Subgroups — C₂×#20265990 #27185895

Order 12928.

1408 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692303	C₂	1	2	yes	maximal, complemented
2	#20265990	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692287	C₂	1	4	yes	maximal, complemented
4	C₂×#15692286	C₂	1	2	yes	maximal, complemented
5	C₂×#15692278	C₂	1	2	yes	maximal, complemented
6	C₂×#15692277	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₁₆×D₁₀₁	C₂	1	2	yes	maximal, complemented
8	C₂×#15692261	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#15692303	—	2	8	no	—
11	#15692303	C₂²	1	16	yes	complemented
12	C₂×#13108075	—	2	2	no	—
13	C₂×#13108075	C₂²	1	4	yes	—
14	C₂×#13108074	—	2	4	no	—
15	C₂×#13108074	C₂²	1	2	yes	—
16	C₂×#13108074	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₀₁	—	2	2	no	—
18	C₂×Q₈×D₁₀₁	C₂²	1	4	yes	—
19	#15692287	C₂²	1	32	yes	complemented
20	#15692286	C₂²	1	16	yes	complemented
21	#15692278	C₂²	1	16	yes	complemented
22	#15692277	C₂²	1	8	yes	complemented
23	C₂×Dic₄₀₄	C₂²	1	1	yes	characteristic
24	C₂×#13108064	C₂²	1	4	yes	—
25	Q₁₆×D₁₀₁	C₂²	1	16	yes	complemented
26	C₂×#13108063	C₂²	1	2	yes	—
27	C₂×#13108061	C₂²	1	4	yes	—
28	C₂×D₈₀₈	C₂²	1	1	yes	characteristic
29	#15692261	C₂²	1	8	yes	complemented
30	C₂×#13108058	C₂²	1	1	yes	characteristic
31	C₂×#13108057	C₂²	1	2	yes	—
32	C₂×C₈×D₁₀₁	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₀₁	C₂²	1	2	yes	—
34	C₂×Q₁₆×C₁₀₁	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₁₀₁	C₂²	1	2	yes	—
36	C₂×Q₁₆×C₁₀₁	C₂²	1	8	yes	complemented
37	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
38	#13108075	—	2	16	no	—
39	#13108075	—	2	8	no	—
40	#13108075	C₂³	1	16	yes	—
41	#13108074	—	2	4	no	—
42	#13108074	—	2	32	no	—
43	#13108074	D₄	1	4	yes	—
44	#13108074	C₂³	1	4	yes	—
45	#13108074	C₂³	1	8	yes	—
46	C₂×Dic₂₀₂	—	2	4	no	—
47	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
48	C₂×Dic₂₀₂	C₂³	1	2	yes	—
49	Q₈×D₁₀₁	—	2	8	no	—
50	Q₈×D₁₀₁	—	2	16	no	—
51	Q₈×D₁₀₁	C₂³	1	16	yes	—
52	C₂×D₄₀₄	—	2	4	no	—
53	C₂×D₄₀₄	D₄	1	1	yes	characteristic
54	C₂×D₄₀₄	C₂³	1	2	yes	—
55	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
56	C₂×C₂²⋊D₁₀₁	D₄	1	2	yes	—
57	C₂×C₄×D₁₀₁	—	2	4	no	—
58	C₂×C₄×D₁₀₁	—	2	2	no	—
59	C₂×C₄×D₁₀₁	—	2	4	no	—
60	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
61	Dic₄₀₄	C₂³	1	4	yes	—
62	#13108064	C₂³	1	16	yes	—
63	#13108063	C₂³	1	8	yes	—
64	#13108061	C₂³	1	16	yes	—
65	D₈₀₈	C₂³	1	4	yes	—
66	#13108058	C₂³	1	4	yes	—
67	#13108057	C₂³	1	8	yes	—
68	C₂×#11717579	C₂³	1	2	yes	—
69	C₈×D₁₀₁	C₂³	1	8	yes	—
70	C₂×Q₈×C₁₀₁	—	2	2	no	—
71	C₂×Q₈×C₁₀₁	—	2	4	no	—
72	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
73	C₂×Q₈×C₁₀₁	C₂³	1	8	yes	—
74	C₂²×C₄×C₁₀₁	—	2	2	no	—
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	Q₁₆×C₁₀₁	C₂³	1	8	yes	—
77	Q₁₆×C₁₀₁	C₂³	1	8	yes	—
78	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₀₁	C₂³	1	4	yes	—
81	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₀₁	—	2	2	no	—
83	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
