Subgroups — C₂×#20268302 #27208386

Order 13184.

1880 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	C₂²×#13110105	C₂	1	1	yes	maximal, characteristic, complemented
2	#20268302	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15693004	C₂	1	2	yes	maximal, complemented
4	C₂×#15693000	C₂	1	2	yes	maximal, complemented
5	C₂×#15692995	C₂	1	2	yes	maximal, complemented
6	C₂×#15692990	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂²⋊D₁₀₃	C₂	1	4	yes	maximal, complemented
8	C₂×#15692980	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13110105	—	2	8	no	—
11	C₂×#13110105	C₂²	1	8	yes	complemented
12	C₂²×Dic₂₀₆	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄₁₂	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₂²⋊D₁₀₃	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₁₀₃	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₀₃	C₂²	1	2	yes	complemented
17	#15693004	C₂²	1	16	yes	complemented
18	#15693000	C₂²	1	16	yes	complemented
19	#15692995	C₂²	1	16	yes	complemented
20	#15692990	C₂²	1	16	yes	complemented
21	C₄×C₂²⋊D₁₀₃	C₂²	1	32	yes	complemented
22	#15692980	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	complemented
24	C₂×Dic₁₀₃⋊C₄	C₂²	1	4	yes	complemented
25	C₂×C₂²⋊Dic₁₀₃	C₂²	1	4	yes	—
26	C₂×C₂²⋊Dic₁₀₃	C₂²	1	2	yes	—
27	C₂×(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	complemented
28	C₂×C₄×Dic₁₀₃	C₂²	1	2	yes	complemented
29	C₂⁵×C₁₀₃	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₀₃	—	2	8	no	—
31	C₂³×C₄×C₁₀₃	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₀₃	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₀₃	C₂²	1	2	yes	—
34	C₂³×C₄×C₁₀₃	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₁₀₃	C₂²	1	8	yes	complemented
36	C₂³×D₁₀₃	C₂³	1	2	yes	—
37	#13110105	—	2	32	no	—
38	C₂×Dic₂₀₆	—	2	4	no	—
39	C₂×Dic₂₀₆	C₂³	1	4	yes	complemented
40	C₂×D₄₁₂	—	2	4	no	—
41	C₂×D₄₁₂	C₂³	1	4	yes	complemented
42	C₂×C₂²⋊D₁₀₃	—	2	8	no	—
43	C₂×C₂²⋊D₁₀₃	—	2	16	no	—
44	C₂×C₂²⋊D₁₀₃	C₂³	1	8	yes	—
45	C₂×C₂²⋊D₁₀₃	C₂³	1	16	yes	—
46	C₂²×Dic₁₀₃	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₃	C₂³	1	4	yes	—
48	C₂×C₄×D₁₀₃	—	2	8	no	—
49	C₂×C₄×D₁₀₃	C₂³	1	8	yes	complemented
50	C₄⋊Dic₁₀₃	C₂³	1	8	yes	complemented
51	Dic₁₀₃⋊C₄	C₂³	1	16	yes	complemented
52	C₂²⋊Dic₁₀₃	C₂³	1	8	yes	—
53	C₂²⋊Dic₁₀₃	C₂³	1	16	yes	—
54	(C₂×D₁₀₃)⋊C₄	C₂³	1	16	yes	complemented
55	C₄×Dic₁₀₃	C₂³	1	8	yes	complemented
56	C₂⁴×C₁₀₃	—	2	4	no	—
57	C₂⁴×C₁₀₃	—	2	8	no	—
58	C₂⁴×C₁₀₃	C₂³	1	4	yes	—
59	C₂⁴×C₁₀₃	C₂³	1	1	yes	characteristic
