Subgroups — C₂×#16864024 #22573607

Order 8576.

1080 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864024	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14524043	C₂	1	2	yes	maximal, complemented
3	C₂×#14524042	C₂	1	2	yes	maximal, complemented
4	C₂×#14524035	C₂	1	2	yes	maximal, complemented
5	C₂×#14524034	C₂	1	2	yes	maximal, complemented
6	C₂×#14524020	C₂	1	4	yes	maximal
7	C₂×#14524002	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₃₄	C₂²	1	4	yes	—
11	C₂³×Dic₆₇	C₂²	1	2	yes	—
12	#14524043	C₂²	1	16	yes	complemented
13	#14524042	C₂²	1	16	yes	complemented
14	#14524035	C₂²	1	16	yes	complemented
15	#14524034	C₂²	1	16	yes	complemented
16	#14524020	C₂²	1	32	yes	—
17	#14524002	C₂²	1	8	yes	—
18	C₂×C₄⋊Dic₆₇	C₂²	1	8	yes	—
19	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
24	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
25	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
26	C₂×C₂²⋊Dic₆₇	C₂²	1	2	yes	—
27	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#117×C₆₇	C₂²	1	8	yes	—
30	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
34	C₂×Dic₁₃₄	—	4	8	no	—
35	C₂×Dic₁₃₄	C₂³	1	16	yes	—
36	C₂²×Dic₆₇	—	4	4	no	—
37	C₂²×Dic₆₇	C₂³	1	2	yes	—
38	C₂²×Dic₆₇	C₂³	1	2	yes	—
39	C₂²×Dic₆₇	C₂³	1	4	yes	—
40	C₂²×Dic₆₇	C₂³	1	8	yes	—
41	C₄⋊Dic₆₇	C₂³	1	8	yes	—
42	C₄⋊Dic₆₇	C₂³	1	8	yes	—
43	C₄⋊Dic₆₇	C₂³	1	4	yes	—
44	C₄⋊Dic₆₇	C₂³	1	4	yes	—
45	C₄⋊Dic₆₇	C₂³	1	16	yes	—
46	Dic₆₇⋊C₄	C₂³	1	16	yes	—
47	Dic₆₇⋊C₄	C₂³	1	16	yes	—
48	C₂²⋊Dic₆₇	C₂³	1	8	yes	—
49	C₄×Dic₆₇	C₂³	1	8	yes	—
50	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	Q₈	1	8	yes	complemented
52	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
57	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
58	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
59	C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
60	C₄²×C₆₇	C₂³	1	8	yes	—
61	C₂³×C₆₇	—	2	2	no	—
62	C₂³×C₆₇	—	2	2	no	—
63	C₂³×C₆₇	C₂×Q₈	1	4	yes	—
64	C₂³×C₆₇	C₂⁴	1	2	yes	—
65	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	2	4	no	—
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
69	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
70	C₂×C₄×C₆₇	C₂×Q₈	1	16	yes	complemented
71	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
72	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
73	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
74	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
75	C₂×Dic₆₇	—	2	8	no	—
76	C₂×Dic₆₇	—	2	4	no	—
77	C₂×Dic₆₇	—	2	4	no	—
78	C₂×Dic₆₇	—	4	8	no	—
79	C₂×Dic₆₇	C₂⁴	1	4	yes	—
80	C₂×Dic₆₇	C₂⁴	1	4	yes	—
81	C₂×Dic₆₇	C₂⁴	1	8	yes	—
82	Dic₁₃₄	—	4	16	no	—
83	C₂²×C₆₇	—	2	8	no	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	—	2	2	no	—
86	C₂²×C₆₇	C₂²×Q₈	1	4	yes	—
87	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
88	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
89	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
90	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
91	C₄×C₆₇	—	2	4	no	—
92	C₄×C₆₇	—	2	4	no	—
93	C₄×C₆₇	C₂²×Q₈	1	4	yes	—
94	C₄×C₆₇	C₂²×Q₈	1	4	yes	—
95	Dic₆₇	—	2	4	no	—
96	Dic₆₇	—	2	8	no	—
97	Dic₆₇	—	2	4	no	—
98	C₂×C₆₇	—	2	4	no	—
99	C₂×C₆₇	#544	1	4	yes	complemented
100	C₂×C₆₇	C₂³×Q₈	1	1	yes	characteristic
101	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
102	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×#544	—	67	1	no	maximal, Sylow
104	C₆₇	C₂×#544	1	1	yes	characteristic, complemented, Sylow
105	#544	—	67	16	no	—
106	C₂×C₄⋊Q₈	—	67	2	no	—
107	C₂×C₄⋊Q₈	—	67	2	no	—
108	C₂×#125	—	67	2	no	—
109	C₂×#125	—	67	2	no	—
