Subgroups — C₂×#20265752 #27184450

Order 12928.

1512 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108075	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265752	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15692253	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15692244	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692243	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692240	C₂	1	4	yes	maximal, complemented
8	C₂×#15692233	C₂	1	4	yes	maximal, complemented
9	C₂×#15692212	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13108075	—	2	8	no	—
12	C₂×#13108075	C₂²	1	8	yes	—
13	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
14	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
15	C₂×Q₈×D₁₀₁	—	2	8	no	—
16	C₂×Q₈×D₁₀₁	C₂²	1	8	yes	complemented
17	C₂²×D₄₀₄	C₂²	1	2	yes	—
18	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
20	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
21	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
22	#15692253	C₂²	1	8	yes	complemented
23	#15692244	C₂²	1	8	yes	complemented
24	#15692243	C₂²	1	8	yes	complemented
25	#15692240	C₂²	1	32	yes	complemented
26	#15692233	C₂²	1	32	yes	complemented
27	#15692212	C₂²	1	8	yes	complemented
28	C₂×#13108053	C₂²	1	2	yes	—
29	C₂×#13108053	C₂²	1	4	yes	—
30	C₂×#13108051	C₂²	1	2	yes	—
31	C₂×#13108051	C₂²	1	4	yes	—
32	C₂×#13108051	C₂²	1	4	yes	—
33	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
34	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₄⋊Q₈×C₁₀₁	C₂²	1	8	yes	complemented
37	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	4	yes	—
38	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂³×D₁₀₁	C₂³	1	2	yes	—
40	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
41	#13108075	—	2	32	no	—
42	C₂×Dic₂₀₂	—	2	4	no	—
43	C₂×Dic₂₀₂	—	2	8	no	—
44	C₂×Dic₂₀₂	C₂³	1	8	yes	—
45	C₂×Dic₂₀₂	C₂³	1	4	yes	—
46	Q₈×D₁₀₁	—	2	32	no	—
47	C₂×D₄₀₄	—	2	8	no	—
48	C₂×D₄₀₄	—	2	4	no	—
49	C₂×D₄₀₄	C₂³	1	8	yes	—
50	C₂×D₄₀₄	C₂³	1	4	yes	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
53	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₀₁	—	2	8	no	—
55	C₂×C₄×D₁₀₁	—	2	8	no	—
56	C₂×C₄×D₁₀₁	D₄	1	8	yes	—
57	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
58	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
59	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
60	#13108053	C₂³	1	8	yes	—
61	#13108053	C₂³	1	16	yes	—
62	#13108051	C₂³	1	8	yes	—
63	#13108051	C₂³	1	16	yes	—
64	#13108051	C₂³	1	16	yes	—
65	C₄×Dic₁₀₁	C₂³	1	4	yes	—
66	C₂×Q₈×C₁₀₁	—	2	4	no	—
67	C₂×Q₈×C₁₀₁	—	2	4	no	—
68	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
69	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
70	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
71	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
72	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₀₁	C₂³	1	16	yes	—
75	C₄²×C₁₀₁	C₂³	1	4	yes	—
76	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
77	C₂²×D₁₀₁	—	2	4	no	—
78	C₂²×D₁₀₁	—	2	4	no	—
79	C₂²×D₁₀₁	—	2	8	no	—
80	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
81	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
82	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
