Subgroups — C₂×#20265892 #27184672

Order 12928.

1240 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265892	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692255	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692241	C₂	1	4	yes	maximal, complemented
4	C₂×#15692229	C₂	1	2	yes	maximal, complemented
5	C₂×#15692226	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692214	C₂	1	2	yes	maximal, complemented
8	C₂×#15692213	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
11	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
13	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
14	#15692255	C₂²	1	8	yes	complemented
15	#15692241	C₂²	1	32	yes	complemented
16	#15692229	C₂²	1	16	yes	complemented
17	#15692226	C₂²	1	16	yes	complemented
18	C₄×D₄₀₄	C₂²	1	8	yes	complemented
19	#15692214	C₂²	1	16	yes	complemented
20	#15692213	C₂²	1	16	yes	complemented
21	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
25	C₂×#13108051	C₂²	1	4	yes	—
26	C₂×#13108051	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
28	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
30	C₄×D₄×C₁₀₁	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂³×D₁₀₁	C₂³	1	2	yes	—
35	C₂×Dic₂₀₂	—	4	4	no	—
36	C₂×Dic₂₀₂	C₂³	1	8	yes	—
37	C₂×D₄₀₄	—	2	4	no	—
38	C₂×D₄₀₄	C₂³	1	4	yes	—
39	C₂×C₂²⋊D₁₀₁	—	4	8	no	—
40	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
43	C₂×C₄×D₁₀₁	—	2	8	no	—
44	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
45	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
46	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
47	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
48	C₂²⋊Dic₁₀₁	C₂³	1	16	yes	—
49	#13108051	C₂³	1	16	yes	—
50	#13108051	C₂³	1	8	yes	—
51	C₄×Dic₁₀₁	C₂³	1	16	yes	—
52	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
53	C₂×D₄×C₁₀₁	—	2	4	no	—
54	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₀₁	—	2	8	no	—
56	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
62	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
63	C₄²×C₁₀₁	C₂³	1	4	yes	—
64	C₂³×C₁₀₁	—	2	4	no	—
65	C₂³×C₁₀₁	—	2	4	no	—
66	C₂³×C₁₀₁	—	2	4	no	—
67	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
68	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
69	C₂²×D₁₀₁	—	2	4	no	—
70	C₂²×D₁₀₁	—	2	4	no	—
71	C₂²×D₁₀₁	—	2	4	no	—
72	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
73	D₄×C₁₀₁	—	2	8	no	—
74	C₂×C₄×C₁₀₁	—	2	4	no	—
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	—	2	16	no	—
77	C₂×C₄×C₁₀₁	#41	1	4	yes	—
78	C₂×C₄×C₁₀₁	#41	1	4	yes	—
79	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
80	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
81	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
82	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
83	C₂²⋊D₁₀₁	—	4	16	no	—
84	C₂×Dic₁₀₁	—	2	4	no	—
85	C₂×Dic₁₀₁	—	2	8	no	—
86	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
87	C₂×Dic₁₀₁	C₂⁴	1	8	yes	—
88	D₄₀₄	—	2	8	no	—
89	C₄×D₁₀₁	—	2	16	no	—
90	Dic₂₀₂	—	4	8	no	—
91	C₂²×C₁₀₁	—	2	4	no	—
92	C₂²×C₁₀₁	—	2	4	no	—
93	C₂²×C₁₀₁	—	2	4	no	—
94	C₂²×C₁₀₁	—	4	8	no	—
95	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
96	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
97	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
98	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
99	C₂×D₁₀₁	—	2	4	no	—
100	C₂×D₁₀₁	—	2	4	no	—
101	C₂×D₁₀₁	—	2	4	no	—
102	C₂×D₁₀₁	—	4	8	no	—
103	C₄×C₁₀₁	—	2	4	no	—
104	C₄×C₁₀₁	—	2	2	no	—
105	C₄×C₁₀₁	C₂×#41	1	4	yes	—
106	C₄×C₁₀₁	C₂×#41	1	4	yes	—
107	Dic₁₀₁	—	2	4	no	—
108	Dic₁₀₁	—	2	8	no	—
109	C₂×C₁₀₁	—	4	4	no	—
110	C₂×C₁₀₁	#555	1	4	yes	complemented
111	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
112	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
113	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
114	D₁₀₁	—	4	4	no	—
115	C₂×#555	—	101	1	no	maximal, Sylow
116	C₁₀₁	C₂×#555	1	1	yes	characteristic, complemented, Sylow
117	#555	—	101	16	no	—
118	C₂×C₄⋊Q₈	—	101	1	no	—
119	C₂×#124	—	101	4	no	—
120	C₂×#121	—	101	2	no	—
121	C₂×#121	—	101	2	no	—
122	C₂×C₄×D₄	—	101	1	no	—
