Subgroups — C₂×#20265723 #27184411

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×#13108074	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265723	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692253	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692251	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15692244	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692242	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692227	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	4	yes	maximal, complemented
9	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13108074	—	2	8	no	—
12	C₂×#13108074	C₂²	1	8	yes	complemented
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
15	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
17	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
19	#15692253	C₂²	1	8	yes	complemented
20	#15692251	C₂²	1	8	yes	complemented
21	#15692244	C₂²	1	8	yes	complemented
22	#15692242	C₂²	1	8	yes	complemented
23	#15692227	C₂²	1	32	yes	complemented
24	C₄×C₂²⋊D₁₀₁	C₂²	1	32	yes	complemented
25	C₂×#13108053	C₂²	1	4	yes	—
26	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
27	C₂×#13108051	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₁₀₁	C₂²	1	8	yes	complemented
32	C₂×#41×C₁₀₁	—	2	8	no	—
33	C₂×#41×C₁₀₁	C₂²	1	8	yes	complemented
34	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
36	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
38	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂³×D₁₀₁	C₂³	1	2	yes	—
40	#13108074	—	2	32	no	—
41	C₂×Dic₂₀₂	—	2	4	no	—
42	C₂×Dic₂₀₂	C₂³	1	4	yes	—
43	C₂×D₄₀₄	—	2	4	no	—
44	C₂×D₄₀₄	C₂³	1	4	yes	—
45	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
46	C₂×C₂²⋊D₁₀₁	—	2	16	no	—
47	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
48	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	—
49	C₂²×Dic₁₀₁	—	2	8	no	—
50	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
53	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
54	C₂×C₄×D₁₀₁	—	2	8	no	—
55	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
56	#13108053	C₂³	1	16	yes	—
57	C₂²⋊Dic₁₀₁	C₂³	1	16	yes	—
58	#13108051	C₂³	1	16	yes	—
59	C₄×Dic₁₀₁	C₂³	1	8	yes	—
60	C₄×Dic₁₀₁	C₂³	1	4	yes	—
61	C₄×Dic₁₀₁	C₂³	1	4	yes	—
62	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
63	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
64	#41×C₁₀₁	—	2	32	no	—
65	C₂×Q₈×C₁₀₁	—	2	4	no	—
66	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
67	C₂×D₄×C₁₀₁	—	2	8	no	—
68	C₂×D₄×C₁₀₁	—	2	4	no	—
69	C₂×D₄×C₁₀₁	C₂³	1	8	yes	—
70	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
71	C₂²×C₄×C₁₀₁	—	2	8	no	—
72	C₂²×C₄×C₁₀₁	D₄	1	8	yes	complemented
73	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
77	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
78	C₂³×C₁₀₁	—	2	4	no	—
79	C₂³×C₁₀₁	—	2	4	no	—
80	C₂³×C₁₀₁	—	2	8	no	—
81	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
82	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
83	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
84	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
85	C₂²×D₁₀₁	—	2	4	no	—
86	C₂²×D₁₀₁	—	2	8	no	—
87	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
88	Q₈×C₁₀₁	—	2	8	no	—
89	D₄×C₁₀₁	—	2	16	no	—
90	D₄×C₁₀₁	—	2	8	no	—
91	C₂×C₄×C₁₀₁	—	2	4	no	—
92	C₂×C₄×C₁₀₁	—	2	16	no	—
93	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
95	C₂×C₄×C₁₀₁	C₂×D₄	1	16	yes	complemented
96	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
97	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
99	C₂²⋊D₁₀₁	—	2	16	no	—
100	C₂²⋊D₁₀₁	—	2	32	no	—
101	C₂×Dic₁₀₁	—	2	4	no	—
102	C₂×Dic₁₀₁	—	2	4	no	—
103	C₂×Dic₁₀₁	—	2	16	no	—
104	C₂×Dic₁₀₁	#41	1	8	yes	complemented
105	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
106	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
107	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
108	D₄₀₄	—	2	8	no	—
109	C₄×D₁₀₁	—	2	16	no	—
110	Dic₂₀₂	—	2	8	no	—
111	C₂²×C₁₀₁	—	2	8	no	—
112	C₂²×C₁₀₁	—	2	4	no	—
113	C₂²×C₁₀₁	—	2	4	no	—
114	C₂²×C₁₀₁	—	2	8	no	—
115	C₂²×C₁₀₁	—	4	8	no	—
116	C₂²×C₁₀₁	C₂²×D₄	1	4	yes	—
117	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
118	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
119	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
120	C₂×D₁₀₁	—	2	4	no	—
121	C₂×D₁₀₁	—	2	8	no	—
122	C₂×D₁₀₁	—	4	8	no	—
123	C₄×C₁₀₁	—	2	4	no	—
124	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
125	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
126	Dic₁₀₁	—	2	4	no	—
127	Dic₁₀₁	—	2	4	no	—
128	Dic₁₀₁	C₂×#41	1	8	yes	complemented
129	C₂×C₁₀₁	—	2	4	no	—
130	C₂×C₁₀₁	—	4	4	no	—
131	C₂×C₁₀₁	#532	1	4	yes	complemented
132	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
133	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
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	D₁₀₁	1	1	yes	characteristic, complemented
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	—	101	1	no	—