60	C₂⁴×C₁₀₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₀₃	—	2	4	no	—
62	C₂²×C₄×C₁₀₃	—	2	8	no	—
63	C₂²×C₄×C₁₀₃	—	2	8	no	—
64	C₂²×C₄×C₁₀₃	—	2	32	no	—
65	C₂²×C₄×C₁₀₃	D₄	1	8	yes	complemented
66	C₂²×C₄×C₁₀₃	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₃	C₂³	1	8	yes	—
68	C₂²×C₄×C₁₀₃	C₂³	1	8	yes	—
69	C₂²×C₄×C₁₀₃	C₂³	1	4	yes	—
70	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₃	C₂³	1	4	yes	—
72	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
73	C₂³×C₁₀₃	—	2	32	no	—
74	C₂³×C₁₀₃	—	2	8	no	—
75	C₂³×C₁₀₃	—	2	4	no	—
76	C₂³×C₁₀₃	—	2	4	no	—
77	C₂³×C₁₀₃	—	2	4	no	—
78	C₂³×C₁₀₃	—	2	16	no	—
79	C₂³×C₁₀₃	C₂×D₄	1	4	yes	—
80	C₂³×C₁₀₃	#41	1	8	yes	complemented
81	C₂³×C₁₀₃	C₂⁴	1	2	yes	—
82	C₂³×C₁₀₃	C₂⁴	1	4	yes	—
83	C₂³×C₁₀₃	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₀₃	—	2	4	no	—
85	C₂²×D₁₀₃	—	2	8	no	—
86	C₂²×D₁₀₃	C₂⁴	1	4	yes	—
87	C₂×C₄×C₁₀₃	—	2	8	no	—
88	C₂×C₄×C₁₀₃	—	2	4	no	—
89	C₂×C₄×C₁₀₃	—	2	16	no	—
90	C₂×C₄×C₁₀₃	—	2	16	no	—
91	C₂×C₄×C₁₀₃	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₁₀₃	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₁₀₃	C₂×D₄	1	16	yes	complemented
94	C₂×C₄×C₁₀₃	C₂⁴	1	4	yes	—
95	C₂×C₄×C₁₀₃	C₂⁴	1	2	yes	—
96	C₂×C₄×C₁₀₃	C₂⁴	1	2	yes	—
97	C₂²⋊D₁₀₃	—	2	16	no	—
98	C₂²⋊D₁₀₃	—	2	32	no	—
99	C₂×Dic₁₀₃	—	2	4	no	—
100	C₂×Dic₁₀₃	—	2	8	no	—
101	C₂×Dic₁₀₃	C₂⁴	1	8	yes	—
102	C₂×Dic₁₀₃	C₂⁴	1	4	yes	—
103	D₄₁₂	—	2	8	no	—
104	C₄×D₁₀₃	—	2	16	no	—
105	Dic₂₀₆	—	2	8	no	—
106	C₂²×C₁₀₃	—	2	32	no	—
107	C₂²×C₁₀₃	—	2	8	no	—
108	C₂²×C₁₀₃	—	2	16	no	—
109	C₂²×C₁₀₃	—	2	4	no	—
110	C₂²×C₁₀₃	—	2	4	no	—
111	C₂²×C₁₀₃	—	2	4	no	—
112	C₂²×C₁₀₃	C₂²×D₄	1	4	yes	—
113	C₂²×C₁₀₃	C₂²×D₄	1	2	yes	—
114	C₂²×C₁₀₃	C₂×#41	1	4	yes	—
115	C₂²×C₁₀₃	C₂×#41	1	8	yes	complemented
116	C₂²×C₁₀₃	C₂⁵	1	1	yes	characteristic
117	C₂×D₁₀₃	—	2	4	no	—
118	C₂×D₁₀₃	—	2	8	no	—
119	C₂×D₁₀₃	—	4	8	no	—
120	C₄×C₁₀₃	—	2	4	no	—
121	C₄×C₁₀₃	C₂²×D₄	1	4	yes	—
122	C₄×C₁₀₃	C₂²×D₄	1	4	yes	—
123	Dic₁₀₃	—	2	4	no	—
124	Dic₁₀₃	—	2	8	no	—
125	C₂×C₁₀₃	—	2	4	no	—
126	C₂×C₁₀₃	—	2	8	no	—
127	C₂×C₁₀₃	#525	1	4	yes	complemented
128	C₂×C₁₀₃	C₂³×D₄	1	1	yes	characteristic
129	C₂×C₁₀₃	C₂²×#41	1	2	yes	—
130	D₁₀₃	—	4	4	no	—
131	C₂×#525	—	103	1	no	maximal, Sylow
132	C₁₀₃	C₂×#525	1	1	yes	characteristic, complemented, Sylow
133	C₂²×#41	—	103	1	no	—
134	C₂⁴×C₄	D₁₀₃	1	1	yes	characteristic, complemented
135	#525	—	103	16	no	—
136	C₂×#123	—	103	2	no	—
137	C₂×C₂²⋊Q₈	—	103	2	no	—
138	C₂×#121	—	103	2	no	—
139	C₂×#120	—	103	2	no	—
140	C₂×C₄×D₄	—	103	4	no	—
141	C₂×#117	—	103	1	no	—
