Subgroups — #27184521

Order 12928.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265796	C₂	1	4	yes	maximal, complemented
2	#20265788	C₂	1	2	yes	maximal, complemented
3	#20265781	C₂	1	2	yes	maximal, complemented
4	#20265772	C₂	1	4	yes	maximal, complemented
5	#20265764	C₂	1	4	yes	maximal, complemented
6	#20265734	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265707	C₂	1	8	yes	maximal, complemented
8	#20265684	C₂	1	2	yes	maximal, complemented
9	#41×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
10	C₂×#15692212	C₂	1	1	yes	maximal, characteristic, complemented
11	#532×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
13	#15692255	C₂²	1	2	yes	complemented
14	#15692250	C₂²	1	8	yes	—
15	#15692248	C₂²	1	8	yes	complemented
16	#15692247	C₂²	1	2	yes	complemented
17	#15692245	C₂²	1	8	yes	complemented
18	#15692241	C₂²	1	8	yes	—
19	#15692239	C₂²	1	8	yes	—
20	#15692236	C₂²	1	4	yes	—
21	#15692236	C₂²	1	8	yes	complemented
22	#15692234	C₂²	1	4	yes	—
23	#15692232	C₂²	1	8	yes	complemented
24	#15692231	C₂²	1	4	yes	—
25	#15692227	C₂²	1	8	yes	—
26	#15692226	C₂²	1	4	yes	—
27	#15692223	C₂²	1	4	yes	—
28	Q₈×Dic₁₀₁	C₂²	1	4	yes	complemented
29	C₄×Dic₂₀₂	C₂²	1	4	yes	complemented
30	D₄×Dic₁₀₁	C₂²	1	8	yes	—
31	D₄×Dic₁₀₁	C₂²	1	4	yes	complemented
32	C₄×C₂²⋊D₁₀₁	C₂²	1	8	yes	—
33	#15692214	C₂²	1	2	yes	—
34	#15692214	C₂²	1	4	yes	complemented
35	#15692212	C₂²	1	2	yes	—
36	#15692212	C₂²	1	2	yes	—
37	#15692212	C₂²	1	4	yes	—
38	C₂×#13108053	C₂²	1	2	yes	—
39	C₂×#13108051	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
41	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	complemented
42	C₂×#41×C₁₀₁	C₂²	1	2	yes	—
43	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
44	#127×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
45	#124×C₁₀₁	C₂²	1	2	yes	complemented
46	#121×C₁₀₁	C₂²	1	4	yes	—
47	C₄×D₄×C₁₀₁	C₂²	1	4	yes	complemented
48	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
49	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
50	C₂×Dic₂₀₂	C₂³	1	4	yes	—
51	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
54	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
55	C₂×C₄×D₁₀₁	—	2	4	no	—
56	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
57	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
58	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
59	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
60	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
61	#13108053	C₂³	1	4	yes	—
62	#13108053	C₂³	1	4	yes	—
63	#13108053	C₂³	1	8	yes	—
64	#13108053	C₂³	1	8	yes	—
65	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
66	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
67	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
68	#13108051	C₂³	1	4	yes	—
69	#13108051	C₂³	1	4	yes	—
70	#13108051	C₂³	1	8	yes	—
71	C₄×Dic₁₀₁	C₂³	1	2	yes	—
72	C₄×Dic₁₀₁	C₂³	1	2	yes	—
73	C₄×Dic₁₀₁	C₂³	1	4	yes	—
74	C₄×Dic₁₀₁	C₂³	1	8	yes	—
75	#41×C₁₀₁	—	2	8	no	—
76	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
77	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
78	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
79	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
80	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
81	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