83	Q₈×C₁₀₁	—	2	8	no	—
84	Q₈×C₁₀₁	—	2	8	no	—
85	C₂×C₄×C₁₀₁	—	2	4	no	—
86	C₂×C₄×C₁₀₁	—	2	4	no	—
87	C₂×C₄×C₁₀₁	—	2	4	no	—
88	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
89	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
90	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
91	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
92	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
93	C₂×Dic₁₀₁	—	2	4	no	—
94	C₂×Dic₁₀₁	—	2	4	no	—
95	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
96	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
97	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
98	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
99	D₄₀₄	—	2	16	no	—
100	D₄₀₄	—	2	8	no	—
101	C₄×D₁₀₁	—	2	16	no	—
102	C₄×D₁₀₁	—	2	16	no	—
103	C₄×D₁₀₁	C₂×D₄	1	16	yes	—
104	Dic₂₀₂	—	2	8	no	—
105	Dic₂₀₂	—	2	16	no	—
106	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
107	C₂²×C₁₀₁	2₋¹⁺⁴	1	4	yes	—
108	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
109	C₂×D₁₀₁	—	2	8	no	—
110	C₂×D₁₀₁	—	2	4	no	—
111	C₂×D₁₀₁	—	2	4	no	—
112	C₂×D₁₀₁	—	2	8	no	—
113	C₂×D₁₀₁	—	4	8	no	—
114	C₂×D₁₀₁	C₂²×D₄	1	4	yes	—
115	C₄×C₁₀₁	—	2	4	no	—
116	C₄×C₁₀₁	—	2	4	no	—
117	C₄×C₁₀₁	—	2	4	no	—
118	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
119	Dic₁₀₁	—	2	4	no	—
120	Dic₁₀₁	—	2	4	no	—
121	Dic₁₀₁	C₂²×D₄	1	4	yes	—
122	C₂×C₁₀₁	#536	1	4	yes	complemented
123	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
124	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	2	yes	—
125	D₁₀₁	—	2	4	no	—
126	D₁₀₁	—	4	4	no	—
127	C₂×#536	—	101	1	no	maximal, Sylow
128	C₁₀₁	C₂×#536	1	1	yes	characteristic, complemented, Sylow
129	C₂²×#41	—	101	1	no	—
130	C₂³×Q₈	—	101	1	no	—
131	#536	—	101	16	no	—
132	C₂×C₄⋊Q₈	—	101	1	no	—
133	C₂×C₄⋊Q₈	D₁₀₁	1	1	yes	characteristic, complemented
134	C₂×#124	—	101	1	no	—
135	C₂×#124	—	101	1	no	—
136	C₂×#123	—	101	4	no	—
137	C₂×C₂²⋊Q₈	—	101	4	no	—
138	C₂×#117	—	101	1	no	—
139	C₂×#41	—	101	8	no	—
140	C₂×#41	—	202	8	no	—
141	C₂²×Q₈	—	101	8	no	—
142	C₂²×Q₈	—	101	2	no	—
143	C₂²×Q₈	—	101	1	no	—
144	C₂²×Q₈	—	202	8	no	—
145	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
146	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
147	C₂²×D₄	—	101	2	no	—
148	C₂²×D₄	—	101	1	no	—
149	C₂³×C₄	—	101	1	no	—
150	C₂³×C₄	—	101	2	no	—
151	C₂³×C₄	—	101	2	no	—
152	C₄⋊Q₈	—	101	8	no	—
153	C₄⋊Q₈	C₂×D₁₀₁	1	8	yes	complemented
154	#124	—	101	8	no	—
155	#124	—	101	8	no	—
156	#123	—	101	32	no	—
157	C₂²⋊Q₈	—	101	32	no	—
158	#117	—	101	8	no	—
159	C₂×C₄⋊C₄	—	101	2	no	—
160	C₂×C₄⋊C₄	—	101	4	no	—
161	C₂×C₄⋊C₄	C₂×D₁₀₁	1	4	yes	—
162	C₂×C₂²⋊C₄	—	101	2	no	—
163	C₂×C₂²⋊C₄	—	101	4	no	—
164	C₂×C₂²⋊C₄	—	101	4	no	—
165	C₂×C₄²	—	101	1	no	—
166	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
167	C₂⁴	—	101	1	no	—
168	C₂⁴	—	101	2	no	—
169	#41	—	202	32	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	101	8	no	—
173	C₂×Q₈	—	101	4	no	—
174	C₂×Q₈	—	202	8	no	—
175	C₂×Q₈	—	202	32	no	—
176	C₂×Q₈	—	202	4	no	—
177	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
178	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
179	C₂×D₄	—	101	8	no	—
180	C₂×D₄	—	101	4	no	—
181	C₂×D₄	—	202	8	no	—
182	C₂×D₄	—	202	4	no	—
