Subgroups — C₂×#20265848 #27184612

Order 12928.

1736 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108074	C₂	1	2	yes	maximal, complemented
2	#20265848	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692251	C₂	1	2	yes	maximal, complemented
5	C₂×#15692229	C₂	1	2	yes	maximal, complemented
6	C₂×#15692228	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15692217	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13108074	—	2	16	no	—
11	C₂×#13108074	C₂²	1	16	yes	complemented
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	C₂²×D₄₀₄	C₂²	1	2	yes	—
14	C₂²×D₄₀₄	C₂²	1	2	yes	—
15	C₂²×D₄₀₄	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
17	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
19	#15692252	C₂²	1	8	yes	complemented
20	#15692251	C₂²	1	16	yes	complemented
21	#15692229	C₂²	1	16	yes	complemented
22	#15692228	C₂²	1	32	yes	complemented
23	C₄×Dic₂₀₂	C₂²	1	8	yes	complemented
24	#15692217	C₂²	1	16	yes	complemented
25	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×#13108053	C₂²	1	2	yes	—
27	C₂×#13108051	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
31	C₄×D₄×C₁₀₁	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂³×D₁₀₁	C₂³	1	4	yes	—
36	#13108074	—	2	64	no	—
37	C₂×Dic₂₀₂	D₄	1	8	yes	complemented
38	C₂×Dic₂₀₂	C₂³	1	4	yes	—
39	C₂×D₄₀₄	—	2	8	no	—
40	C₂×D₄₀₄	—	2	8	no	—
41	C₂×D₄₀₄	—	4	4	no	—
42	C₂×D₄₀₄	C₂³	1	8	yes	—
43	C₂×D₄₀₄	C₂³	1	8	yes	—
44	C₂×D₄₀₄	C₂³	1	8	yes	—
45	C₂×C₂²⋊D₁₀₁	—	2	16	no	—
46	C₂×C₂²⋊D₁₀₁	—	4	8	no	—
47	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	—
48	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
51	C₂×C₄×D₁₀₁	—	2	16	no	—
52	C₂×C₄×D₁₀₁	C₂³	1	16	yes	—
53	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
54	#13108053	C₂³	1	8	yes	—
55	#13108051	C₂³	1	16	yes	—
56	C₄×Dic₁₀₁	C₂³	1	8	yes	—
57	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
58	C₂×D₄×C₁₀₁	—	2	4	no	—
59	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
60	C₂²×C₄×C₁₀₁	—	2	8	no	—
61	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
64	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
67	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
68	C₄²×C₁₀₁	C₂³	1	4	yes	—
69	C₂³×C₁₀₁	—	2	4	no	—
70	C₂³×C₁₀₁	—	2	4	no	—
71	C₂³×C₁₀₁	—	2	4	no	—
72	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
73	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
74	C₂²×D₁₀₁	—	2	8	no	—
75	C₂²×D₁₀₁	—	2	8	no	—
76	C₂²×D₁₀₁	—	2	8	no	—
77	C₂²×D₁₀₁	C₂⁴	1	8	yes	—
78	D₄×C₁₀₁	—	2	8	no	—
79	C₂×C₄×C₁₀₁	—	2	4	no	—
80	C₂×C₄×C₁₀₁	—	2	2	no	—
81	C₂×C₄×C₁₀₁	—	2	16	no	—
82	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₁₀₁	#41	1	4	yes	complemented
84	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
85	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
86	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
88	C₂²⋊D₁₀₁	—	2	32	no	—
89	C₂²⋊D₁₀₁	—	4	16	no	—
90	C₂×Dic₁₀₁	—	2	4	no	—
91	C₂×Dic₁₀₁	C₂×D₄	1	8	yes	—
92	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
93	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
94	D₄₀₄	—	2	16	no	—
95	D₄₀₄	—	2	16	no	—
96	D₄₀₄	—	4	8	no	—
97	C₄×D₁₀₁	—	2	32	no	—
98	Dic₂₀₂	C₂×D₄	1	16	yes	complemented
99	C₂²×C₁₀₁	—	2	4	no	—
100	C₂²×C₁₀₁	—	2	4	no	—
101	C₂²×C₁₀₁	—	2	4	no	—
102	C₂²×C₁₀₁	—	4	8	no	—
103	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
104	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
105	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
106	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
107	C₂×D₁₀₁	—	2	8	no	—
108	C₂×D₁₀₁	—	2	8	no	—
109	C₂×D₁₀₁	—	2	8	no	—
110	C₂×D₁₀₁	—	4	16	no	—
111	C₄×C₁₀₁	—	2	4	no	—
112	C₄×C₁₀₁	—	2	2	no	—
113	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
114	C₄×C₁₀₁	C₂×#41	1	4	yes	complemented
115	Dic₁₀₁	—	2	4	no	—
116	Dic₁₀₁	C₂²×D₄	1	8	yes	—
117	C₂×C₁₀₁	—	4	4	no	—
118	C₂×C₁₀₁	#550	1	4	yes	complemented
119	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
120	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
121	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
122	D₁₀₁	—	4	8	no	—
123	C₂×#550	—	101	1	no	maximal, Sylow
124	C₁₀₁	C₂×#550	1	1	yes	characteristic, complemented, Sylow
125	C₂²×#41	—	101	2	no	—
126	#550	—	101	16	no	—
127	C₂×#127	—	101	2	no	—
128	C₂×#127	—	101	1	no	—
129	C₂×#121	—	101	2	no	—
130	C₂×#121	—	101	4	no	—
131	C₂×C₄×Q₈	—	101	1	no	—
132	C₂×C₄×D₄	—	101	2	no	—
133	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
134	C₂×#41	—	101	16	no	—