144	C₂×#121	—	101	4	no	—
145	C₂×C₄×D₄	—	101	4	no	—
146	C₂²×C₄²	—	101	1	no	—
147	C₂×#41	—	2	8	no	—
148	C₂×#41	—	101	8	no	—
149	C₂×#41	—	202	8	no	—
150	C₂×#41	C₂×D₁₀₁	1	8	yes	complemented
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	1	no	—
156	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
157	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
158	C₂³×C₄	—	101	2	no	—
159	C₂³×C₄	—	101	2	no	—
160	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
161	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
162	C₄⋊Q₈	—	101	8	no	—
163	#127	—	101	8	no	—
164	#124	—	101	8	no	—
165	#124	—	101	8	no	—
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	—	101	4	no	—
171	C₂×C₄²	—	101	1	no	—
172	C₂×C₄²	—	101	1	no	—
173	C₂×C₄²	—	101	8	no	—
174	C₂×C₄²	—	101	2	no	—
175	C₂⁴	—	101	2	no	—
176	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
177	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
178	#41	—	2	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	8	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	101	16	no	—
187	C₂×D₄	—	101	8	no	—
188	C₂×D₄	—	101	4	no	—
189	C₂×D₄	—	202	16	no	—
190	C₂×D₄	—	202	8	no	—
191	C₂×D₄	—	202	4	no	—
192	C₂×D₄	C₂²×D₁₀₁	1	8	yes	—
193	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	101	2	no	—
196	C₂²×C₄	—	101	2	no	—
197	C₂²×C₄	—	101	8	no	—
198	C₂²×C₄	—	101	2	no	—
199	C₂²×C₄	—	101	8	no	—
200	C₂²×C₄	—	202	8	no	—
201	C₂²×C₄	—	202	8	no	—
202	C₂²×C₄	C₂²⋊D₁₀₁	1	8	yes	complemented
203	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
204	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
205	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₀₁	1	4	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	—	101	16	no	—
211	C₄²	—	101	8	no	—
212	C₄²	—	101	4	no	—
213	C₄²	—	101	4	no	—
214	C₂³	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	8	no	—
217	C₂³	—	101	4	no	—
218	C₂³	—	202	4	no	—
219	C₂³	—	202	8	no	—
220	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
221	C₂³	C₂³×D₁₀₁	1	4	yes	—
222	C₂³	C₂³×D₁₀₁	1	2	yes	—
223	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	Q₈	—	202	8	no	—
226	D₄	—	2	16	no	—
227	D₄	—	2	8	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	16	no	—
233	C₂×C₄	—	101	8	no	—
234	C₂×C₄	—	101	4	no	—
235	C₂×C₄	—	101	4	no	—
236	C₂×C₄	—	101	4	no	—
237	C₂×C₄	—	202	16	no	—
238	C₂×C₄	—	202	4	no	—
239	C₂×C₄	—	202	4	no	—
240	C₂×C₄	—	202	16	no	—
241	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
242	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
243	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	16	yes	complemented
244	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
245	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
246	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	8	no	—
251	C₂²	—	4	8	no	—
252	C₂²	—	404	8	no	—
253	C₂²	—	202	4	no	—
254	C₂²	—	202	8	no	—
255	C₂²	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
256	C₂²	C₂²×C₂²⋊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	—	101	8	no	—
261	C₄	—	202	4	no	—
262	C₄	—	202	4	no	—
263	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
264	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
265	C₂	—	2	4	no	—
266	C₂	—	4	4	no	—
267	C₂	—	404	4	no	—
268	C₂	#20265723	1	4	yes	direct product factor
269	C₂	C₂³×C₂²⋊D₁₀₁	1	1	yes	characteristic
270	C₂	C₂×#41×D₁₀₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂²×#41×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×#41	D₁₀₁
hypocenter	C₁₀₁	C₂×#532
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108074	C₂	semidirect
2. #20265723	C₂	direct (N × H)
3. C₂×#15692253	C₂	semidirect
4. C₂×#15692251	C₂	semidirect
5. C₂×#15692244	C₂	semidirect
6. C₂×#15692242	C₂	semidirect
7. C₂×#15692227	C₂	semidirect
8. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
9. C₂²×C₄×Dic₁₀₁	C₂	semidirect
10. C₂²×#41×C₁₀₁	C₂	semidirect
12. C₂×#13108074	C₂²	semidirect
19. #15692253	C₂²	semidirect
20. #15692251	C₂²	semidirect
21. #15692244	C₂²	semidirect
22. #15692242	C₂²	semidirect
23. #15692227	C₂²	semidirect
24. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
31. C₂×C₄×Dic₁₀₁	C₂²	semidirect
33. C₂×#41×C₁₀₁	C₂²	semidirect
72. C₂²×C₄×C₁₀₁	D₄	semidirect
95. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
104. C₂×Dic₁₀₁	#41	semidirect
128. Dic₁₀₁	C₂×#41	semidirect
131. C₂×C₁₀₁	#532	semidirect
136. C₁₀₁	C₂×#532	semidirect
138. C₂²×#41	D₁₀₁	semidirect
150. C₂×#41	C₂×D₁₀₁	semidirect
202. C₂²×C₄	C₂²⋊D₁₀₁	semidirect
243. C₂×C₄	C₂×C₂²⋊D₁₀₁	semidirect
268. C₂	#20265723	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×#13108074	2
#20265723	2
C₂×#15692253	2
C₂×#15692251	2
C₂×#15692244	2
C₂×#15692242	2
C₂×#15692227	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂²×C₄×Dic₁₀₁	2
C₂²×#41×C₁₀₁	2
C₂×#532	101

Subgroups — C2×#20265723 #27184411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265723 #27184411