Subgroups — C₂×#20265861 #27184638

Order 12928.

1124 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265861	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692248	C₂	1	2	yes	maximal, complemented
3	C₂×#15692246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692245	C₂	1	2	yes	maximal, complemented
5	C₂×#15692237	C₂	1	2	yes	maximal
6	C₂×#15692231	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#15692214	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692213	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂²⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	#15692248	C₂²	1	16	yes	complemented
17	#15692246	C₂²	1	8	yes	complemented
18	#15692245	C₂²	1	16	yes	complemented
19	#15692237	C₂²	1	16	yes	—
20	#15692231	C₂²	1	8	yes	complemented
21	Q₈×Dic₁₀₁	C₂²	1	8	yes	complemented
22	C₄×C₂²⋊D₁₀₁	C₂²	1	8	yes	—
23	#15692214	C₂²	1	8	yes	complemented
24	#15692213	C₂²	1	16	yes	complemented
25	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
29	C₂×#13108053	C₂²	1	2	yes	—
30	C₂×#13108053	C₂²	1	1	yes	characteristic
31	C₂×#13108053	C₂²	1	2	yes	—
32	C₂×#13108053	C₂²	1	2	yes	—
33	C₄⋊C₄×D₁₀₁	C₂²	1	8	yes	complemented
34	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×#13108051	C₂²	1	1	yes	characteristic
37	C₂×#13108051	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂²⋊Q₈×C₁₀₁	C₂²	1	8	yes	complemented
44	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
47	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
49	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
51	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
55	C₂×C₄×D₁₀₁	—	2	8	no	—
56	C₂×C₄×D₁₀₁	—	2	4	no	—
57	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
58	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
59	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
60	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
61	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
62	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
63	#13108053	C₂³	1	8	yes	—
64	#13108053	C₂³	1	4	yes	—
65	#13108053	C₂³	1	8	yes	—
66	#13108053	C₂³	1	8	yes	—
67	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
68	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
69	#13108051	C₂³	1	4	yes	—
70	#13108051	C₂³	1	8	yes	—
71	C₄×Dic₁₀₁	C₂³	1	8	yes	—
72	C₄×Dic₁₀₁	C₂³	1	8	yes	—
73	C₄×Dic₁₀₁	C₂³	1	4	yes	—
74	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₁₀₁	—	2	4	no	—
76	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
77	C₂²×C₄×C₁₀₁	—	2	4	no	—
78	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
79	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
82	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
84	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
85	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
86	C₂³×C₁₀₁	—	2	2	no	—
87	C₂³×C₁₀₁	—	2	2	no	—
88	C₂³×C₁₀₁	—	2	2	no	—
89	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
90	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
91	C₂²×D₁₀₁	—	2	2	no	—
92	C₂²×D₁₀₁	—	2	2	no	—
93	C₂²×D₁₀₁	#41	1	4	yes	—
94	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
95	Q₈×C₁₀₁	—	2	8	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	4	no	—
98	C₂×C₄×C₁₀₁	—	2	4	no	—
99	C₂×C₄×C₁₀₁	—	2	8	no	—
100	C₂×C₄×C₁₀₁	#41	1	4	yes	—
101	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