135	C₂×#41	—	202	16	no	—
136	C₂²×Q₈	—	101	1	no	—
137	C₂²×D₄	—	101	4	no	—
138	C₂²×D₄	—	101	4	no	—
139	C₂²×D₄	—	101	2	no	—
140	C₂²×D₄	—	101	2	no	—
141	C₂²×D₄	—	101	2	no	—
142	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
143	C₂³×C₄	—	101	4	no	—
144	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
145	#127	—	101	16	no	—
146	#127	—	101	8	no	—
147	#121	—	101	16	no	—
148	#121	—	101	32	no	—
149	C₄×Q₈	—	101	8	no	—
150	C₄×D₄	—	101	16	no	—
151	C₄×D₄	C₂×D₁₀₁	1	8	yes	complemented
152	C₂×C₄⋊C₄	—	101	2	no	—
153	C₂×C₄⋊C₄	—	101	1	no	—
154	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	—	101	4	no	—
156	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
157	C₂×C₄²	—	101	2	no	—
158	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
159	C₂⁴	—	101	4	no	—
160	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
161	#41	—	202	64	no	—
162	C₂×Q₈	—	101	8	no	—
163	C₂×Q₈	—	101	4	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	101	16	no	—
166	C₂×D₄	—	101	16	no	—
167	C₂×D₄	—	101	8	no	—
168	C₂×D₄	—	101	8	no	—
169	C₂×D₄	—	101	8	no	—
170	C₂×D₄	—	404	4	no	—
171	C₂×D₄	—	404	8	no	—
172	C₂×D₄	—	202	16	no	—
173	C₂×D₄	—	202	8	no	—
174	C₂×D₄	—	202	8	no	—
175	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	101	2	no	—
178	C₂²×C₄	—	101	2	no	—
179	C₂²×C₄	—	101	16	no	—
180	C₂²×C₄	—	202	16	no	—
181	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
182	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
184	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
186	C₄⋊C₄	—	101	8	no	—
187	C₄⋊C₄	—	101	4	no	—
188	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
189	C₂²⋊C₄	—	101	16	no	—
190	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
191	C₄²	—	101	8	no	—
192	C₄²	C₂²×D₁₀₁	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	101	8	no	—
197	C₂³	—	202	8	no	—
198	C₂³	—	202	8	no	—
199	C₂³	—	202	8	no	—
200	C₂³	C₂³×D₁₀₁	1	4	yes	—
201	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
202	Q₈	—	101	16	no	—
203	D₄	—	2	8	no	—
204	D₄	—	404	8	no	—
205	D₄	—	404	16	no	—
206	D₄	—	202	32	no	—
207	D₄	—	202	16	no	—
208	D₄	—	202	16	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	101	8	no	—
213	C₂×C₄	—	101	4	no	—
214	C₂×C₄	—	101	4	no	—
215	C₂×C₄	—	202	32	no	—
216	C₂×C₄	—	202	4	no	—
217	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
218	C₂×C₄	#13108074	1	4	yes	complemented
219	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
220	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
221	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
222	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	8	no	—
227	C₂²	—	404	16	no	—
228	C₂²	—	202	8	no	—
229	C₂²	—	202	8	no	—
230	C₂²	—	202	8	no	—
231	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
232	C₂²	C₂×#13108074	1	2	yes	—
233	C₂²	#15692301	1	2	yes	—
234	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	2	2	no	—
237	C₄	—	101	8	no	—
238	C₄	—	202	4	no	—
239	C₄	C₂×D₄×D₁₀₁	1	4	yes	—
240	C₄	C₂×#13108074	1	4	yes	complemented
241	C₂	—	4	4	no	—
242	C₂	—	404	8	no	—
243	C₂	#20265848	1	4	yes	direct product factor
244	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
245	C₂	C₂²×#13108074	1	1	yes	characteristic
246	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×#550
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. C₂²×#13108074	C₂	semidirect
2. #20265848	C₂	direct (N × H)
3. C₂×#15692252	C₂	semidirect
4. C₂×#15692251	C₂	semidirect
5. C₂×#15692229	C₂	semidirect
6. C₂×#15692228	C₂	semidirect
7. C₂×C₄×Dic₂₀₂	C₂	semidirect
8. C₂×#15692217	C₂	semidirect
9. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
11. C₂×#13108074	C₂²	semidirect
19. #15692252	C₂²	semidirect
20. #15692251	C₂²	semidirect
21. #15692229	C₂²	semidirect
22. #15692228	C₂²	semidirect
23. C₄×Dic₂₀₂	C₂²	semidirect
24. #15692217	C₂²	semidirect
31. C₄×D₄×C₁₀₁	C₂²	semidirect
37. C₂×Dic₂₀₂	D₄	semidirect
83. C₂×C₄×C₁₀₁	#41	semidirect
98. Dic₂₀₂	C₂×D₄	semidirect
114. C₄×C₁₀₁	C₂×#41	semidirect
118. C₂×C₁₀₁	#550	semidirect
124. C₁₀₁	C₂×#550	semidirect
133. C₂×C₄×D₄	D₁₀₁	semidirect
151. C₄×D₄	C₂×D₁₀₁	semidirect
218. C₂×C₄	#13108074	semidirect
240. C₄	C₂×#13108074	semidirect
243. C₂	#20265848	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13108074	2
#20265848	2
C₂×#15692252	2
C₂×#15692251	2
C₂×#15692229	2
C₂×#15692228	2
C₂×C₄×Dic₂₀₂	2
C₂×#15692217	2
C₂×C₄×D₄×C₁₀₁	2
C₂×#550	101

Subgroups — C2×#20265848 #27184612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265848 #27184612