Subgroups — C₂×#20265917 #27185249

Order 12928.

1344 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265917	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692251	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692250	C₂	1	2	yes	maximal, complemented
4	C₂×#15692237	C₂	1	2	yes	maximal, complemented
5	C₂×#15692230	C₂	1	2	yes	maximal, complemented
6	C₂×#15692229	C₂	1	2	yes	maximal, complemented
7	C₂×#15692228	C₂	1	2	yes	maximal, complemented
8	C₂×#15692225	C₂	1	2	yes	maximal, complemented
9	C₂×#15692223	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#123×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×D₄₀₄	C₂²	1	2	yes	—
13	C₂²×D₄₀₄	C₂²	1	2	yes	—
14	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	complemented
18	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	complemented
19	#15692251	C₂²	1	8	yes	complemented
20	#15692250	C₂²	1	16	yes	complemented
21	#15692237	C₂²	1	16	yes	complemented
22	#15692230	C₂²	1	16	yes	complemented
23	#15692229	C₂²	1	16	yes	complemented
24	#15692228	C₂²	1	16	yes	complemented
25	#15692225	C₂²	1	16	yes	complemented
26	#15692223	C₂²	1	8	yes	complemented
27	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
28	C₂×#13108053	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×#13108051	C₂²	1	2	yes	—
31	C₂×#13108051	C₂²	1	2	yes	—
32	C₂×#13108051	C₂²	1	2	yes	—
33	C₂×#13108051	C₂²	1	2	yes	complemented
34	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
35	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
37	#123×C₁₀₁	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₀₁	—	4	4	no	—
42	C₂³×D₁₀₁	—	4	2	no	—
43	C₂³×D₁₀₁	C₂³	1	2	yes	—
44	C₂³×D₁₀₁	C₂³	1	2	yes	—
45	C₂³×D₁₀₁	C₂³	1	4	yes	—
46	C₂×D₄₀₄	—	4	4	no	—
47	C₂×D₄₀₄	—	4	4	no	—
48	C₂×D₄₀₄	—	4	2	no	—
49	C₂×D₄₀₄	C₂³	1	8	yes	—
50	C₂×D₄₀₄	C₂³	1	8	yes	—
51	C₂×D₄₀₄	C₂³	1	4	yes	complemented
52	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
53	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
54	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
55	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
56	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	complemented
58	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
59	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
60	C₂×C₄×D₁₀₁	—	4	4	no	—
61	C₂×C₄×D₁₀₁	C₂³	1	8	yes	complemented
62	C₄⋊Dic₁₀₁	C₂³	1	8	yes	complemented
63	#13108053	C₂³	1	8	yes	complemented
64	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
65	#13108051	C₂³	1	8	yes	—
66	#13108051	C₂³	1	8	yes	—
67	#13108051	C₂³	1	8	yes	—
68	#13108051	C₂³	1	8	yes	complemented
69	C₄×Dic₁₀₁	C₂³	1	4	yes	complemented
70	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₀₁	—	4	2	no	—
73	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
74	C₂²×C₄×C₁₀₁	—	4	2	no	—
75	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
76	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
77	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	complemented
79	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	complemented
80	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
81	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	complemented
82	C₂³×C₁₀₁	—	2	2	no	—
83	C₂³×C₁₀₁	—	2	2	no	—
84	C₂³×C₁₀₁	—	2	4	no	—
85	C₂³×C₁₀₁	—	2	4	no	—
86	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
87	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
88	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
89	C₂²×D₁₀₁	—	2	4	no	—
90	C₂²×D₁₀₁	—	2	4	no	—
91	C₂²×D₁₀₁	—	2	4	no	—
92	C₂²×D₁₀₁	—	2	4	no	—
93	C₂²×D₁₀₁	—	2	4	no	—
94	C₂²×D₁₀₁	—	2	4	no	—
95	C₂²×D₁₀₁	—	4	4	no	—
96	C₂²×D₁₀₁	—	4	8	no	—
97	C₂²×D₁₀₁	—	4	16	no	—
98	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
99	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
100	D₄×C₁₀₁	—	4	4	no	—
101	C₂×C₄×C₁₀₁	—	2	4	no	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	—	2	4	no	—
104	C₂×C₄×C₁₀₁	—	4	4	no	—
105	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
106	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
107	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
108	C₂²⋊D₁₀₁	—	4	8	no	—
109	C₂²⋊D₁₀₁	—	4	8	no	—
110	C₂²⋊D₁₀₁	—	4	8	no	—
111	C₂×Dic₁₀₁	—	2	4	no	—
112	C₂×Dic₁₀₁	—	2	4	no	—
113	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
114	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
115	D₄₀₄	—	4	8	no	—
116	D₄₀₄	—	4	8	no	—
117	D₄₀₄	—	4	4	no	—
118	C₄×D₁₀₁	—	4	8	no	—
119	C₂²×C₁₀₁	—	2	4	no	—
120	C₂²×C₁₀₁	—	2	4	no	—
121	C₂²×C₁₀₁	—	2	2	no	—
122	C₂²×C₁₀₁	—	2	2	no	—
123	C₂²×C₁₀₁	—	4	4	no	—
124	C₂²×C₁₀₁	—	4	4	no	—
125	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
126	C₂²×C₁₀₁	H(5,𝔽₂)	1	4	yes	—
