Subgroups — C₂×#20265726 #27184414

Order 12928.

1728 conjugacy classes of subgroups.

The subgroup lattice has 272 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, complemented
2	C₂²×#13108073	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265726	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15692253	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15692251	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692243	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692228	C₂	1	4	yes	maximal, complemented
8	C₂×#15692215	C₂	1	4	yes	maximal, complemented
9	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13108075	—	2	8	no	—
12	C₂×#13108075	C₂²	1	8	yes	complemented
13	C₂×#13108073	—	2	8	no	—
14	C₂×#13108073	C₂²	1	8	yes	complemented
15	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
16	C₂²×D₄₀₄	C₂²	1	2	yes	—
17	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
18	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
20	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
21	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
22	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
23	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
24	#15692253	C₂²	1	8	yes	complemented
25	#15692251	C₂²	1	8	yes	complemented
26	#15692243	C₂²	1	8	yes	complemented
27	#15692228	C₂²	1	32	yes	complemented
28	#15692215	C₂²	1	32	yes	complemented
29	C₂×#13108053	C₂²	1	4	yes	—
30	C₂×#13108051	C₂²	1	4	yes	—
31	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
32	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
33	C₄²×D₁₀₁	C₂²	1	8	yes	complemented
34	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	#124×C₁₀₁	C₂²	1	8	yes	complemented
37	C₂×C₂²⋊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	#13108073	—	2	32	no	—
43	C₂×Dic₂₀₂	—	2	4	no	—
44	C₂×Dic₂₀₂	C₂³	1	4	yes	—
45	C₂×D₄₀₄	—	2	8	no	—
46	C₂×D₄₀₄	—	2	4	no	—
47	C₂×D₄₀₄	C₂³	1	8	yes	—
48	C₂×D₄₀₄	C₂³	1	4	yes	—
49	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
50	C₂×C₂²⋊D₁₀₁	—	2	16	no	—
51	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
52	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	—
53	C₂²×Dic₁₀₁	—	2	8	no	—
54	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
55	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
56	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
58	C₂×C₄×D₁₀₁	—	2	8	no	—
59	C₂×C₄×D₁₀₁	—	2	8	no	—
60	C₂×C₄×D₁₀₁	D₄	1	8	yes	complemented
61	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
62	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
63	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
64	#13108053	C₂³	1	16	yes	—
65	#13108051	C₂³	1	16	yes	—
66	C₄×Dic₁₀₁	C₂³	1	8	yes	—
67	C₄×Dic₁₀₁	C₂³	1	4	yes	—
68	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
69	C₂×Q₈×C₁₀₁	—	2	4	no	—
70	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
71	C₂×D₄×C₁₀₁	—	2	4	no	—
72	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
73	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₀₁	C₂³	1	16	yes	—
77	C₄²×C₁₀₁	C₂³	1	4	yes	—
78	C₂³×C₁₀₁	—	2	4	no	—
79	C₂³×C₁₀₁	—	2	8	no	—
80	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
81	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
82	C₂²×D₁₀₁	—	2	4	no	—
83	C₂²×D₁₀₁	—	2	4	no	—
84	C₂²×D₁₀₁	—	2	8	no	—
85	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
