Subgroups — C₂×#20265889 #27184669

Order 12928.

1240 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265889	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692244	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692241	C₂	1	2	yes	maximal, complemented
5	C₂×#15692228	C₂	1	2	yes	maximal, complemented
6	C₂×#15692227	C₂	1	2	yes	maximal, complemented
7	C₂×#15692215	C₂	1	2	yes	maximal, complemented
8	C₂×#15692212	C₂	1	2	yes	maximal, complemented
9	C₂×#15692211	C₂	1	2	yes	maximal, complemented
10	C₂×#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
12	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
15	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	#15692253	C₂²	1	8	yes	complemented
18	#15692244	C₂²	1	8	yes	complemented
19	#15692241	C₂²	1	16	yes	complemented
20	#15692228	C₂²	1	16	yes	complemented
21	#15692227	C₂²	1	16	yes	complemented
22	#15692215	C₂²	1	16	yes	complemented
23	#15692212	C₂²	1	16	yes	complemented
24	#15692211	C₂²	1	16	yes	complemented
25	C₂×#13108053	C₂²	1	2	yes	—
26	C₂×#13108053	C₂²	1	2	yes	—
27	C₂×#13108053	C₂²	1	2	yes	—
28	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
29	C₂×#13108051	C₂²	1	2	yes	—
30	C₂×#13108051	C₂²	1	2	yes	—
31	C₂×#13108051	C₂²	1	2	yes	—
32	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
34	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	#124×C₁₀₁	C₂²	1	8	yes	complemented
38	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
40	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
41	C₂³×D₁₀₁	C₂³	1	2	yes	—
42	C₂×Dic₂₀₂	—	4	2	no	—
43	C₂×Dic₂₀₂	C₂³	1	4	yes	—
44	C₂×D₄₀₄	—	4	2	no	—
45	C₂×D₄₀₄	C₂³	1	4	yes	—
46	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
47	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
48	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
49	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
50	C₂²×Dic₁₀₁	—	2	8	no	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
53	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
54	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
55	C₂×C₄×D₁₀₁	—	2	8	no	—
56	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
57	#13108053	C₂³	1	8	yes	—
58	#13108053	C₂³	1	8	yes	—
59	#13108053	C₂³	1	8	yes	—
60	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
61	#13108051	C₂³	1	8	yes	—
62	#13108051	C₂³	1	8	yes	—
63	#13108051	C₂³	1	8	yes	—
64	C₄×Dic₁₀₁	C₂³	1	4	yes	—
65	C₄×Dic₁₀₁	C₂³	1	8	yes	—
66	C₄×Dic₁₀₁	C₂³	1	4	yes	—
67	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₀₁	—	4	2	no	—
69	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
70	C₂×D₄×C₁₀₁	—	4	2	no	—
71	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
72	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
77	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
78	C₄²×C₁₀₁	C₂³	1	4	yes	—
79	C₂³×C₁₀₁	—	2	4	no	—
80	C₂³×C₁₀₁	—	2	4	no	—
81	C₂³×C₁₀₁	—	2	4	no	—
82	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