104	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
105	C₂²⋊D₁₀₁	—	2	8	no	—
106	C₂×Dic₁₀₁	—	2	4	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	—	2	4	no	—
111	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
112	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
113	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
114	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
115	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
116	C₄×D₁₀₁	—	2	16	no	—
117	C₄×D₁₀₁	—	2	8	no	—
118	C₂²×C₁₀₁	—	2	2	no	—
119	C₂²×C₁₀₁	—	2	2	no	—
120	C₂²×C₁₀₁	—	2	2	no	—
121	C₂²×C₁₀₁	—	4	4	no	—
122	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
123	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
124	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
125	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
126	C₂×D₁₀₁	—	2	8	no	—
127	C₂×D₁₀₁	—	2	2	no	—
128	C₂×D₁₀₁	—	2	2	no	—
129	C₂×D₁₀₁	C₂×#41	1	4	yes	—
130	C₄×C₁₀₁	—	2	4	no	—
131	C₄×C₁₀₁	—	2	2	no	—
132	C₄×C₁₀₁	—	2	4	no	—
133	C₄×C₁₀₁	C₂×#41	1	4	yes	—
134	Dic₁₀₁	—	2	4	no	—
135	Dic₁₀₁	—	2	2	no	—
136	Dic₁₀₁	—	2	2	no	—
137	Dic₁₀₁	—	2	2	no	—
138	Dic₁₀₁	—	2	4	no	—
139	C₂×C₁₀₁	—	4	2	no	—
140	C₂×C₁₀₁	#552	1	4	yes	complemented
141	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
142	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
143	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
144	D₁₀₁	—	2	4	no	—
145	C₂×#552	—	101	1	no	maximal, Sylow
146	C₁₀₁	C₂×#552	1	1	yes	characteristic, complemented, Sylow
147	#552	—	101	16	no	—
148	C₂×#126	—	101	2	no	—
149	C₂×#125	—	101	2	no	—
150	C₂×#125	—	101	1	no	—
151	C₂×#123	—	101	2	no	—
152	C₂×C₂²⋊Q₈	—	101	1	no	—
153	C₂×C₂²⋊Q₈	D₁₀₁	1	1	yes	characteristic, complemented
154	C₂×C₄×Q₈	—	101	1	no	—
155	C₂×C₄×D₄	—	101	1	no	—
156	C₂×#117	—	101	1	no	—
157	C₂×#117	—	101	2	no	—
158	C₂²×C₄⋊C₄	—	101	1	no	—
159	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
160	C₂²×D₄	—	101	1	no	—
161	C₂³×C₄	—	101	2	no	—
162	C₂³×C₄	—	101	1	no	—
163	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
164	#126	—	101	16	no	—
165	#125	—	101	16	no	—
166	#125	—	101	8	no	—
167	#123	—	101	16	no	—
168	C₂²⋊Q₈	—	101	8	no	—
169	C₂²⋊Q₈	C₂×D₁₀₁	1	8	yes	complemented
170	C₄×Q₈	—	101	8	no	—
171	C₄×D₄	—	101	8	no	—
172	#117	—	101	8	no	—
173	#117	—	101	16	no	—
174	C₂×C₄⋊C₄	—	101	1	no	—
175	C₂×C₄⋊C₄	—	101	2	no	—
176	C₂×C₄⋊C₄	—	101	1	no	—
177	C₂×C₄⋊C₄	—	101	2	no	—
178	C₂×C₄⋊C₄	—	101	1	no	—
179	C₂×C₄⋊C₄	—	101	2	no	—
180	C₂×C₄⋊C₄	—	101	2	no	—
181	C₂×C₄⋊C₄	—	101	2	no	—
182	C₂×C₄⋊C₄	—	101	8	no	—
183	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
185	C₂×C₂²⋊C₄	—	101	2	no	—
186	C₂×C₂²⋊C₄	—	101	1	no	—
187	C₂×C₂²⋊C₄	—	101	1	no	—
188	C₂×C₂²⋊C₄	—	101	2	no	—
189	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
190	C₂×C₄²	—	101	2	no	—
191	C₂×C₄²	—	101	2	no	—
192	C₂×C₄²	—	101	1	no	—
193	C₂⁴	—	101	1	no	—
194	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
197	C₂×D₄	—	101	4	no	—
198	C₂×D₄	—	202	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	101	1	no	—
201	C₂²×C₄	—	101	2	no	—
202	C₂²×C₄	—	101	2	no	—
203	C₂²×C₄	—	101	1	no	—
204	C₂²×C₄	—	101	1	no	—
205	C₂²×C₄	—	101	8	no	—
206	C₂²×C₄	—	101	4	no	—
207	C₂²×C₄	—	202	8	no	—
208	C₂²×C₄	—	202	4	no	—
209	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
210	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