110	C₂×C₂²⋊Q₈	—	67	4	no	—
111	C₂×#117	—	67	1	no	—
112	C₂×#117	D₆₇	1	1	yes	characteristic
113	C₂²×C₄⋊C₄	—	67	1	no	—
114	C₂²×Q₈	—	67	4	no	—
115	C₂³×C₄	—	67	2	no	—
116	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
117	C₄⋊Q₈	—	67	16	no	—
118	C₄⋊Q₈	—	67	16	no	—
119	#125	—	67	16	no	—
120	#125	—	67	16	no	—
121	C₂²⋊Q₈	—	67	32	no	—
122	#117	—	67	8	no	—
123	#117	C₂×D₆₇	1	8	yes	—
124	C₂×C₄⋊C₄	—	67	2	no	—
125	C₂×C₄⋊C₄	—	67	2	no	—
126	C₂×C₄⋊C₄	—	67	8	no	—
127	C₂×C₄⋊C₄	—	67	1	no	—
128	C₂×C₄⋊C₄	—	67	1	no	—
129	C₂×C₄⋊C₄	—	67	4	no	—
130	C₂×C₄⋊C₄	—	67	4	no	—
131	C₂×C₄⋊C₄	—	67	4	no	—
132	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
134	C₂×C₂²⋊C₄	—	67	2	no	—
135	C₂×C₂²⋊C₄	C₂×D₆₇	1	2	yes	—
136	C₂×C₄²	—	67	2	no	—
137	C₂×C₄²	C₂×D₆₇	1	2	yes	—
138	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
139	C₂×Q₈	—	268	8	no	—
140	C₂×Q₈	—	67	16	no	—
141	C₂²×C₄	—	268	4	no	—
142	C₂²×C₄	—	67	8	no	—
143	C₂²×C₄	—	67	2	no	—
144	C₂²×C₄	—	67	2	no	—
145	C₂²×C₄	—	67	4	no	—
146	C₂²×C₄	Dic₁₃₄	1	8	yes	complemented
147	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
148	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
149	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
152	C₄⋊C₄	—	67	8	no	—
153	C₄⋊C₄	—	67	8	no	—
154	C₄⋊C₄	—	67	4	no	—
155	C₄⋊C₄	—	67	4	no	—
156	C₄⋊C₄	—	67	16	no	—
157	C₄⋊C₄	—	67	16	no	—
158	C₄⋊C₄	—	67	16	no	—
159	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
160	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
161	C₂²⋊C₄	—	67	8	no	—
162	C₂²⋊C₄	C₂²×D₆₇	1	8	yes	—
163	C₄²	—	67	8	no	—
164	C₄²	C₂²×D₆₇	1	8	yes	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	C₂×Dic₁₃₄	1	4	yes	—
168	C₂³	C₂³×D₆₇	1	2	yes	—
169	C₂³	C₂³×D₆₇	1	1	yes	characteristic
170	Q₈	—	268	16	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	268	8	no	—
174	C₂×C₄	—	67	4	no	—
175	C₂×C₄	—	67	4	no	—
176	C₂×C₄	—	67	8	no	—
177	C₂×C₄	—	134	8	no	—
178	C₂×C₄	—	134	4	no	—
179	C₂×C₄	—	134	4	no	—
180	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
181	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
182	C₂×C₄	C₂×Dic₁₃₄	1	16	yes	complemented
183	C₂×C₄	C₂³×D₆₇	1	4	yes	—
184	C₂×C₄	C₂³×D₆₇	1	4	yes	—
185	C₂×C₄	C₂³×D₆₇	1	2	yes	—
186	C₂×C₄	C₂³×D₆₇	1	2	yes	—
187	C₂²	—	2	8	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	C₂²×Dic₁₃₄	1	4	yes	—
191	C₂²	C₂²×Dic₁₃₄	1	2	yes	—
192	C₂²	#14524089	1	2	yes	—
193	C₂²	#14524090	1	2	yes	—
194	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	4	no	—
197	C₄	—	134	8	no	—
198	C₄	—	134	4	no	—
199	C₄	—	134	4	no	—
200	C₄	C₂²×Dic₁₃₄	1	4	yes	—
201	C₄	C₂²×Dic₁₃₄	1	4	yes	—
202	C₂	—	2	4	no	—
203	C₂	#16864024	1	4	yes	direct product factor
204	C₂	C₂³×Dic₁₃₄	1	1	yes	characteristic
205	C₂	C₂×#14524089	1	1	yes	characteristic
206	C₂	C₂×#14524090	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁵
Frattini	C₂²	C₂⁴×D₆₇
Fitting	C₂×#117×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂×#117	D₆₇
hypocenter	C₆₇	C₂×#544
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #16864024	C₂	direct (N × H)
2. C₂×#14524043	C₂	semidirect
3. C₂×#14524042	C₂	semidirect
4. C₂×#14524035	C₂	semidirect
5. C₂×#14524034	C₂	semidirect
12. #14524043	C₂²	semidirect
13. #14524042	C₂²	semidirect
14. #14524035	C₂²	semidirect
15. #14524034	C₂²	semidirect
51. C₂²×C₄×C₆₇	Q₈	semidirect
70. C₂×C₄×C₆₇	C₂×Q₈	semidirect
99. C₂×C₆₇	#544	semidirect
104. C₆₇	C₂×#544	semidirect
146. C₂²×C₄	Dic₁₃₄	semidirect
182. C₂×C₄	C₂×Dic₁₃₄	semidirect
203. C₂	#16864024	direct (N × H)

Subgroups — C2×#16864024 #22573607

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16864024 #22573607