86	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
87	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
88	Q₈×C₁₀₁	—	2	8	no	—
89	D₄×C₁₀₁	—	2	8	no	—
90	C₂×C₄×C₁₀₁	—	2	4	no	—
91	C₂×C₄×C₁₀₁	—	2	4	no	—
92	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
94	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
95	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
96	C₂²⋊D₁₀₁	—	2	16	no	—
97	C₂²⋊D₁₀₁	—	2	32	no	—
98	C₂×Dic₁₀₁	—	2	4	no	—
99	C₂×Dic₁₀₁	—	2	16	no	—
100	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
101	C₂×Dic₁₀₁	#41	1	8	yes	complemented
102	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
103	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
104	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
105	D₄₀₄	—	2	16	no	—
106	D₄₀₄	—	2	8	no	—
107	C₄×D₁₀₁	—	2	16	no	—
108	C₄×D₁₀₁	—	2	16	no	—
109	C₄×D₁₀₁	C₂×D₄	1	16	yes	complemented
110	Dic₂₀₂	—	2	8	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	—	2	8	no	—
113	C₂²×C₁₀₁	—	4	8	no	—
114	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
115	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
116	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
117	C₂×D₁₀₁	—	2	8	no	—
118	C₂×D₁₀₁	—	2	4	no	—
119	C₂×D₁₀₁	—	2	4	no	—
120	C₂×D₁₀₁	—	2	8	no	—
121	C₂×D₁₀₁	—	4	8	no	—
122	C₂×D₁₀₁	C₂²×D₄	1	4	yes	—
123	C₄×C₁₀₁	—	2	4	no	—
124	C₄×C₁₀₁	—	2	4	no	—
125	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
126	Dic₁₀₁	—	2	4	no	—
127	Dic₁₀₁	C₂²×D₄	1	4	yes	—
128	Dic₁₀₁	C₂×#41	1	8	yes	complemented
129	C₂×C₁₀₁	—	4	4	no	—
130	C₂×C₁₀₁	#532	1	4	yes	complemented
131	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
132	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
133	D₁₀₁	—	2	4	no	—
134	D₁₀₁	—	4	4	no	—
135	C₂×#532	—	101	1	no	maximal, Sylow
136	C₁₀₁	C₂×#532	1	1	yes	characteristic, complemented, Sylow
137	C₂²×#41	—	101	1	no	—
138	C₂²×#41	—	101	1	no	—
139	#532	—	101	16	no	—
140	C₂×C₄⋊Q₈	—	101	1	no	—
141	C₂×#127	—	101	1	no	—
142	C₂×#124	—	101	1	no	—
143	C₂×#124	D₁₀₁	1	1	yes	characteristic, complemented
144	C₂×#121	—	101	4	no	—
145	C₂×C₄×D₄	—	101	4	no	—
146	C₂²×C₄²	—	101	1	no	—
147	C₂×#41	—	101	8	no	—
148	C₂×#41	—	101	8	no	—
149	C₂×#41	—	202	8	no	—
150	C₂×#41	—	202	8	no	—
151	C₂²×Q₈	—	101	1	no	—
152	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
153	C₂²×D₄	—	101	4	no	—
154	C₂²×D₄	—	101	2	no	—
155	C₂²×D₄	—	101	2	no	—
156	C₂²×D₄	—	101	1	no	—
157	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
158	C₂³×C₄	—	101	1	no	—
159	C₂³×C₄	—	101	2	no	—
160	C₂³×C₄	—	101	2	no	—
161	C₂³×C₄	—	101	2	no	—
162	C₄⋊Q₈	—	101	8	no	—
163	#127	—	101	8	no	—
164	#124	—	101	8	no	—
165	#124	C₂×D₁₀₁	1	8	yes	complemented
166	#121	—	101	32	no	—
167	C₄×D₄	—	101	32	no	—
168	C₂×C₄⋊C₄	—	101	4	no	—
169	C₂×C₂²⋊C₄	—	101	4	no	—
170	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	4	yes	—
171	C₂×C₄²	—	101	2	no	—
172	C₂×C₄²	—	101	1	no	—
173	C₂×C₄²	—	101	8	no	—
174	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
175	C₂⁴	—	101	1	no	—
176	C₂⁴	—	101	2	no	—
177	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
178	#41	—	202	32	no	—
179	#41	—	202	32	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	101	4	no	—
182	C₂×Q₈	—	202	4	no	—
183	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	101	16	no	—
186	C₂×D₄	—	101	8	no	—
187	C₂×D₄	—	101	8	no	—
188	C₂×D₄	—	101	4	no	—
189	C₂×D₄	—	202	8	no	—