212	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
213	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
214	C₄⋊C₄	—	101	8	no	—
215	C₄⋊C₄	—	101	4	no	—
216	C₄⋊C₄	—	101	8	no	—
217	C₄⋊C₄	—	101	8	no	—
218	C₄⋊C₄	—	101	4	no	—
219	C₄⋊C₄	—	101	4	no	—
220	C₄⋊C₄	—	101	8	no	—
221	C₄⋊C₄	—	101	8	no	—
222	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
223	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
224	C₂²⋊C₄	—	101	8	no	—
225	C₂²⋊C₄	—	101	4	no	—
226	C₂²⋊C₄	—	101	4	no	—
227	C₂²⋊C₄	—	101	8	no	—
228	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
229	C₄²	—	101	8	no	—
230	C₄²	—	101	8	no	—
231	C₄²	—	101	4	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	101	4	no	—
236	C₂³	—	101	2	no	—
237	C₂³	—	202	2	no	—
238	C₂³	—	202	2	no	—
239	C₂³	C₂³×D₁₀₁	1	2	yes	—
240	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
241	Q₈	—	2	8	no	—
242	D₄	—	202	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	101	2	no	—
248	C₂×C₄	—	101	4	no	—
249	C₂×C₄	—	101	4	no	—
250	C₂×C₄	—	101	2	no	—
251	C₂×C₄	—	101	2	no	—
252	C₂×C₄	—	202	16	no	—
253	C₂×C₄	—	202	4	no	—
254	C₂×C₄	—	202	2	no	—
255	C₂×C₄	—	202	2	no	—
256	C₂×C₄	—	202	2	no	—
257	C₂×C₄	—	202	4	no	—
258	C₂×C₄	—	202	8	no	—
259	C₂×C₄	#13108073	1	4	yes	—
260	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
261	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
262	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
263	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	101	4	no	—
269	C₂²	—	202	2	no	—
270	C₂²	—	202	2	no	—
271	C₂²	—	202	8	no	—
272	C₂²	#41×D₁₀₁	1	2	yes	—
273	C₂²	C₂×#13108073	1	2	yes	—
274	C₂²	#15692303	1	2	yes	—
275	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	202	4	no	—
280	C₄	—	202	2	no	—
281	C₄	—	202	2	no	—
282	C₄	—	202	2	no	—
283	C₄	—	202	4	no	—
284	C₄	C₂×#13108073	1	4	yes	—
285	C₂	—	4	2	no	—
286	C₂	—	202	4	no	—
287	C₂	#20265861	1	4	yes	direct product factor
288	C₂	C₂×#41×D₁₀₁	1	1	yes	characteristic
289	C₂	C₂²×#13108073	1	1	yes	characteristic
290	C₂	C₂×#15692303	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₂²⋊Q₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×C₂²⋊D₁₀₁	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#552
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20265861	C₂	direct (N × H)
2. C₂×#15692248	C₂	semidirect
3. C₂×#15692246	C₂	semidirect
4. C₂×#15692245	C₂	semidirect
6. C₂×#15692231	C₂	semidirect
7. C₂×Q₈×Dic₁₀₁	C₂	semidirect
9. C₂×#15692214	C₂	semidirect
10. C₂×#15692213	C₂	semidirect
11. C₂×C₄⋊C₄×D₁₀₁	C₂	semidirect
12. C₂×C₂²⋊Q₈×C₁₀₁	C₂	semidirect
16. #15692248	C₂²	semidirect
17. #15692246	C₂²	semidirect
18. #15692245	C₂²	semidirect
20. #15692231	C₂²	semidirect
21. Q₈×Dic₁₀₁	C₂²	semidirect
23. #15692214	C₂²	semidirect
24. #15692213	C₂²	semidirect
33. C₄⋊C₄×D₁₀₁	C₂²	semidirect
43. C₂²⋊Q₈×C₁₀₁	C₂²	semidirect
140. C₂×C₁₀₁	#552	semidirect
146. C₁₀₁	C₂×#552	semidirect
153. C₂×C₂²⋊Q₈	D₁₀₁	semidirect
169. C₂²⋊Q₈	C₂×D₁₀₁	semidirect
287. C₂	#20265861	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265861	2
C₂×#15692248	2
C₂×#15692246	2
C₂×#15692245	2
C₂×#15692237	2
C₂×#15692231	2
C₂×Q₈×Dic₁₀₁	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692214	2
C₂×#15692213	2
C₂×C₄⋊C₄×D₁₀₁	2
C₂×C₂²⋊Q₈×C₁₀₁	2
C₂×#552	101

Subgroups — C2×#20265861 #27184638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265861 #27184638