Subgroups — C₂×#20265755 #27184454

Order 12928.

1640 conjugacy classes of subgroups.

The subgroup lattice has 272 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	1	yes	maximal, characteristic, complemented
2	#20265755	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692234	C₂	1	2	yes	maximal, complemented
4	C₂×#15692231	C₂	1	2	yes	maximal, complemented
5	C₂×#15692229	C₂	1	2	yes	maximal, complemented
6	C₂×#15692227	C₂	1	4	yes	maximal, complemented
7	C₂×#15692226	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13108074	—	2	8	no	—
11	C₂×#13108074	C₂²	1	8	yes	complemented
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
16	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
18	#15692234	C₂²	1	16	yes	complemented
19	#15692231	C₂²	1	16	yes	complemented
20	#15692229	C₂²	1	16	yes	complemented
21	#15692227	C₂²	1	32	yes	complemented
22	#15692226	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₀₁	C₂²	1	8	yes	complemented
26	C₂×#13108053	C₂²	1	4	yes	—
27	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
28	C₂×#13108051	C₂²	1	4	yes	—
29	C₂×#41×C₁₀₁	—	2	8	no	—
30	C₂×#41×C₁₀₁	C₂²	1	8	yes	complemented
31	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
34	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
35	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂³×D₁₀₁	C₂³	1	2	yes	—
37	#13108074	—	2	32	no	—
38	C₂×Dic₂₀₂	—	2	4	no	—
39	C₂×Dic₂₀₂	C₂³	1	4	yes	—
40	C₂×D₄₀₄	—	2	4	no	—
41	C₂×D₄₀₄	C₂³	1	4	yes	—
42	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
43	C₂×C₂²⋊D₁₀₁	—	4	8	no	—
44	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
45	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	—
46	C₂²×Dic₁₀₁	—	4	4	no	—
47	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
50	C₂×C₄×D₁₀₁	—	2	8	no	—
51	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
52	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
53	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
54	#13108053	C₂³	1	16	yes	—
55	C₂²⋊Dic₁₀₁	C₂³	1	16	yes	—
56	#13108051	C₂³	1	16	yes	—
57	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
58	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
59	#41×C₁₀₁	—	2	32	no	—
60	C₂×Q₈×C₁₀₁	—	2	4	no	—
61	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
62	C₂×D₄×C₁₀₁	—	2	8	no	—
63	C₂×D₄×C₁₀₁	—	2	4	no	—
64	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
65	C₂×D₄×C₁₀₁	C₂³	1	8	yes	—
66	C₂²×C₄×C₁₀₁	—	2	8	no	—
67	C₂²×C₄×C₁₀₁	D₄	1	8	yes	complemented
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
73	C₂³×C₁₀₁	—	2	2	no	—
74	C₂³×C₁₀₁	—	2	2	no	—
75	C₂³×C₁₀₁	—	2	4	no	—
76	C₂³×C₁₀₁	—	2	4	no	—
77	C₂³×C₁₀₁	—	2	4	no	—
78	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
79	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
80	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
81	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
82	C₂²×D₁₀₁	—	2	4	no	—
83	C₂²×D₁₀₁	—	2	4	no	—
84	C₂²×D₁₀₁	—	2	4	no	—
85	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
86	Q₈×C₁₀₁	—	2	8	no	—
87	D₄×C₁₀₁	—	2	16	no	—
88	D₄×C₁₀₁	—	2	8	no	—
89	C₂×C₄×C₁₀₁	—	2	4	no	—
90	C₂×C₄×C₁₀₁	—	2	16	no	—
91	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₁₀₁	C₂×D₄	1	16	yes	complemented
94	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
