Subgroups — C₂×#20265899 #27184680

Order 12928.

1112 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265899	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692250	C₂	1	2	yes	maximal, complemented
4	C₂×#15692249	C₂	1	2	yes	maximal, complemented
5	C₂×#15692244	C₂	1	1	yes	maximal, characteristic
6	C₂×#15692243	C₂	1	1	yes	maximal, characteristic
7	C₂×#15692233	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692220	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692217	C₂	1	1	yes	maximal, characteristic
11	C₂×#15692212	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
15	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	#15692253	C₂²	1	8	yes	complemented
18	#15692250	C₂²	1	16	yes	complemented
19	#15692249	C₂²	1	16	yes	complemented
20	#15692244	C₂²	1	8	yes	—
21	#15692243	C₂²	1	8	yes	—
22	#15692233	C₂²	1	16	yes	complemented
23	C₄×Dic₂₀₂	C₂²	1	8	yes	complemented
24	#15692220	C₂²	1	8	yes	complemented
25	#15692217	C₂²	1	8	yes	—
26	#15692212	C₂²	1	16	yes	complemented
27	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×#13108053	C₂²	1	1	yes	characteristic
29	C₂×#13108053	C₂²	1	2	yes	—
30	C₂×#13108053	C₂²	1	2	yes	—
31	C₂×#13108053	C₂²	1	1	yes	characteristic
32	C₂×#13108053	C₂²	1	2	yes	—
33	C₂×#13108051	C₂²	1	2	yes	—
34	C₂×#13108051	C₂²	1	2	yes	—
35	C₂×#13108051	C₂²	1	2	yes	—
36	C₂×#13108051	C₂²	1	2	yes	—
37	C₂×#13108051	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₁₀₁	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
45	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
46	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
47	C₂³×D₁₀₁	C₂³	1	2	yes	—
48	C₂×Dic₂₀₂	—	2	4	no	—
49	C₂×Dic₂₀₂	—	2	4	no	—
50	C₂×Dic₂₀₂	C₂³	1	4	yes	—
51	C₂×Dic₂₀₂	C₂³	1	4	yes	—
52	C₂×D₄₀₄	—	2	4	no	—
53	C₂×D₄₀₄	C₂³	1	4	yes	—
54	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
55	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₀₁	—	2	8	no	—
60	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
61	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
62	#13108053	C₂³	1	4	yes	—
63	#13108053	C₂³	1	8	yes	—
64	#13108053	C₂³	1	8	yes	—
65	#13108053	C₂³	1	4	yes	—
66	#13108053	C₂³	1	8	yes	—
67	#13108051	C₂³	1	8	yes	—
68	#13108051	C₂³	1	8	yes	—
69	#13108051	C₂³	1	8	yes	—
70	#13108051	C₂³	1	8	yes	—
71	#13108051	C₂³	1	8	yes	—
72	C₄×Dic₁₀₁	C₂³	1	8	yes	—
73	C₄×Dic₁₀₁	C₂³	1	4	yes	—
74	C₄×Dic₁₀₁	C₂³	1	4	yes	—
75	C₂×Q₈×C₁₀₁	—	2	4	no	—
76	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
77	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
83	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
84	C₄²×C₁₀₁	C₂³	1	8	yes	—
85	C₄²×C₁₀₁	C₂³	1	4	yes	—
86	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
87	C₂²×D₁₀₁	—	2	4	no	—
88	C₂²×D₁₀₁	—	2	4	no	—
89	C₂²×D₁₀₁	—	2	4	no	—
90	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
91	Q₈×C₁₀₁	—	2	8	no	—
92	C₂×C₄×C₁₀₁	—	2	4	no	—
93	C₂×C₄×C₁₀₁	—	2	4	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	#41	1	4	yes	—
97	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
99	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
101	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	—	2	4	no	—
104	C₂×Dic₁₀₁	—	2	2	no	—
