	C₄×Dic₁₃₄	C₂²	1	8	yes	complemented
20	C₄×D₂₆₈	C₂²	1	16	yes	complemented
21	#14524005	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	—
25	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	—
26	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
28	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	C₂³×D₆₇	C₂³	1	4	yes	—
35	C₂×Dic₁₃₄	—	2	4	no	—
36	C₂×Dic₁₃₄	C₂³	1	4	yes	—
37	C₂×D₂₆₈	—	2	8	no	—
38	C₂×D₂₆₈	—	2	8	no	—
39	C₂×D₂₆₈	—	4	4	no	—
40	C₂×D₂₆₈	C₂³	1	8	yes	—
41	C₂×D₂₆₈	C₂³	1	8	yes	—
42	C₂×D₂₆₈	C₂³	1	8	yes	—
43	C₂²×Dic₆₇	C₂³	1	2	yes	—
44	C₂²×Dic₆₇	C₂³	1	2	yes	—
45	C₂×C₄×D₆₇	—	4	8	no	—
46	C₂×C₄×D₆₇	C₂³	1	16	yes	—
47	C₄⋊Dic₆₇	C₂³	1	4	yes	—
48	Dic₆₇⋊C₄	C₂³	1	8	yes	—
49	(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	—
50	(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	—
51	(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	—
52	C₄×Dic₆₇	C₂³	1	8	yes	—
53	C₂×Q₈×C₆₇	—	2	4	no	—
54	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
61	C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
62	C₄²×C₆₇	C₂³	1	8	yes	—
63	C₄²×C₆₇	C₂³	1	4	yes	—
64	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂²×D₆₇	—	2	8	no	—
66	C₂²×D₆₇	—	2	8	no	—
67	C₂²×D₆₇	—	2	8	no	—
68	C₂²×D₆₇	C₂⁴	1	8	yes	—
69	Q₈×C₆₇	—	2	8	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	#41	1	4	yes	—
74	C₂×C₄×C₆₇	#41	1	4	yes	—
75	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
76	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
77	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
78	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
79	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
80	C₂×Dic₆₇	—	2	4	no	—
81	C₂×Dic₆₇	—	2	4	no	—
82	C₂×Dic₆₇	C₂⁴	1	4	yes	—
83	C₂×Dic₆₇	C₂⁴	1	4	yes	—
84	D₂₆₈	—	2	16	no	—
85	D₂₆₈	—	2	16	no	—
86	D₂₆₈	—	4	8	no	—
87	C₄×D₆₇	—	4	16	no	—
88	Dic₁₃₄	—	2	8	no	—
89	C₂²×C₆₇	C₂×#41	1	2	yes	—
90	C₂²×C₆₇	C₂×#41	1	2	yes	—
91	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
92	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
93	C₂×D₆₇	—	2	8	no	—
94	C₂×D₆₇	—	2	8	no	—
95	C₂×D₆₇	—	2	8	no	—
96	C₂×D₆₇	—	4	16	no	—
97	C₄×C₆₇	—	2	4	no	—
98	C₄×C₆₇	—	2	4	no	—
99	C₄×C₆₇	—	2	2	no	—
100	C₄×C₆₇	C₂×#41	1	4	yes	—
101	C₄×C₆₇	C₂×#41	1	4	yes	—
102	Dic₆₇	—	2	4	no	—
103	Dic₆₇	—	2	4	no	—
104	C₂×C₆₇	#559	1	4	yes	complemented
105	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
106	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
107	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
108	D₆₇	—	4	8	no	—
109	C₂×#559	—	67	1	no	maximal, Sylow
110	C₆₇	C₂×#559	1	1	yes	characteristic, complemented, Sylow
111	#559	—	67	16	no	—
112	C₂×#127	—	67	1	no	—
113	C₂×#124	—	67	2	no	—
114	C₂×#124	—	67	2	no	—
115	C₂×#123	—	67	4	no	—
116	C₂×C₄×Q₈	—	67	1	no	—
117	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic, complemented
118	C₂×C₄×D₄	—	67	2	no	—
119	C₂×C₄×D₄	—	67	2	no	—
120	C₂²×Q₈	—	67	1	no	—
121	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
122	C₂²×D₄	—	67	2	no	—
123	C₂²×D₄	—	67	2	no	—
124	C₂²×D₄	—	67	2	no	—
125	C₂³×C₄	—	67	4	no	—
126	#127	—	67	8	no	—
127	#124	—	67	16	no	—
128	#124	—	67	16	no	—
129	#123	—	67	32	no	—
130	C₄×Q₈	—	67	8	no	—