84	Q₈×C₁₀₁	—	2	4	no	—
85	Q₈×C₁₀₁	—	2	8	no	—
86	Q₈×C₁₀₁	C₂⁴	1	8	yes	—
87	C₂×C₄×C₁₀₁	—	2	4	no	—
88	C₂×C₄×C₁₀₁	—	2	8	no	—
89	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
93	C₂²⋊D₁₀₁	—	2	8	no	—
94	C₂²⋊D₁₀₁	C₂×D₄	1	8	yes	—
95	C₂×Dic₁₀₁	—	2	2	no	—
96	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
97	D₄₀₄	—	2	16	no	—
98	D₄₀₄	—	2	2	no	—
99	D₄₀₄	C₂×D₄	1	4	yes	—
100	D₄₀₄	C₂⁴	1	4	yes	—
101	C₄×D₁₀₁	—	2	16	no	—
102	C₄×D₁₀₁	—	2	8	no	—
103	C₄×D₁₀₁	—	2	16	no	—
104	C₄×D₁₀₁	C₂×D₄	1	4	yes	—
105	C₄×D₁₀₁	C₂⁴	1	4	yes	—
106	Dic₂₀₂	—	2	2	no	—
107	Dic₂₀₂	—	2	16	no	—
108	Dic₂₀₂	C₂×D₄	1	4	yes	—
109	Dic₂₀₂	C₂⁴	1	4	yes	—
110	C₈×C₁₀₁	C₂⁴	1	2	yes	—
111	C₈×C₁₀₁	C₂⁴	1	2	yes	—
112	#11717579	C₂⁴	1	4	yes	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
115	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₁₀₁	—	2	4	no	—
117	C₂×D₁₀₁	—	2	4	no	—
118	C₂×D₁₀₁	—	4	4	no	—
119	C₂×D₁₀₁	C₂²×D₄	1	4	yes	—
120	C₄×C₁₀₁	—	2	8	no	—
121	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
123	C₄×C₁₀₁	C₂⁵	1	1	yes	characteristic
124	Dic₁₀₁	—	2	4	no	—
125	Dic₁₀₁	C₂²×D₄	1	4	yes	—
126	C₂×C₁₀₁	—	2	2	no	—
127	C₂×C₁₀₁	#578	1	2	yes	complemented
128	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
129	D₁₀₁	—	2	4	no	—
130	D₁₀₁	—	4	4	no	—
131	C₂×#578	—	101	1	no	maximal, Sylow
132	C₁₀₁	C₂×#578	1	1	yes	characteristic, complemented, Sylow
133	C₂×2₋¹⁺⁴	—	101	2	no	—
134	#578	—	101	16	no	—
135	C₂×#137	—	101	2	no	—
136	C₂×#137	—	101	4	no	—
137	C₂×#135	—	101	2	no	—
138	C₂×#135	—	101	1	no	—
139	C₂²×Q₁₆	—	101	2	no	—
140	C₂²×Q₁₆	D₁₀₁	1	1	yes	characteristic, complemented
141	C₂×#131	—	101	1	no	—
142	2₋¹⁺⁴	—	101	16	no	—
143	2₋¹⁺⁴	—	202	8	no	—
144	C₂×#41	—	101	1	no	—
145	C₂×#41	—	101	4	no	—
146	C₂×#41	—	101	2	no	—
147	C₂×#41	—	202	4	no	—
148	C₂×#41	—	202	2	no	—
149	C₂²×Q₈	—	101	4	no	—
150	C₂²×Q₈	—	202	2	no	—
151	C₂²×Q₈	C₂×D₁₀₁	1	2	yes	—
152	#137	—	101	16	no	—
153	#137	—	101	32	no	—
154	#135	—	101	16	no	—
155	#135	—	101	8	no	—
156	C₂×Q₁₆	—	101	4	no	—
157	C₂×Q₁₆	—	101	16	no	—
158	C₂×Q₁₆	—	101	1	no	—
159	C₂×Q₁₆	C₂×D₁₀₁	1	2	yes	—
160	C₂×Q₁₆	C₂×D₁₀₁	1	2	yes	—
161	C₂×Q₁₆	C₂×D₁₀₁	1	8	yes	complemented
162	C₂×SD₁₆	—	101	4	no	—
163	C₂×SD₁₆	—	101	2	no	—
164	C₂×D₈	—	101	1	no	—
165	#131	—	101	8	no	—
166	C₂×M₁₆	—	101	1	no	—
167	C₂×M₁₆	—	101	2	no	—
168	C₂²×C₈	—	101	2	no	—
169	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
170	#41	—	101	4	no	—
171	#41	—	101	4	no	—
172	#41	—	101	16	no	—
173	#41	—	101	8	no	—
174	#41	—	202	32	no	—
175	#41	—	202	8	no	—
176	#41	—	202	16	no	—
177	#41	—	202	4	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	101	1	no	—
181	C₂×Q₈	—	101	16	no	—
182	C₂×Q₈	—	101	2	no	—
183	C₂×Q₈	—	202	4	no	—
184	C₂×Q₈	—	202	8	no	—
185	C₂×Q₈	—	202	16	no	—
186	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
187	C₂×Q₈	C₂²×D₁₀₁	1	8	yes	—
188	C₂×D₄	—	101	2	no	—
189	C₂×D₄	—	101	1	no	—
190	C₂×D₄	—	101	2	no	—
191	C₂×D₄	—	202	4	no	—
192	C₂×D₄	—	202	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	202	4	no	—
196	C₂²×C₄	—	202	4	no	—
197	C₂²×C₄	—	202	2	no	—