105	C₂×Dic₁₀₁	—	2	2	no	—
106	C₂×Dic₁₀₁	#41	1	4	yes	—
107	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
108	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
109	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
110	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
111	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
112	D₄₀₄	—	2	8	no	—
113	C₄×D₁₀₁	—	2	16	no	—
114	Dic₂₀₂	—	2	8	no	—
115	Dic₂₀₂	—	2	8	no	—
116	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
117	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
118	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
119	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
120	C₂×D₁₀₁	—	2	4	no	—
121	C₂×D₁₀₁	—	2	4	no	—
122	C₂×D₁₀₁	—	2	4	no	—
123	C₂×D₁₀₁	—	4	8	no	—
124	C₄×C₁₀₁	—	2	2	no	—
125	C₄×C₁₀₁	—	2	4	no	—
126	C₄×C₁₀₁	—	2	4	no	—
127	C₄×C₁₀₁	—	2	2	no	—
128	C₄×C₁₀₁	C₂×#41	1	4	yes	—
129	Dic₁₀₁	—	2	2	no	—
130	Dic₁₀₁	—	2	4	no	—
131	Dic₁₀₁	—	2	2	no	—
132	Dic₁₀₁	—	2	2	no	—
133	Dic₁₀₁	C₂×#41	1	4	yes	—
134	C₂×C₁₀₁	#556	1	4	yes	complemented
135	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
136	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
137	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
138	D₁₀₁	—	4	4	no	—
139	C₂×#556	—	101	1	no	maximal, Sylow
140	C₁₀₁	C₂×#556	1	1	yes	characteristic, complemented, Sylow
141	#556	—	101	16	no	—
142	C₂×C₄⋊Q₈	—	101	1	no	—
143	C₂×#126	—	101	2	no	—
144	C₂×#126	—	101	2	no	—
145	C₂×#124	—	101	1	no	—
146	C₂×#124	—	101	1	no	—
147	C₂×C₂²⋊Q₈	—	101	2	no	—
148	C₂×C₄×Q₈	—	101	1	no	—
149	C₂×C₄×Q₈	—	101	1	no	—
150	C₂×C₄×Q₈	D₁₀₁	1	1	yes	characteristic, complemented
151	C₂×C₄×D₄	—	101	1	no	—
152	C₂×#117	—	101	2	no	—
153	C₂²×Q₈	—	101	1	no	—
154	C₂²×Q₈	—	101	1	no	—
155	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
156	C₂²×D₄	—	101	1	no	—
157	C₂³×C₄	—	101	2	no	—
158	C₄⋊Q₈	—	101	8	no	—
159	#126	—	101	16	no	—
160	#126	—	101	16	no	—
161	#124	—	101	8	no	—
162	#124	—	101	8	no	—
163	C₂²⋊Q₈	—	101	16	no	—
164	C₄×Q₈	—	101	8	no	—
165	C₄×Q₈	—	101	8	no	—
166	C₄×Q₈	C₂×D₁₀₁	1	8	yes	complemented
167	C₄×D₄	—	101	8	no	—
168	#117	—	101	16	no	—
169	C₂×C₄⋊C₄	—	101	1	no	—
170	C₂×C₄⋊C₄	—	101	2	no	—
171	C₂×C₄⋊C₄	—	101	2	no	—
172	C₂×C₄⋊C₄	—	101	1	no	—
173	C₂×C₄⋊C₄	—	101	1	no	—
174	C₂×C₄⋊C₄	—	101	2	no	—
175	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
177	C₂×C₂²⋊C₄	—	101	2	no	—
178	C₂×C₂²⋊C₄	—	101	2	no	—
179	C₂×C₂²⋊C₄	—	101	2	no	—
180	C₂×C₂²⋊C₄	—	101	2	no	—
181	C₂×C₂²⋊C₄	—	101	2	no	—
182	C₂×C₄²	—	101	2	no	—
183	C₂×C₄²	—	101	1	no	—
184	C₂×C₄²	—	101	1	no	—
185	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
186	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
187	C₂⁴	—	101	2	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	101	4	no	—
190	C₂×Q₈	—	101	4	no	—
191	C₂×Q₈	—	202	4	no	—
192	C₂×Q₈	—	202	4	no	—
193	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
194	C₂×D₄	—	101	4	no	—
195	C₂×D₄	—	202	4	no	—
196	C₂²×C₄	—	101	1	no	—
197	C₂²×C₄	—	101	1	no	—
198	C₂²×C₄	—	101	1	no	—
199	C₂²×C₄	—	101	1	no	—
200	C₂²×C₄	—	101	2	no	—
201	C₂²×C₄	—	101	8	no	—
202	C₂²×C₄	—	202	8	no	—
203	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
204	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