123	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
124	C₂×#117	—	101	2	no	—
125	C₂×#117	—	101	2	no	—
126	C₂²×Q₈	—	101	2	no	—
127	C₂²×D₄	—	101	4	no	—
128	C₂²×D₄	—	101	1	no	—
129	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
130	C₂³×C₄	—	101	2	no	—
131	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
132	C₄⋊Q₈	—	101	8	no	—
133	#124	—	101	32	no	—
134	#121	—	101	16	no	—
135	#121	—	101	16	no	—
136	C₄×D₄	—	101	8	no	—
137	C₄×D₄	C₂×D₁₀₁	1	8	yes	complemented
138	#117	—	101	16	no	—
139	#117	—	101	16	no	—
140	C₂×C₄⋊C₄	—	101	1	no	—
141	C₂×C₄⋊C₄	—	101	2	no	—
142	C₂×C₄⋊C₄	—	101	2	no	—
143	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	—	101	4	no	—
145	C₂×C₂²⋊C₄	—	101	4	no	—
146	C₂×C₂²⋊C₄	—	101	2	no	—
147	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
148	C₂×C₄²	—	101	4	no	—
149	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
150	C₂⁴	—	101	2	no	—
151	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
152	C₂×Q₈	—	101	8	no	—
153	C₂×Q₈	—	404	4	no	—
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	—	101	16	no	—
156	C₂×D₄	—	101	4	no	—
157	C₂×D₄	—	404	8	no	—
158	C₂×D₄	—	202	4	no	—
159	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	101	2	no	—
162	C₂²×C₄	—	101	4	no	—
163	C₂²×C₄	—	101	8	no	—
164	C₂²×C₄	—	202	8	no	—
165	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
166	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
170	C₄⋊C₄	—	101	4	no	—
171	C₄⋊C₄	—	101	8	no	—
172	C₄⋊C₄	—	101	8	no	—
173	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
174	C₂²⋊C₄	—	101	16	no	—
175	C₂²⋊C₄	—	101	16	no	—
176	C₂²⋊C₄	—	101	8	no	—
177	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
178	C₄²	—	101	16	no	—
179	C₄²	C₂²×D₁₀₁	1	4	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	101	4	no	—
184	C₂³	—	202	4	no	—
185	C₂³	—	202	4	no	—
186	C₂³	—	202	4	no	—
187	C₂³	C₂³×D₁₀₁	1	4	yes	—
188	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
189	Q₈	—	404	8	no	—
190	D₄	—	2	8	no	—
191	D₄	—	404	16	no	—
192	D₄	—	202	8	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	101	4	no	—
197	C₂×C₄	—	101	8	no	—
198	C₂×C₄	—	202	4	no	—
199	C₂×C₄	—	202	8	no	—
200	C₂×C₄	—	202	16	no	—
201	C₂×C₄	#13108073	1	4	yes	—
202	C₂×C₄	#13108074	1	4	yes	—
203	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
204	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
205	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
206	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	8	no	—
211	C₂²	—	404	8	no	—
212	C₂²	—	202	4	no	—
213	C₂²	—	202	4	no	—
214	C₂²	—	202	4	no	—
215	C₂²	C₂×#13108073	1	2	yes	—
216	C₂²	C₂×#13108074	1	2	yes	—
217	C₂²	#15692301	1	2	yes	—
218	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	2	2	no	—
221	C₄	—	202	4	no	—
222	C₄	—	202	8	no	—
223	C₄	C₂×#13108073	1	4	yes	—
224	C₄	C₂×#13108074	1	4	yes	—
225	C₂	—	4	4	no	—
226	C₂	—	404	4	no	—
227	C₂	#20265892	1	4	yes	direct product factor
228	C₂	C₂²×#13108073	1	1	yes	characteristic
229	C₂	C₂²×#13108074	1	1	yes	characteristic
230	C₂	C₂×#15692301	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#555
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20265892	C₂	direct (N × H)
2. C₂×#15692255	C₂	semidirect
3. C₂×#15692241	C₂	semidirect
4. C₂×#15692229	C₂	semidirect
5. C₂×#15692226	C₂	semidirect
6. C₂×C₄×D₄₀₄	C₂	semidirect
7. C₂×#15692214	C₂	semidirect
8. C₂×#15692213	C₂	semidirect
9. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
14. #15692255	C₂²	semidirect
15. #15692241	C₂²	semidirect
16. #15692229	C₂²	semidirect
17. #15692226	C₂²	semidirect
18. C₄×D₄₀₄	C₂²	semidirect
19. #15692214	C₂²	semidirect
20. #15692213	C₂²	semidirect
30. C₄×D₄×C₁₀₁	C₂²	semidirect
110. C₂×C₁₀₁	#555	semidirect
116. C₁₀₁	C₂×#555	semidirect
123. C₂×C₄×D₄	D₁₀₁	semidirect
137. C₄×D₄	C₂×D₁₀₁	semidirect
227. C₂	#20265892	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20265892	2
C₂×#15692255	2
C₂×#15692241	2
C₂×#15692229	2
C₂×#15692226	2
C₂×C₄×D₄₀₄	2
C₂×#15692214	2
C₂×#15692213	2
C₂×C₄×D₄×C₁₀₁	2
C₂×#555	101

Subgroups — C2×#20265892 #27184672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265892 #27184672