83	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₀₁	—	2	4	no	—
85	C₂²×D₁₀₁	—	2	4	no	—
86	C₂²×D₁₀₁	—	2	4	no	—
87	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
88	Q₈×C₁₀₁	—	4	4	no	—
89	D₄×C₁₀₁	—	4	4	no	—
90	C₂×C₄×C₁₀₁	—	2	4	no	—
91	C₂×C₄×C₁₀₁	—	2	2	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
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	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
98	C₂²⋊D₁₀₁	—	2	16	no	—
99	C₂²⋊D₁₀₁	—	4	8	no	—
100	C₂×Dic₁₀₁	—	2	4	no	—
101	C₂×Dic₁₀₁	—	2	4	no	—
102	C₂×Dic₁₀₁	—	2	16	no	—
103	C₂×Dic₁₀₁	#41	1	8	yes	—
104	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
105	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
106	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
107	D₄₀₄	—	4	4	no	—
108	C₄×D₁₀₁	—	2	16	no	—
109	Dic₂₀₂	—	4	4	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	C₂×#41	1	4	yes	—
115	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
117	C₂×D₁₀₁	—	2	4	no	—
118	C₂×D₁₀₁	—	2	4	no	—
119	C₂×D₁₀₁	—	2	4	no	—
120	C₂×D₁₀₁	—	4	8	no	—
121	C₄×C₁₀₁	—	2	4	no	—
122	C₄×C₁₀₁	—	2	2	no	—
123	C₄×C₁₀₁	—	2	2	no	—
124	C₄×C₁₀₁	—	2	2	no	—
125	Dic₁₀₁	—	2	4	no	—
126	Dic₁₀₁	—	2	4	no	—
127	Dic₁₀₁	C₂×#41	1	8	yes	—
128	C₂×C₁₀₁	—	4	4	no	—
129	C₂×C₁₀₁	#555	1	4	yes	complemented
130	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
131	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
132	D₁₀₁	—	4	4	no	—
133	C₂×#555	—	101	1	no	maximal, Sylow
134	C₁₀₁	C₂×#555	1	1	yes	characteristic, complemented, Sylow
135	#555	—	101	16	no	—
136	C₂×C₄⋊Q₈	—	101	1	no	—
137	C₂×#124	—	101	1	no	—
138	C₂×#124	—	101	2	no	—
139	C₂×#124	D₁₀₁	1	1	yes	characteristic, complemented
140	C₂×#121	—	101	2	no	—
141	C₂×#121	—	101	2	no	—
142	C₂×C₄×D₄	—	101	2	no	—
143	C₂×#117	—	101	2	no	—
144	C₂×#117	—	101	2	no	—
145	C₂²×Q₈	—	101	1	no	—
146	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
147	C₂²×D₄	—	101	2	no	—
148	C₂²×D₄	—	101	2	no	—
149	C₂²×D₄	—	101	1	no	—
150	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
151	C₂³×C₄	—	101	2	no	—
152	C₂³×C₄	—	101	2	no	—
153	C₄⋊Q₈	—	101	8	no	—
154	#124	—	101	8	no	—
155	#124	—	101	16	no	—
156	#124	C₂×D₁₀₁	1	8	yes	complemented
157	#121	—	101	16	no	—
158	#121	—	101	16	no	—
159	C₄×D₄	—	101	16	no	—
160	#117	—	101	16	no	—
161	#117	—	101	16	no	—
162	C₂×C₄⋊C₄	—	101	2	no	—
163	C₂×C₄⋊C₄	—	101	2	no	—
164	C₂×C₄⋊C₄	—	101	2	no	—
165	C₂×C₂²⋊C₄	—	101	2	no	—
166	C₂×C₂²⋊C₄	—	101	2	no	—
167	C₂×C₂²⋊C₄	—	101	2	no	—
168	C₂×C₂²⋊C₄	—	101	2	no	—
169	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
171	C₂×C₄²	—	101	1	no	—
172	C₂×C₄²	—	101	2	no	—
173	C₂×C₄²	—	101	1	no	—
174	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
175	C₂⁴	—	101	2	no	—
176	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	101	4	no	—
179	C₂×Q₈	—	404	2	no	—
180	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	101	8	no	—
183	C₂×D₄	—	101	8	no	—
184	C₂×D₄	—	101	4	no	—
185	C₂×D₄	—	404	2	no	—
186	C₂×D₄	—	404	4	no	—
187	C₂×D₄	—	202	8	no	—