142	C₂⁵	C₂×D₁₀₃	1	1	yes	characteristic
143	C₂×#41	—	103	8	no	—
144	C₂×#41	—	206	8	no	—
145	C₂²×Q₈	—	103	1	no	—
146	C₂²×D₄	—	103	4	no	—
147	C₂²×D₄	—	103	2	no	—
148	C₂²×D₄	—	103	1	no	—
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	103	2	no	—
151	C₂³×C₄	C₂×D₁₀₃	1	2	yes	—
152	C₂³×C₄	C₂×D₁₀₃	1	1	yes	characteristic
153	C₂³×C₄	C₂×D₁₀₃	1	2	yes	—
154	C₂³×C₄	C₂×D₁₀₃	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₁₀₃	1	8	yes	complemented
156	#123	—	103	16	no	—
157	C₂²⋊Q₈	—	103	16	no	—
158	#121	—	103	16	no	—
159	#120	—	103	16	no	—
160	C₄×D₄	—	103	32	no	—
161	#117	—	103	8	no	—
162	C₂×C₄⋊C₄	—	103	2	no	—
163	C₂×C₄⋊C₄	—	103	4	no	—
164	C₂×C₂²⋊C₄	—	103	2	no	—
165	C₂×C₂²⋊C₄	—	103	4	no	—
166	C₂×C₂²⋊C₄	—	103	4	no	—
167	C₂×C₄²	—	103	2	no	—
168	C₂⁴	—	2	4	no	—
169	C₂⁴	—	2	8	no	—
170	C₂⁴	—	103	2	no	—
171	C₂⁴	C₂²×D₁₀₃	1	4	yes	—
172	C₂⁴	C₂²×D₁₀₃	1	1	yes	characteristic
173	C₂⁴	C₂²×D₁₀₃	1	2	yes	—
174	#41	—	206	32	no	—
175	C₂×Q₈	—	103	4	no	—
176	C₂×Q₈	—	206	4	no	—
177	C₂×D₄	—	103	16	no	—
178	C₂×D₄	—	103	8	no	—
179	C₂×D₄	—	103	4	no	—
180	C₂×D₄	—	206	8	no	—
181	C₂×D₄	—	206	16	no	—
182	C₂×D₄	—	206	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	103	4	no	—
188	C₂²×C₄	—	103	2	no	—
189	C₂²×C₄	—	103	8	no	—
190	C₂²×C₄	—	206	8	no	—
191	C₂²×C₄	C₂²⋊D₁₀₃	1	8	yes	complemented
192	C₂²×C₄	C₂²×D₁₀₃	1	2	yes	—
193	C₂²×C₄	C₂²×D₁₀₃	1	8	yes	—
194	C₂²×C₄	C₂²×D₁₀₃	1	8	yes	—
195	C₂²×C₄	C₂²×D₁₀₃	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₀₃	1	4	yes	—
198	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
199	C₄⋊C₄	—	103	16	no	—
200	C₄⋊C₄	—	103	8	no	—
201	C₂²⋊C₄	—	103	8	no	—
202	C₂²⋊C₄	—	103	16	no	—
203	C₂²⋊C₄	—	103	16	no	—
204	C₄²	—	103	8	no	—
205	C₂³	—	2	32	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	16	no	—
211	C₂³	—	103	4	no	—
212	C₂³	—	206	4	no	—
213	C₂³	—	206	8	no	—
214	C₂³	C₂×C₂²⋊D₁₀₃	1	4	yes	—
215	C₂³	#13110105	1	8	yes	complemented
216	C₂³	C₂³×D₁₀₃	1	2	yes	—
217	C₂³	C₂³×D₁₀₃	1	4	yes	—
218	C₂³	C₂³×D₁₀₃	1	1	yes	characteristic
219	Q₈	—	206	8	no	—
220	D₄	—	206	16	no	—
221	D₄	—	206	32	no	—
222	D₄	—	206	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	16	no	—
227	C₂×C₄	—	103	8	no	—
228	C₂×C₄	—	103	4	no	—
229	C₂×C₄	—	206	4	no	—
230	C₂×C₄	—	206	8	no	—
231	C₂×C₄	—	206	16	no	—
232	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	16	yes	complemented
235	C₂×C₄	C₂³×D₁₀₃	1	4	yes	—
236	C₂×C₄	C₂³×D₁₀₃	1	2	yes	—
237	C₂×C₄	C₂³×D₁₀₃	1	2	yes	—
238	C₂²	—	2	32	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	16	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	412	8	no	—