198	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
199	Q₁₆	—	101	16	no	—
200	Q₁₆	—	101	4	no	—
201	Q₁₆	C₂²×D₁₀₁	1	8	yes	—
202	Q₁₆	C₂²×D₁₀₁	1	8	yes	—
203	SD₁₆	—	101	16	no	—
204	SD₁₆	—	101	8	no	—
205	D₈	—	101	4	no	—
206	M₁₆	—	101	4	no	—
207	M₁₆	—	101	8	no	—
208	C₂×C₈	—	101	2	no	—
209	C₂×C₈	—	101	8	no	—
210	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₁₀₁	1	4	yes	—
213	C₂³	—	101	2	no	—
214	C₂³	—	202	2	no	—
215	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
216	Q₈	—	2	4	no	—
217	Q₈	—	2	8	no	—
218	Q₈	—	101	4	no	—
219	Q₈	—	101	4	no	—
220	Q₈	—	202	16	no	—
221	Q₈	—	202	2	no	—
222	Q₈	C₂³×D₁₀₁	1	8	yes	—
223	D₄	—	101	8	no	—
224	D₄	—	101	4	no	—
225	D₄	—	101	4	no	—
226	D₄	—	202	16	no	—
227	D₄	—	202	8	no	—
228	D₄	—	202	2	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	101	2	no	—
232	C₂×C₄	—	101	4	no	—
233	C₂×C₄	—	101	4	no	—
234	C₂×C₄	—	202	16	no	—
235	C₂×C₄	—	202	16	no	—
236	C₂×C₄	—	202	2	no	—
237	C₂×C₄	—	202	8	no	—
238	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
239	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
241	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
242	C₈	—	101	4	no	—
243	C₈	C₂³×D₁₀₁	1	2	yes	—
244	C₈	C₂³×D₁₀₁	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	101	4	no	—
247	C₂²	—	404	4	no	—
248	C₂²	—	202	4	no	—
249	C₂²	—	202	4	no	—
250	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
251	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
252	C₄	—	2	8	no	—
253	C₄	—	101	4	no	—
254	C₄	—	202	4	no	—
255	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
256	C₄	C₂×D₄×D₁₀₁	1	2	yes	—
257	C₄	C₂⁴×D₁₀₁	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	404	4	no	—
260	C₂	—	202	4	no	—
261	C₂	#20265990	1	2	yes	direct product factor
262	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator, cosocle	C₄×C₁₀₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₁
Fitting	C₂²×Q₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	C₂²×Q₁₆	D₁₀₁
hypocenter	C₁₀₁	C₂×#578
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15692303	C₂	semidirect
2. #20265990	C₂	direct (N × H)
3. C₂×#15692287	C₂	semidirect
4. C₂×#15692286	C₂	semidirect
5. C₂×#15692278	C₂	semidirect
6. C₂×#15692277	C₂	semidirect
7. C₂×Q₁₆×D₁₀₁	C₂	semidirect
8. C₂×#15692261	C₂	semidirect
9. C₂²×Q₁₆×C₁₀₁	C₂	semidirect
11. #15692303	C₂²	semidirect
19. #15692287	C₂²	semidirect
20. #15692286	C₂²	semidirect
21. #15692278	C₂²	semidirect
22. #15692277	C₂²	semidirect
25. Q₁₆×D₁₀₁	C₂²	semidirect
29. #15692261	C₂²	semidirect
36. C₂×Q₁₆×C₁₀₁	C₂²	semidirect
127. C₂×C₁₀₁	#578	semidirect
132. C₁₀₁	C₂×#578	semidirect
140. C₂²×Q₁₆	D₁₀₁	semidirect
161. C₂×Q₁₆	C₂×D₁₀₁	semidirect
261. C₂	#20265990	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15692303	2
#20265990	2
C₂×#15692287	2
C₂×#15692286	2
C₂×#15692278	2
C₂×#15692277	2
C₂×Q₁₆×D₁₀₁	2
C₂×#15692261	2
C₂²×Q₁₆×C₁₀₁	2
C₂×#578	101

Subgroups — C2×#20265990 #27185895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265990 #27185895