188	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
189	C₂²×C₄	—	101	2	no	—
190	C₂²×C₄	—	101	8	no	—
191	C₂²×C₄	—	101	2	no	—
192	C₂²×C₄	—	101	2	no	—
193	C₂²×C₄	—	101	8	no	—
194	C₂²×C₄	—	202	8	no	—
195	C₂²×C₄	—	202	8	no	—
196	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
200	C₄⋊C₄	—	101	8	no	—
201	C₄⋊C₄	—	101	8	no	—
202	C₄⋊C₄	—	101	8	no	—
203	C₂²⋊C₄	—	101	8	no	—
204	C₂²⋊C₄	—	101	8	no	—
205	C₂²⋊C₄	—	101	8	no	—
206	C₂²⋊C₄	—	101	8	no	—
207	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
208	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
209	C₄²	—	101	4	no	—
210	C₄²	—	101	8	no	—
211	C₄²	—	101	4	no	—
212	C₄²	C₂²×D₁₀₁	1	4	yes	—
213	C₂³	—	2	4	no	—
214	C₂³	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	101	4	no	—
217	C₂³	—	202	4	no	—
218	C₂³	—	202	4	no	—
219	C₂³	—	202	4	no	—
220	C₂³	C₂³×D₁₀₁	1	4	yes	—
221	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
222	Q₈	—	4	4	no	—
223	Q₈	—	404	4	no	—
224	D₄	—	4	4	no	—
225	D₄	—	404	4	no	—
226	D₄	—	404	8	no	—
227	D₄	—	202	16	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	101	8	no	—
233	C₂×C₄	—	101	4	no	—
234	C₂×C₄	—	101	4	no	—
235	C₂×C₄	—	101	4	no	—
236	C₂×C₄	—	202	16	no	—
237	C₂×C₄	—	202	16	no	—
238	C₂×C₄	—	202	4	no	—
239	C₂×C₄	—	202	4	no	—
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	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	404	8	no	—
249	C₂²	—	202	4	no	—
250	C₂²	—	202	4	no	—
251	C₂²	—	202	4	no	—
252	C₂²	#41×D₁₀₁	1	4	yes	—
253	C₂²	#15692300	1	2	yes	—
254	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	101	8	no	—
260	C₄	—	202	4	no	—
261	C₄	—	202	4	no	—
262	C₂	—	4	4	no	—
263	C₂	—	404	4	no	—
264	C₂	#20265889	1	4	yes	direct product factor
265	C₂	C₂×#41×D₁₀₁	1	2	yes	—
266	C₂	C₂×#15692300	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#124×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#124	D₁₀₁
hypocenter	C₁₀₁	C₂×#555
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. #20265889	C₂	direct (N × H)
2. C₂×#15692253	C₂	semidirect
3. C₂×#15692244	C₂	semidirect
4. C₂×#15692241	C₂	semidirect
5. C₂×#15692228	C₂	semidirect
6. C₂×#15692227	C₂	semidirect
7. C₂×#15692215	C₂	semidirect
8. C₂×#15692212	C₂	semidirect
9. C₂×#15692211	C₂	semidirect
10. C₂×#124×C₁₀₁	C₂	semidirect
17. #15692253	C₂²	semidirect
18. #15692244	C₂²	semidirect
19. #15692241	C₂²	semidirect
20. #15692228	C₂²	semidirect
21. #15692227	C₂²	semidirect
22. #15692215	C₂²	semidirect
23. #15692212	C₂²	semidirect
24. #15692211	C₂²	semidirect
37. #124×C₁₀₁	C₂²	semidirect
129. C₂×C₁₀₁	#555	semidirect
134. C₁₀₁	C₂×#555	semidirect
139. C₂×#124	D₁₀₁	semidirect
156. #124	C₂×D₁₀₁	semidirect
264. C₂	#20265889	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265889	2
C₂×#15692253	2
C₂×#15692244	2
C₂×#15692241	2
C₂×#15692228	2
C₂×#15692227	2
C₂×#15692215	2
C₂×#15692212	2
C₂×#15692211	2
C₂×#124×C₁₀₁	2
C₂×#555	101

Subgroups — C2×#20265889 #27184669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265889 #27184669