127	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
128	C₂×D₁₀₁	—	2	4	no	—
129	C₂×D₁₀₁	—	2	4	no	—
130	C₂×D₁₀₁	—	2	4	no	—
131	C₂×D₁₀₁	—	2	4	no	—
132	C₂×D₁₀₁	—	2	4	no	—
133	C₂×D₁₀₁	—	2	4	no	—
134	C₂×D₁₀₁	—	4	16	no	—
135	C₂×D₁₀₁	—	4	8	no	—
136	C₂×D₁₀₁	—	4	8	no	—
137	C₄×C₁₀₁	—	2	4	no	—
138	C₄×C₁₀₁	—	2	4	no	—
139	C₄×C₁₀₁	—	2	2	no	—
140	Dic₁₀₁	—	2	4	no	—
141	Dic₁₀₁	—	2	4	no	—
142	C₂×C₁₀₁	—	4	2	no	—
143	C₂×C₁₀₁	—	4	2	no	—
144	C₂×C₁₀₁	#560	1	4	yes	complemented
145	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	2	yes	—
147	D₁₀₁	—	4	4	no	—
148	D₁₀₁	—	4	4	no	—
149	C₂×#560	—	101	1	no	maximal, Sylow
150	C₁₀₁	C₂×#560	1	1	yes	characteristic, complemented, Sylow
151	#560	—	101	16	no	—
152	C₂×#127	—	101	1	no	—
153	C₂×#126	—	101	2	no	—
154	C₂×#123	—	101	2	no	—
155	C₂×#123	D₁₀₁	1	1	yes	characteristic, complemented
156	C₂×#121	—	101	2	no	—
157	C₂×#121	—	101	2	no	—
158	C₂×#121	—	101	2	no	—
159	C₂×#120	—	101	1	no	—
160	C₂×#120	—	101	2	no	—
161	C₂⁵	—	101	1	no	—
162	C₂²×D₄	—	101	2	no	—
163	C₂²×D₄	—	101	2	no	—
164	C₂²×D₄	—	101	2	no	—
165	C₂²×D₄	—	101	2	no	—
166	C₂²×D₄	—	101	2	no	—
167	C₂²×D₄	—	101	1	no	—
168	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
169	C₂³×C₄	—	101	2	no	—
170	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
171	#127	—	101	8	no	—
172	#126	—	101	16	no	—
173	#123	—	101	16	no	—
174	#123	C₂×D₁₀₁	1	8	yes	complemented
175	#121	—	101	16	no	—
176	#121	—	101	16	no	—
177	#121	—	101	16	no	—
178	#120	—	101	8	no	—
179	#120	—	101	16	no	—
180	C₂×C₄⋊C₄	—	101	2	no	—
181	C₂×C₄⋊C₄	—	101	2	no	—
182	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	complemented
183	C₂×C₂²⋊C₄	—	101	1	no	—
184	C₂×C₂²⋊C₄	—	101	2	no	—
185	C₂×C₂²⋊C₄	—	101	2	no	—
186	C₂×C₂²⋊C₄	—	101	2	no	—
187	C₂×C₂²⋊C₄	—	101	2	no	—
188	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
189	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
190	C₂×C₄²	—	101	1	no	—
191	C₂⁴	—	101	4	no	—
192	C₂⁴	—	101	2	no	—
193	C₂⁴	—	101	2	no	—
194	C₂⁴	—	404	2	no	—
195	C₂⁴	—	404	4	no	—
196	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
197	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	101	8	no	—
200	C₂×D₄	—	101	8	no	—
201	C₂×D₄	—	101	8	no	—
202	C₂×D₄	—	101	8	no	—
203	C₂×D₄	—	101	8	no	—
204	C₂×D₄	—	101	4	no	—
205	C₂×D₄	—	404	4	no	—
206	C₂×D₄	—	404	2	no	—
207	C₂×D₄	—	404	4	no	—
208	C₂×D₄	—	404	4	no	—
209	C₂×D₄	—	404	4	no	—
210	C₂×D₄	—	404	4	no	—
211	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	101	2	no	—
214	C₂²×C₄	—	101	2	no	—
215	C₂²×C₄	—	101	8	no	—
216	C₂²×C₄	—	404	4	no	—
217	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
219	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
220	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	complemented
221	C₄⋊C₄	—	101	8	no	—
222	C₄⋊C₄	—	101	8	no	—
223	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	complemented
224	C₂²⋊C₄	—	101	4	no	—
225	C₂²⋊C₄	—	101	8	no	—
226	C₂²⋊C₄	—	101	8	no	—
227	C₂²⋊C₄	—	101	8	no	—
228	C₂²⋊C₄	—	101	8	no	—
229	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
230	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	complemented
231	C₄²	—	101	4	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	101	4	no	—
237	C₂³	—	101	4	no	—
238	C₂³	—	404	4	no	—
239	C₂³	—	404	8	no	—
240	C₂³	—	404	16	no	—
241	C₂³	—	202	4	no	—
242	C₂³	—	202	4	no	—
243	C₂³	—	202	4	no	—
244	C₂³	—	202	4	no	—
245	C₂³	—	202	4	no	—
246	C₂³	—	202	4	no	—
247	C₂³	C₂³×D₁₀₁	1	2	yes	—
248	C₂³	C₂³×D₁₀₁	1	2	yes	—
249	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
250	D₄	—	4	4	no	—
251	D₄	—	404	8	no	—
252	D₄	—	404	4	no	—
253	D₄	—	404	8	no	—
254	D₄	—	404	8	no	—
255	D₄	—	404	8	no	—
256	D₄	—	404	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	101	4	no	—
262	C₂×C₄	—	101	4	no	—
263	C₂×C₄	—	404	8	no	—
264	C₂×C₄	—	202	4	no	—
265	C₂×C₄	—	202	4	no	—
266	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
267	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
268	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	404	16	no	—
276	C₂²	—	404	8	no	—
277	C₂²	—	404	8	no	—
278	C₂²	—	202	4	no	—
279	C₂²	—	202	4	no	—
280	C₂²	—	202	4	no	—
281	C₂²	—	202	4	no	—
282	C₂²	—	202	4	no	—
283	C₂²	—	202	4	no	—
284	C₂²	#15692300	1	2	yes	—
285	C₂²	#15692301	1	4	yes	—
286	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	202	4	no	—
291	C₄	—	202	4	no	—
292	C₂	—	4	2	no	—
293	C₂	—	4	2	no	—
294	C₂	—	404	4	no	—
295	C₂	—	404	4	no	—
296	C₂	#20265917	1	4	yes	direct product factor
297	C₂	C₂×#15692300	1	1	yes	characteristic
298	C₂	C₂×#15692301	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#123×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#123	D₁₀₁
hypocenter	C₁₀₁	C₂×#560
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#123