131	C₄×Q₈	C₂×D₆₇	1	8	yes	complemented
132	C₄×D₄	—	67	16	no	—
133	C₄×D₄	—	67	16	no	—
134	C₂×C₄⋊C₄	—	67	1	no	—
135	C₂×C₄⋊C₄	—	67	2	no	—
136	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
138	C₂×C₂²⋊C₄	—	67	4	no	—
139	C₂×C₂²⋊C₄	—	67	4	no	—
140	C₂×C₂²⋊C₄	—	67	4	no	—
141	C₂×C₄²	—	67	2	no	—
142	C₂×C₄²	C₂×D₆₇	1	2	yes	—
143	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
144	C₂⁴	—	67	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	67	4	no	—
147	C₂×Q₈	—	134	4	no	—
148	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
149	C₂×D₄	—	268	4	no	—
150	C₂×D₄	—	67	8	no	—
151	C₂×D₄	—	67	8	no	—
152	C₂×D₄	—	67	8	no	—
153	C₂×D₄	—	134	8	no	—
154	C₂×D₄	—	134	8	no	—
155	C₂²×C₄	—	268	8	no	—
156	C₂²×C₄	—	67	16	no	—
157	C₂²×C₄	—	67	2	no	—
158	C₂²×C₄	—	67	2	no	—
159	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
161	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
164	C₄⋊C₄	—	67	4	no	—
165	C₄⋊C₄	—	67	8	no	—
166	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
167	C₄⋊C₄	C₂²×D₆₇	1	8	yes	—
168	C₂²⋊C₄	—	67	16	no	—
169	C₂²⋊C₄	—	67	16	no	—
170	C₂²⋊C₄	—	67	16	no	—
171	C₄²	—	67	8	no	—
172	C₄²	C₂²×D₆₇	1	8	yes	—
173	C₄²	C₂²×D₆₇	1	4	yes	—
174	C₂³	—	67	8	no	—
175	C₂³	—	134	8	no	—
176	C₂³	—	134	8	no	—
177	C₂³	—	134	8	no	—
178	C₂³	C₂³×D₆₇	1	1	yes	characteristic
179	Q₈	—	2	8	no	—
180	Q₈	—	134	8	no	—
181	D₄	—	268	8	no	—
182	D₄	—	134	16	no	—
183	D₄	—	134	16	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	268	16	no	—
188	C₂×C₄	—	67	4	no	—
189	C₂×C₄	—	67	4	no	—
190	C₂×C₄	—	134	4	no	—
191	C₂×C₄	—	134	4	no	—
192	C₂×C₄	#11761628	1	4	yes	—
193	C₂×C₄	#11761629	1	4	yes	—
194	C₂×C₄	C₂³×D₆₇	1	4	yes	—
195	C₂×C₄	C₂³×D₆₇	1	4	yes	—
196	C₂×C₄	C₂³×D₆₇	1	2	yes	—
197	C₂×C₄	C₂³×D₆₇	1	2	yes	—
198	C₂×C₄	C₂³×D₆₇	1	2	yes	—
199	C₂²	—	268	16	no	—
200	C₂²	—	134	8	no	—
201	C₂²	—	134	8	no	—
202	C₂²	—	134	8	no	—
203	C₂²	C₂×#11761628	1	2	yes	—
204	C₂²	C₂×#11761629	1	2	yes	—
205	C₂²	#14524089	1	2	yes	—
206	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	134	4	no	—
211	C₄	—	134	4	no	—
212	C₄	C₂×#11761628	1	4	yes	—
213	C₄	C₂×#11761629	1	4	yes	—
214	C₂	—	268	8	no	—
215	C₂	#16864131	1	4	yes	direct product factor
216	C₂	C₂²×#11761628	1	1	yes	characteristic
217	C₂	C₂²×#11761629	1	1	yes	characteristic
218	C₂	C₂×#14524089	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁵
Frattini	C₂²	C₂⁴×D₆₇
Fitting	C₂×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×#559
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #16864131	C₂	direct (N × H)
3. C₂×#14524032	C₂	semidirect
4. C₂×#14524031	C₂	semidirect
5. C₂×#14524028	C₂	semidirect
6. C₂×C₄×Dic₁₃₄	C₂	semidirect
7. C₂×C₄×D₂₆₈	C₂	semidirect
8. C₂×#14524005	C₂	semidirect
9. C₂×C₄×Q₈×C₆₇	C₂	semidirect
16. #14524032	C₂²	semidirect
17. #14524031	C₂²	semidirect
18. #14524028	C₂²	semidirect
19. C₄×Dic₁₃₄	C₂²	semidirect
20. C₄×D₂₆₈	C₂²	semidirect
21. #14524005	C₂²	semidirect
29. C₄×Q₈×C₆₇	C₂²	semidirect
104. C₂×C₆₇	#559	semidirect
110. C₆₇	C₂×#559	semidirect
117. C₂×C₄×Q₈	D₆₇	semidirect
131. C₄×Q₈	C₂×D₆₇	semidirect
215. C₂	#16864131	direct (N × H)

Subgroups — C2×#16864131 #22573797

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16864131 #22573797