245	C₂²	—	206	4	no	—
246	C₂²	—	206	8	no	—
247	C₂²	C₂²×C₂²⋊D₁₀₃	1	4	yes	—
248	C₂²	C₂²×C₂²⋊D₁₀₃	1	2	yes	—
249	C₂²	C₂×#13110105	1	4	yes	—
250	C₂²	C₂×#13110105	1	8	yes	complemented
251	C₂²	C₂⁴×D₁₀₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	206	4	no	—
254	C₄	—	206	8	no	—
255	C₄	C₂²×C₂²⋊D₁₀₃	1	4	yes	—
256	C₄	C₂²×C₂²⋊D₁₀₃	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	2	8	no	—
259	C₂	—	412	4	no	—
260	C₂	#20268302	1	4	yes	direct product factor
261	C₂	C₂³×C₂²⋊D₁₀₃	1	1	yes	characteristic
262	C₂	C₂²×#13110105	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₃
commutator, cosocle	C₂²×C₁₀₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₃
Fitting	C₂⁴×C₄×C₁₀₃	C₂
socle	C₂³×C₁₀₃	C₂⁴
hypercenter	C₂⁴×C₄	D₁₀₃
hypocenter	C₁₀₃	C₂×#525
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13110105	C₂	semidirect
2. #20268302	C₂	direct (N × H)
3. C₂×#15693004	C₂	semidirect
4. C₂×#15693000	C₂	semidirect
5. C₂×#15692995	C₂	semidirect
6. C₂×#15692990	C₂	semidirect
7. C₂×C₄×C₂²⋊D₁₀₃	C₂	semidirect
8. C₂×#15692980	C₂	semidirect
9. C₂⁴×C₄×C₁₀₃	C₂	semidirect
11. C₂×#13110105	C₂²	semidirect
12. C₂²×Dic₂₀₆	C₂²	semidirect
13. C₂²×D₄₁₂	C₂²	semidirect
16. C₂²×C₄×D₁₀₃	C₂²	semidirect
17. #15693004	C₂²	semidirect
18. #15693000	C₂²	semidirect
19. #15692995	C₂²	semidirect
20. #15692990	C₂²	semidirect
21. C₄×C₂²⋊D₁₀₃	C₂²	semidirect
22. #15692980	C₂²	semidirect
23. C₂×C₄⋊Dic₁₀₃	C₂²	semidirect
24. C₂×Dic₁₀₃⋊C₄	C₂²	semidirect
27. C₂×(C₂×D₁₀₃)⋊C₄	C₂²	semidirect
28. C₂×C₄×Dic₁₀₃	C₂²	semidirect
35. C₂³×C₄×C₁₀₃	C₂²	semidirect
39. C₂×Dic₂₀₆	C₂³	semidirect
41. C₂×D₄₁₂	C₂³	semidirect
49. C₂×C₄×D₁₀₃	C₂³	semidirect
50. C₄⋊Dic₁₀₃	C₂³	semidirect
51. Dic₁₀₃⋊C₄	C₂³	semidirect
54. (C₂×D₁₀₃)⋊C₄	C₂³	semidirect
55. C₄×Dic₁₀₃	C₂³	semidirect
65. C₂²×C₄×C₁₀₃	D₄	semidirect
80. C₂³×C₁₀₃	#41	semidirect
93. C₂×C₄×C₁₀₃	C₂×D₄	semidirect
115. C₂²×C₁₀₃	C₂×#41	semidirect
127. C₂×C₁₀₃	#525	semidirect
132. C₁₀₃	C₂×#525	semidirect
134. C₂⁴×C₄	D₁₀₃	semidirect
155. C₂³×C₄	C₂×D₁₀₃	semidirect
191. C₂²×C₄	C₂²⋊D₁₀₃	semidirect
215. C₂³	#13110105	semidirect
234. C₂×C₄	C₂×C₂²⋊D₁₀₃	semidirect
250. C₂²	C₂×#13110105	semidirect
260. C₂	#20268302	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×#13110105	2
#20268302	2
C₂×#15693004	2
C₂×#15693000	2
C₂×#15692995	2
C₂×#15692990	2
C₂×C₄×C₂²⋊D₁₀₃	2
C₂×#15692980	2
C₂⁴×C₄×C₁₀₃	2
C₂×#525	103

Subgroups — C2×#20268302 #27208386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20268302 #27208386