95	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
96	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
97	C₂²⋊D₁₀₁	—	2	16	no	—
98	C₂²⋊D₁₀₁	—	4	16	no	—
99	C₂×Dic₁₀₁	—	2	4	no	—
100	C₂×Dic₁₀₁	—	2	8	no	—
101	C₂×Dic₁₀₁	—	4	8	no	—
102	C₂×Dic₁₀₁	C₂⁴	1	8	yes	—
103	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
104	D₄₀₄	—	2	8	no	—
105	C₄×D₁₀₁	—	2	16	no	—
106	Dic₂₀₂	—	2	8	no	—
107	C₂²×C₁₀₁	—	2	8	no	—
108	C₂²×C₁₀₁	—	2	2	no	—
109	C₂²×C₁₀₁	—	2	2	no	—
110	C₂²×C₁₀₁	—	2	4	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	—	2	4	no	—
113	C₂²×C₁₀₁	—	4	8	no	—
114	C₂²×C₁₀₁	C₂²×D₄	1	4	yes	—
115	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
116	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
117	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
118	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₀₁	—	2	4	no	—
120	C₂×D₁₀₁	—	2	4	no	—
121	C₂×D₁₀₁	—	2	4	no	—
122	C₂×D₁₀₁	—	4	8	no	—
123	C₄×C₁₀₁	—	2	4	no	—
124	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
125	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
126	Dic₁₀₁	—	2	4	no	—
127	Dic₁₀₁	—	2	8	no	—
128	C₂×C₁₀₁	—	2	4	no	—
129	C₂×C₁₀₁	—	4	4	no	—
130	C₂×C₁₀₁	#537	1	4	yes	complemented
131	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
132	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
134	D₁₀₁	—	4	4	no	—
135	C₂×#537	—	101	1	no	maximal, Sylow
136	C₁₀₁	C₂×#537	1	1	yes	characteristic, complemented, Sylow
137	C₂²×#41	—	101	1	no	—
138	C₂²×#41	D₁₀₁	1	1	yes	characteristic, complemented
139	#537	—	101	16	no	—
140	C₂×C₂²⋊Q₈	—	101	2	no	—
141	C₂×C₂²⋊Q₈	—	101	2	no	—
142	C₂×#121	—	101	2	no	—
143	C₂×#121	—	101	4	no	—
144	C₂×#121	—	101	2	no	—
145	C₂²×C₄⋊C₄	—	101	1	no	—
146	C₂×#41	—	2	8	no	—
147	C₂×#41	—	101	8	no	—
148	C₂×#41	—	202	8	no	—
149	C₂×#41	C₂×D₁₀₁	1	8	yes	complemented
150	C₂²×Q₈	—	101	1	no	—
151	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
152	C₂²×D₄	—	101	4	no	—
153	C₂²×D₄	—	101	2	no	—
154	C₂²×D₄	—	101	1	no	—
155	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
156	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
157	C₂³×C₄	—	101	2	no	—
158	C₂³×C₄	—	101	2	no	—
159	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
160	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
161	C₂²⋊Q₈	—	101	16	no	—
162	C₂²⋊Q₈	—	101	16	no	—
163	#121	—	101	16	no	—
164	#121	—	101	32	no	—
165	#121	—	101	16	no	—
166	C₂×C₄⋊C₄	—	101	2	no	—
167	C₂×C₄⋊C₄	—	101	2	no	—
168	C₂×C₄⋊C₄	—	101	8	no	—
169	C₂×C₄⋊C₄	—	101	4	no	—
170	C₂×C₂²⋊C₄	—	101	4	no	—
171	C₂×C₂²⋊C₄	—	101	4	no	—
172	C₂⁴	—	101	2	no	—
173	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
174	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
175	#41	—	2	32	no	—
176	#41	—	202	32	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	101	4	no	—
179	C₂×Q₈	—	202	4	no	—
180	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
181	C₂×D₄	—	2	8	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	101	16	no	—
184	C₂×D₄	—	101	8	no	—
185	C₂×D₄	—	101	4	no	—
186	C₂×D₄	—	404	8	no	—
187	C₂×D₄	—	202	8	no	—
188	C₂×D₄	—	202	4	no	—
189	C₂×D₄	C₂²×D₁₀₁	1	8	yes	—
190	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	101	8	no	—
193	C₂²×C₄	—	101	4	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	8	no	—
196	C₂²×C₄	—	404	4	no	—
197	C₂²×C₄	—	202	8	no	—
198	C₂²×C₄	C₂²⋊D₁₀₁	1	8	yes	complemented