206	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
208	C₄⋊C₄	—	101	4	no	—
209	C₄⋊C₄	—	101	8	no	—
210	C₄⋊C₄	—	101	8	no	—
211	C₄⋊C₄	—	101	4	no	—
212	C₄⋊C₄	—	101	8	no	—
213	C₄⋊C₄	—	101	4	no	—
214	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
215	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
216	C₂²⋊C₄	—	101	8	no	—
217	C₂²⋊C₄	—	101	8	no	—
218	C₂²⋊C₄	—	101	8	no	—
219	C₂²⋊C₄	—	101	8	no	—
220	C₂²⋊C₄	—	101	8	no	—
221	C₄²	—	101	8	no	—
222	C₄²	—	101	4	no	—
223	C₄²	—	101	4	no	—
224	C₄²	C₂²×D₁₀₁	1	8	yes	—
225	C₄²	C₂²×D₁₀₁	1	4	yes	—
226	C₂³	—	101	4	no	—
227	C₂³	—	202	4	no	—
228	C₂³	—	202	4	no	—
229	C₂³	—	202	4	no	—
230	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	Q₈	—	202	8	no	—
233	Q₈	—	202	8	no	—
234	D₄	—	202	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	101	4	no	—
240	C₂×C₄	—	101	2	no	—
241	C₂×C₄	—	101	2	no	—
242	C₂×C₄	—	101	2	no	—
243	C₂×C₄	—	101	2	no	—
244	C₂×C₄	—	101	4	no	—
245	C₂×C₄	—	202	16	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	—	202	4	no	—
248	C₂×C₄	—	202	2	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	#13108074	1	4	yes	—
251	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
252	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
253	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
254	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
255	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
256	C₂²	—	404	8	no	—
257	C₂²	—	202	4	no	—
258	C₂²	—	202	4	no	—
259	C₂²	—	202	4	no	—
260	C₂²	#41×D₁₀₁	1	2	yes	—
261	C₂²	C₂×#13108074	1	2	yes	—
262	C₂²	#15692303	1	2	yes	—
263	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₄	—	101	4	no	—
269	C₄	—	202	2	no	—
270	C₄	—	202	4	no	—
271	C₄	—	202	2	no	—
272	C₄	—	202	2	no	—
273	C₄	C₂×#13108074	1	4	yes	—
274	C₂	—	404	4	no	—
275	C₂	#20265899	1	4	yes	direct product factor
276	C₂	C₂×#41×D₁₀₁	1	1	yes	characteristic
277	C₂	C₂²×#13108074	1	1	yes	characteristic
278	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×C₄×Q₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×D₄₀₄	C₂²
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#556
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20265899	C₂	direct (N × H)
2. C₂×#15692253	C₂	semidirect
3. C₂×#15692250	C₂	semidirect
4. C₂×#15692249	C₂	semidirect
7. C₂×#15692233	C₂	semidirect
8. C₂×C₄×Dic₂₀₂	C₂	semidirect
9. C₂×#15692220	C₂	semidirect
11. C₂×#15692212	C₂	semidirect
12. C₂×C₄×Q₈×C₁₀₁	C₂	semidirect
17. #15692253	C₂²	semidirect
18. #15692250	C₂²	semidirect
19. #15692249	C₂²	semidirect
22. #15692233	C₂²	semidirect
23. C₄×Dic₂₀₂	C₂²	semidirect
24. #15692220	C₂²	semidirect
26. #15692212	C₂²	semidirect
42. C₄×Q₈×C₁₀₁	C₂²	semidirect
134. C₂×C₁₀₁	#556	semidirect
140. C₁₀₁	C₂×#556	semidirect
150. C₂×C₄×Q₈	D₁₀₁	semidirect
166. C₄×Q₈	C₂×D₁₀₁	semidirect
275. C₂	#20265899	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265899	2
C₂×#15692253	2
C₂×#15692250	2
C₂×#15692249	2
C₂×#15692244	2
C₂×#15692243	2
C₂×#15692233	2
C₂×C₄×Dic₂₀₂	2
C₂×#15692220	2
C₂×#15692217	2
C₂×#15692212	2
C₂×C₄×Q₈×C₁₀₁	2
C₂×#556	101

Subgroups — C2×#20265899 #27184680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265899 #27184680