82	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
84	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
85	C₄²×C₁₀₁	C₂³	1	2	yes	—
86	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
87	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
88	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
89	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
90	C₂²×D₁₀₁	—	2	2	no	—
91	C₂²×D₁₀₁	—	4	2	no	—
92	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
93	Q₈×C₁₀₁	—	2	4	no	—
94	D₄×C₁₀₁	—	2	8	no	—
95	D₄×C₁₀₁	—	2	4	no	—
96	D₄×C₁₀₁	—	4	4	no	—
97	C₂×C₄×C₁₀₁	—	2	8	no	—
98	C₂×C₄×C₁₀₁	—	4	2	no	—
99	C₂×C₄×C₁₀₁	#41	1	4	yes	—
100	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
101	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₁₀₁	—	4	8	no	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	4	8	no	—
108	C₂×Dic₁₀₁	—	4	2	no	—
109	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
110	C₂×Dic₁₀₁	C₂⁴	1	8	yes	—
111	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
112	C₄×D₁₀₁	—	2	4	no	—
113	C₄×D₁₀₁	—	2	4	no	—
114	Dic₂₀₂	—	4	4	no	—
115	C₂²×C₁₀₁	—	2	2	no	—
116	C₂²×C₁₀₁	—	2	4	no	—
117	C₂²×C₁₀₁	—	2	8	no	—
118	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
119	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
120	C₂×D₁₀₁	—	2	2	no	—
121	C₂×D₁₀₁	—	2	4	no	—
122	C₂×D₁₀₁	—	4	4	no	—
123	C₄×C₁₀₁	—	2	4	no	—
124	C₄×C₁₀₁	—	2	2	no	—
125	C₄×C₁₀₁	—	2	2	no	—
126	C₄×C₁₀₁	C₂×#41	1	2	yes	—
127	C₄×C₁₀₁	C₂×#41	1	2	yes	—
128	Dic₁₀₁	—	2	2	no	—
129	Dic₁₀₁	—	2	8	no	—
130	Dic₁₀₁	—	2	4	no	—
131	C₂×C₁₀₁	—	2	2	no	—
132	C₂×C₁₀₁	—	4	4	no	—
133	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
134	C₂×C₁₀₁	#585	1	2	yes	—
135	D₁₀₁	—	4	2	no	—
136	#3455	—	101	1	no	maximal, Sylow
137	C₁₀₁	#3455	1	1	yes	characteristic, complemented, Sylow
138	#542	—	101	4	no	—
139	#541	—	101	2	no	—
140	#540	—	101	2	no	—
141	#539	—	101	4	no	—
142	#538	—	101	4	no	—
143	#533	—	101	1	no	—
144	#532	D₁₀₁	1	1	yes	characteristic, complemented
145	#529	—	101	8	no	—
146	#525	—	101	2	no	—
147	C₄×#41	—	101	2	no	—
148	C₂×#117	—	101	1	no	—
149	C₂×#41	C₂×D₁₀₁	1	2	yes	—
150	C₂³×C₄	—	101	1	no	—
151	C₄⋊Q₈	—	101	2	no	—
152	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
153	#127	C₂×D₁₀₁	1	1	yes	characteristic, complemented
154	#126	—	101	8	no	—
155	#126	—	101	8	no	—
156	#125	—	101	8	no	—
157	#125	—	101	2	no	—
158	#124	—	101	8	no	—
159	#124	C₂×D₁₀₁	1	2	yes	complemented
160	#123	—	101	4	no	—
161	#123	—	101	8	no	—
162	#123	—	101	8	no	—
163	C₂²⋊Q₈	—	101	4	no	—
164	C₂²⋊Q₈	—	101	4	no	—
165	C₂²⋊Q₈	—	101	8	no	—
166	#121	—	101	4	no	—
167	#121	—	101	8	no	—
168	#121	C₂×D₁₀₁	1	4	yes	—
169	#120	—	101	4	no	—
170	C₄×Q₈	—	101	4	no	—
171	C₄×Q₈	—	101	4	no	—
172	C₄×D₄	—	101	8	no	—
173	C₄×D₄	—	101	4	no	—
174	C₄×D₄	—	101	8	no	—
175	C₄×D₄	C₂×D₁₀₁	1	4	yes	complemented
176	#117	—	101	2	no	—
177	#117	—	101	4	no	—
178	#117	—	101	4	no	—
179	#117	—	101	2	no	—
180	#117	—	101	2	no	—
181	C₂×C₄⋊C₄	—	101	2	no	—
182	C₂×C₂²⋊C₄	—	101	2	no	—
183	C₂×C₄²	—	101	1	no	—
184	C₂×C₄²	—	101	4	no	—
185	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
186	C₂⁴	—	101	1	no	—
187	#41	—	2	8	no	—
188	C₂×Q₈	—	101	4	no	—
189	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
190	C₂×D₄	—	101	8	no	—
191	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
192	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