199	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
200	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
204	C₄⋊C₄	—	101	16	no	—
205	C₄⋊C₄	—	101	8	no	—
206	C₄⋊C₄	—	101	8	no	—
207	C₂²⋊C₄	—	101	16	no	—
208	C₂²⋊C₄	—	101	16	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	4	no	—
214	C₂³	—	101	4	no	—
215	C₂³	—	202	4	no	—
216	C₂³	—	202	4	no	—
217	C₂³	—	202	4	no	—
218	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
219	C₂³	C₂³×D₁₀₁	1	4	yes	—
220	C₂³	C₂³×D₁₀₁	1	2	yes	—
221	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
222	Q₈	—	2	8	no	—
223	Q₈	—	202	8	no	—
224	D₄	—	2	16	no	—
225	D₄	—	2	8	no	—
226	D₄	—	404	16	no	—
227	D₄	—	202	16	no	—
228	D₄	—	202	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	101	8	no	—
232	C₂×C₄	—	101	4	no	—
233	C₂×C₄	—	404	8	no	—
234	C₂×C₄	—	202	4	no	—
235	C₂×C₄	—	202	8	no	—
236	C₂×C₄	—	202	16	no	—
237	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
238	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
239	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	16	yes	complemented
240	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
241	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
242	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	404	8	no	—
251	C₂²	—	202	4	no	—
252	C₂²	—	202	4	no	—
253	C₂²	—	202	4	no	—
254	C₂²	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
255	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
256	C₂²	#15692301	1	2	yes	—
257	C₂²	#15692302	1	2	yes	—
258	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
259	C₄	—	2	4	no	—
260	C₄	—	202	4	no	—
261	C₄	—	202	8	no	—
262	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
263	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
264	C₂	—	2	4	no	—
265	C₂	—	4	4	no	—
266	C₂	—	404	4	no	—
267	C₂	#20265755	1	4	yes	direct product factor
268	C₂	C₂³×C₂²⋊D₁₀₁	1	1	yes	characteristic
269	C₂	C₂×#15692301	1	1	yes	characteristic
270	C₂	C₂×#15692302	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂²×#41×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×#41	D₁₀₁
hypocenter	C₁₀₁	C₂×#537
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108074	C₂	semidirect
2. #20265755	C₂	direct (N × H)
3. C₂×#15692234	C₂	semidirect
4. C₂×#15692231	C₂	semidirect
5. C₂×#15692229	C₂	semidirect
6. C₂×#15692227	C₂	semidirect
7. C₂×#15692226	C₂	semidirect
8. C₂²×C₄⋊Dic₁₀₁	C₂	semidirect
9. C₂²×#41×C₁₀₁	C₂	semidirect
11. C₂×#13108074	C₂²	semidirect
18. #15692234	C₂²	semidirect
19. #15692231	C₂²	semidirect
20. #15692229	C₂²	semidirect
21. #15692227	C₂²	semidirect
22. #15692226	C₂²	semidirect
25. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
30. C₂×#41×C₁₀₁	C₂²	semidirect
67. C₂²×C₄×C₁₀₁	D₄	semidirect
93. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
130. C₂×C₁₀₁	#537	semidirect
136. C₁₀₁	C₂×#537	semidirect
138. C₂²×#41	D₁₀₁	semidirect
149. C₂×#41	C₂×D₁₀₁	semidirect
198. C₂²×C₄	C₂²⋊D₁₀₁	semidirect
239. C₂×C₄	C₂×C₂²⋊D₁₀₁	semidirect
267. C₂	#20265755	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
#20265755	2
C₂×#15692234	2
C₂×#15692231	2
C₂×#15692229	2
C₂×#15692227	2
C₂×#15692226	2
C₂²×C₄⋊Dic₁₀₁	2
C₂²×#41×C₁₀₁	2
C₂×#537	101

Subgroups — C2×#20265755 #27184454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265755 #27184454