190	C₂×D₄	—	202	16	no	—
191	C₂×D₄	—	202	8	no	—
192	C₂×D₄	—	202	4	no	—
193	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	1	no	—
196	C₂²×C₄	—	101	8	no	—
197	C₂²×C₄	—	101	4	no	—
198	C₂²×C₄	—	101	8	no	—
199	C₂²×C₄	—	101	2	no	—
200	C₂²×C₄	—	101	8	no	—
201	C₂²×C₄	—	101	8	no	—
202	C₂²×C₄	—	202	8	no	—
203	C₂²×C₄	—	202	8	no	—
204	C₂²×C₄	—	202	8	no	—
205	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
206	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
208	C₄⋊C₄	—	101	16	no	—
209	C₂²⋊C₄	—	101	16	no	—
210	C₂²⋊C₄	C₂²×D₁₀₁	1	16	yes	—
211	C₄²	—	101	8	no	—
212	C₄²	—	101	4	no	—
213	C₄²	C₂²×D₁₀₁	1	4	yes	—
214	C₂³	—	2	4	no	—
215	C₂³	—	2	8	no	—
216	C₂³	—	101	4	no	—
217	C₂³	—	101	2	no	—
218	C₂³	—	101	4	no	—
219	C₂³	—	202	4	no	—
220	C₂³	—	202	8	no	—
221	C₂³	—	202	4	no	—
222	C₂³	C₂³×D₁₀₁	1	4	yes	—
223	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	Q₈	—	202	8	no	—
226	D₄	—	2	8	no	—
227	D₄	—	202	16	no	—
228	D₄	—	202	32	no	—
229	D₄	—	202	16	no	—
230	D₄	—	202	8	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	101	4	no	—
234	C₂×C₄	—	101	8	no	—
235	C₂×C₄	—	101	4	no	—
236	C₂×C₄	—	101	2	no	—
237	C₂×C₄	—	101	16	no	—
238	C₂×C₄	—	101	4	no	—
239	C₂×C₄	—	202	16	no	—
240	C₂×C₄	—	202	16	no	—
241	C₂×C₄	—	202	16	no	—
242	C₂×C₄	—	202	4	no	—
243	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
244	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
245	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
246	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	101	4	no	—
251	C₂²	—	404	8	no	—
252	C₂²	—	202	4	no	—
253	C₂²	—	202	8	no	—
254	C₂²	—	202	4	no	—
255	C₂²	—	202	8	no	—
256	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
257	C₂²	#41×D₁₀₁	1	4	yes	—
258	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	101	4	no	—
262	C₄	—	101	8	no	—
263	C₄	—	202	4	no	—
264	C₄	C₂×D₄×D₁₀₁	1	4	yes	—
265	C₂	—	4	4	no	—
266	C₂	—	404	4	no	—
267	C₂	—	202	4	no	—
268	C₂	#20265726	1	4	yes	direct product factor
269	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
270	C₂	C₂×#41×D₁₀₁	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#124×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#124	D₁₀₁
hypocenter	C₁₀₁	C₂×#532
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108075	C₂	semidirect
2. C₂²×#13108073	C₂	semidirect
3. #20265726	C₂	direct (N × H)
4. C₂×#15692253	C₂	semidirect
5. C₂×#15692251	C₂	semidirect
6. C₂×#15692243	C₂	semidirect
7. C₂×#15692228	C₂	semidirect
8. C₂×#15692215	C₂	semidirect
9. C₂×C₄²×D₁₀₁	C₂	semidirect
10. C₂×#124×C₁₀₁	C₂	semidirect
12. C₂×#13108075	C₂²	semidirect
14. C₂×#13108073	C₂²	semidirect
24. #15692253	C₂²	semidirect
25. #15692251	C₂²	semidirect
26. #15692243	C₂²	semidirect
27. #15692228	C₂²	semidirect
28. #15692215	C₂²	semidirect
33. C₄²×D₁₀₁	C₂²	semidirect
36. #124×C₁₀₁	C₂²	semidirect
60. C₂×C₄×D₁₀₁	D₄	semidirect
101. C₂×Dic₁₀₁	#41	semidirect
109. C₄×D₁₀₁	C₂×D₄	semidirect
128. Dic₁₀₁	C₂×#41	semidirect
130. C₂×C₁₀₁	#532	semidirect
136. C₁₀₁	C₂×#532	semidirect
143. C₂×#124	D₁₀₁	semidirect
165. #124	C₂×D₁₀₁	semidirect
268. C₂	#20265726	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×#13108073	2
#20265726	2
C₂×#15692253	2
C₂×#15692251	2
C₂×#15692243	2
C₂×#15692228	2
C₂×#15692215	2
C₂×C₄²×D₁₀₁	2
C₂×#124×C₁₀₁	2
C₂×#532	101

Subgroups — C2×#20265726 #27184414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265726 #27184414