193	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	8	no	—
196	C₂²×C₄	—	101	1	no	—
197	C₂²×C₄	—	101	2	no	—
198	C₂²×C₄	—	101	2	no	—
199	C₂²×C₄	—	202	4	no	—
200	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
201	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
203	C₄⋊C₄	—	101	4	no	—
204	C₄⋊C₄	—	101	4	no	—
205	C₄⋊C₄	—	101	8	no	—
206	C₄⋊C₄	—	101	8	no	—
207	C₄⋊C₄	—	101	4	no	—
208	C₄⋊C₄	—	101	8	no	—
209	C₄⋊C₄	—	101	4	no	—
210	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
211	C₂²⋊C₄	—	101	4	no	—
212	C₂²⋊C₄	—	101	8	no	—
213	C₂²⋊C₄	—	101	8	no	—
214	C₂²⋊C₄	—	101	4	no	—
215	C₂²⋊C₄	—	101	4	no	—
216	C₂²⋊C₄	—	101	8	no	—
217	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
218	C₄²	—	101	2	no	—
219	C₄²	—	101	8	no	—
220	C₄²	—	101	2	no	—
221	C₄²	—	101	4	no	—
222	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
223	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
224	C₄²	C₂²×D₁₀₁	1	2	yes	—
225	C₂³	—	101	2	no	—
226	C₂³	—	404	2	no	—
227	C₂³	—	202	2	no	—
228	C₂³	C₂³×D₁₀₁	1	4	yes	—
229	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	404	4	no	—
232	D₄	—	2	8	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	404	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	101	4	no	—
239	C₂×C₄	—	101	8	no	—
240	C₂×C₄	—	101	2	no	—
241	C₂×C₄	—	404	2	no	—
242	C₂×C₄	—	404	8	no	—
243	C₂×C₄	—	202	2	no	—
244	C₂×C₄	—	202	4	no	—
245	C₂×C₄	—	202	4	no	—
246	C₂×C₄	#13108073	1	4	yes	—
247	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
248	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
249	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
250	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	404	4	no	—
256	C₂²	—	202	2	no	—
257	C₂²	—	202	4	no	—
258	C₂²	C₂×#13108073	1	2	yes	—
259	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	202	2	no	—
264	C₄	—	202	4	no	—
265	C₄	—	202	8	no	—
266	C₄	C₂×#13108073	1	2	yes	—
267	C₄	C₂×#13108073	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	—	404	2	no	—
271	C₂	C₂²×#13108073	1	1	yes	characteristic
272	C₂	#20266013	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	#532×C₁₀₁	C₂
hypercenter	#532	D₁₀₁
hypocenter	C₁₀₁	#3455
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #20265796	C₂	semidirect
2. #20265788	C₂	semidirect
3. #20265781	C₂	semidirect
4. #20265772	C₂	semidirect
5. #20265764	C₂	semidirect
6. #20265734	C₂	semidirect
7. #20265707	C₂	semidirect
8. #20265684	C₂	semidirect
9. #41×Dic₁₀₁	C₂	semidirect
10. C₂×#15692212	C₂	semidirect
11. #532×C₁₀₁	C₂	semidirect
13. #15692255	C₂²	semidirect
15. #15692248	C₂²	semidirect
16. #15692247	C₂²	semidirect
17. #15692245	C₂²	semidirect
21. #15692236	C₂²	semidirect
23. #15692232	C₂²	semidirect
28. Q₈×Dic₁₀₁	C₂²	semidirect
29. C₄×Dic₂₀₂	C₂²	semidirect
31. D₄×Dic₁₀₁	C₂²	semidirect
34. #15692214	C₂²	semidirect
41. C₂×C₄×Dic₁₀₁	C₂²	semidirect
43. C₄⋊Q₈×C₁₀₁	C₂²	semidirect
44. #127×C₁₀₁	C₂²	semidirect
45. #124×C₁₀₁	C₂²	semidirect
47. C₄×D₄×C₁₀₁	C₂²	semidirect
137. C₁₀₁	#3455	semidirect
144. #532	D₁₀₁	semidirect
152. C₄⋊Q₈	C₂×D₁₀₁	semidirect
153. #127	C₂×D₁₀₁	semidirect
159. #124	C₂×D₁₀₁	semidirect
175. C₄×D₄	C₂×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265796	2
#20265788	2
#20265781	2
#20265772	2
#20265764	2
#20265734	2
#20265707	2
#20265684	2
#41×Dic₁₀₁	2
C₂×#15692212	2
#532×C₁₀₁	2
#3455	101