183	C₂²×C₄	—	101	1	no	—
184	C₂²×C₄	—	101	8	no	—
185	C₂²×C₄	—	101	4	no	—
186	C₂²×C₄	—	101	2	no	—
187	C₂²×C₄	—	101	2	no	—
188	C₂²×C₄	—	101	8	no	—
189	C₂²×C₄	—	101	8	no	—
190	C₂²×C₄	—	202	8	no	—
191	C₂²×C₄	—	202	8	no	—
192	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
193	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
194	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
195	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
196	C₄⋊C₄	—	101	8	no	—
197	C₄⋊C₄	—	101	16	no	—
198	C₄⋊C₄	C₂²×D₁₀₁	1	16	yes	—
199	C₂²⋊C₄	—	101	8	no	—
200	C₂²⋊C₄	—	101	16	no	—
201	C₂²⋊C₄	—	101	16	no	—
202	C₄²	—	101	4	no	—
203	C₄²	C₂²×D₁₀₁	1	4	yes	—
204	C₂³	—	101	4	no	—
205	C₂³	—	101	2	no	—
206	C₂³	—	101	4	no	—
207	C₂³	—	202	4	no	—
208	C₂³	—	202	8	no	—
209	C₂³	—	202	4	no	—
210	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
211	Q₈	—	2	8	no	—
212	Q₈	—	2	8	no	—
213	Q₈	—	202	16	no	—
214	Q₈	—	202	8	no	—
215	D₄	—	202	16	no	—
216	D₄	—	202	8	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	101	4	no	—
221	C₂×C₄	—	101	2	no	—
222	C₂×C₄	—	101	16	no	—
223	C₂×C₄	—	101	4	no	—
224	C₂×C₄	—	101	4	no	—
225	C₂×C₄	—	202	16	no	—
226	C₂×C₄	—	202	16	no	—
227	C₂×C₄	—	202	4	no	—
228	C₂×C₄	—	202	4	no	—
229	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
230	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
231	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
232	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
233	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
234	C₂²	—	101	4	no	—
235	C₂²	—	404	8	no	—
236	C₂²	—	202	4	no	—
237	C₂²	—	202	8	no	—
238	C₂²	—	202	8	no	—
239	C₂²	—	202	4	no	—
240	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
241	C₂²	#15692303	1	4	yes	—
242	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	101	4	no	—
247	C₄	—	202	4	no	—
248	C₄	—	202	4	no	—
249	C₄	C₂×D₄×D₁₀₁	1	4	yes	—
250	C₂	—	404	4	no	—
251	C₂	—	202	4	no	—
252	C₂	#20265752	1	4	yes	direct product factor
253	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
254	C₂	C₂×#15692303	1	2	yes	—

	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×#13108075	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#536
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
2. C₂²×Q₈×D₁₀₁	C₂	semidirect
3. #20265752	C₂	direct (N × H)
4. C₂×#15692253	C₂	semidirect
5. C₂×#15692244	C₂	semidirect
6. C₂×#15692243	C₂	semidirect
7. C₂×#15692240	C₂	semidirect
8. C₂×#15692233	C₂	semidirect
9. C₂×#15692212	C₂	semidirect
10. C₂×C₄⋊Q₈×C₁₀₁	C₂	semidirect
16. C₂×Q₈×D₁₀₁	C₂²	semidirect
22. #15692253	C₂²	semidirect
23. #15692244	C₂²	semidirect
24. #15692243	C₂²	semidirect
25. #15692240	C₂²	semidirect
26. #15692233	C₂²	semidirect
27. #15692212	C₂²	semidirect
36. C₄⋊Q₈×C₁₀₁	C₂²	semidirect
122. C₂×C₁₀₁	#536	semidirect
128. C₁₀₁	C₂×#536	semidirect
133. C₂×C₄⋊Q₈	D₁₀₁	semidirect
153. C₄⋊Q₈	C₂×D₁₀₁	semidirect
252. C₂	#20265752	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×#13108075	2
C₂²×Q₈×D₁₀₁	2
#20265752	2
C₂×#15692253	2
C₂×#15692244	2
C₂×#15692243	2
C₂×#15692240	2
C₂×#15692233	2
C₂×#15692212	2
C₂×C₄⋊Q₈×C₁₀₁	2
C₂×#536	101

Subgroups — C2×#20265752 #27184450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265752 #27184450