Complements

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

N	H ≅ G/N	type
1. #20265917	C₂	direct (N × H)
2. C₂×#15692251	C₂	semidirect
3. C₂×#15692250	C₂	semidirect
4. C₂×#15692237	C₂	semidirect
5. C₂×#15692230	C₂	semidirect
6. C₂×#15692229	C₂	semidirect
7. C₂×#15692228	C₂	semidirect
8. C₂×#15692225	C₂	semidirect
9. C₂×#15692223	C₂	semidirect
10. C₂×#123×C₁₀₁	C₂	semidirect
14. C₂²×D₄₀₄	C₂²	semidirect
17. C₂²×C₂²⋊D₁₀₁	C₂²	semidirect
18. C₂²×C₄×D₁₀₁	C₂²	semidirect
19. #15692251	C₂²	semidirect
20. #15692250	C₂²	semidirect
21. #15692237	C₂²	semidirect
22. #15692230	C₂²	semidirect
23. #15692229	C₂²	semidirect
24. #15692228	C₂²	semidirect
25. #15692225	C₂²	semidirect
26. #15692223	C₂²	semidirect
27. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
28. C₂×#13108053	C₂²	semidirect
33. C₂×#13108051	C₂²	semidirect
34. C₂×C₄×Dic₁₀₁	C₂²	semidirect
36. C₂³×C₄×C₁₀₁	C₂²	semidirect
37. #123×C₁₀₁	C₂²	semidirect
38. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
40. C₂×C₂²⋊C₄×C₁₀₁	C₂²	semidirect
51. C₂×D₄₀₄	C₂³	semidirect
57. C₂×C₂²⋊D₁₀₁	C₂³	semidirect
61. C₂×C₄×D₁₀₁	C₂³	semidirect
62. C₄⋊Dic₁₀₁	C₂³	semidirect
63. #13108053	C₂³	semidirect
68. #13108051	C₂³	semidirect
69. C₄×Dic₁₀₁	C₂³	semidirect
78. C₂²×C₄×C₁₀₁	C₂³	semidirect
79. C₄⋊C₄×C₁₀₁	C₂³	semidirect
81. C₂²⋊C₄×C₁₀₁	C₂³	semidirect
144. C₂×C₁₀₁	#560	semidirect
150. C₁₀₁	C₂×#560	semidirect
155. C₂×#123	D₁₀₁	semidirect
170. C₂³×C₄	C₂×D₁₀₁	semidirect
174. #123	C₂×D₁₀₁	semidirect
182. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
189. C₂×C₂²⋊C₄	C₂×D₁₀₁	semidirect
220. C₂²×C₄	C₂²×D₁₀₁	semidirect
223. C₄⋊C₄	C₂²×D₁₀₁	semidirect
230. C₂²⋊C₄	C₂²×D₁₀₁	semidirect
296. C₂	#20265917	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265917	2
C₂×#15692251	2
C₂×#15692250	2
C₂×#15692237	2
C₂×#15692230	2
C₂×#15692229	2
C₂×#15692228	2
C₂×#15692225	2
C₂×#15692223	2
C₂×#123×C₁₀₁	2
C₂×#560	101

Subgroups — C2×#20265917 #27185249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265917 #27185249