Subgroups — #27187853

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13108084	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266211	C₂	1	2	yes	maximal, complemented
4	#20266210	C₂	1	2	yes	maximal, complemented
5	#20266207	C₂	1	2	yes	maximal, complemented
6	#20266206	C₂	1	2	yes	maximal, complemented
7	#20266183	C₂	1	1	yes	maximal, characteristic, complemented
8	#20266150	C₂	1	1	yes	maximal, characteristic, complemented
9	#20266149	C₂	1	1	yes	maximal, characteristic, complemented
10	#20266141	C₂	1	1	yes	maximal, characteristic, complemented
11	#20265685	C₂	1	1	yes	maximal, characteristic
12	#15692342	—	2	4	no	—
13	C₂×#13108084	—	2	4	no	—
14	C₂×#13108084	C₂²	1	2	yes	—
15	C₂×#13108084	C₂²	1	2	yes	—
16	C₂×#13108084	C₂²	1	1	yes	characteristic, complemented
17	C₂×#13108083	C₂²	1	1	yes	characteristic
18	C₂×#13108083	C₂²	1	1	yes	characteristic
19	C₂×#13108083	C₂²	1	1	yes	characteristic, complemented
20	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
21	C₂²×#11717581	C₂²	1	1	yes	characteristic
22	C₂²×#11717581	C₂²	1	1	yes	characteristic
23	C₂²×#11717581	C₂²	1	1	yes	characteristic, complemented
24	#15692322	C₂²	1	2	yes	complemented
25	C₄⋊#11717581	C₂²	1	2	yes	complemented
26	#15692312	C₂²	1	2	yes	—
27	#15692312	C₂²	1	2	yes	—
28	#15692312	C₂²	1	2	yes	complemented
29	C₂²⋊#11717581	C₂²	1	2	yes	complemented
30	#15692310	C₂²	1	2	yes	complemented
31	C₄×#11717581	C₂²	1	2	yes	complemented
32	C₂×#13108073	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
34	#15692238	C₄	1	1	yes	characteristic
35	#15692233	C₄	1	2	yes	—
36	#15692228	C₄	1	2	yes	—
37	#15692225	C₄	1	1	yes	characteristic
38	#15692223	C₄	1	1	yes	characteristic
39	#15692217	C₂²	1	2	yes	—
40	#15692215	C₂²	1	2	yes	—
41	#15692211	C₂²	1	1	yes	characteristic
42	#123×C₁₀₁	C₄	1	1	yes	characteristic
43	#13108084	—	2	4	no	—
44	#13108084	—	2	4	no	—
45	#13108084	—	2	2	no	—
46	#13108083	—	2	2	no	—
47	#13108083	—	2	2	no	—
48	#13108083	—	2	2	no	—
49	D₁₀₁⋊C₈	—	2	2	no	—
50	C₂×#11717581	—	2	2	no	—
51	C₂×#11717581	—	2	2	no	—
52	C₂×#11717581	D₄	1	4	yes	complemented
53	C₂×#11717581	C₂³	1	2	yes	—
54	C₂×#11717581	C₂³	1	2	yes	—
55	C₂×#11717581	C₂³	1	1	yes	characteristic
56	C₂×#11717581	C₂³	1	1	yes	characteristic
57	C₂×#11717581	C₂³	1	1	yes	characteristic
58	C₂×#11717581	C₂³	1	1	yes	characteristic
59	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
60	#13108073	—	2	2	no	—
61	#13108073	—	2	2	no	—
62	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
63	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
67	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₀₁	—	2	2	no	—
70	C₂×C₄×D₁₀₁	—	2	2	no	—
71	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
72	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
73	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
74	#13108053	C₂×C₄	1	2	yes	—
75	#13108053	C₂×C₄	1	2	yes	—
76	C₂²⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
77	#13108051	C₂×C₄	1	2	yes	—
78	#13108051	C₂×C₄	1	2	yes	—
79	#13108051	C₂×C₄	1	2	yes	—
80	C₄×Dic₁₀₁	C₂³	1	2	yes	—
81	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
84	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
86	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
87	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₀₁	—	2	2	no	—
89	C₂²×D₁₀₁	—	4	2	no	—
90	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
92	D₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊D₁₀₁	—	2	2	no	—
98	C₂²⋊D₁₀₁	—	2	2	no	—
99	C₂²⋊D₁₀₁	—	4	2	no	—
100	C₂×Dic₁₀₁	—	2	1	no	—
101	C₂×Dic₁₀₁	—	2	1	no	—
102	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
103	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
106	C₂×Dic₁₀₁	#41	1	2	yes	—
107	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
108	D₄₀₄	—	4	2	no	—
109	C₄×D₁₀₁	—	2	4	no	—
110	C₄×D₁₀₁	—	2	2	no	—
111	C₄×D₁₀₁	—	2	1	no	—
112	C₄×D₁₀₁	—	2	2	no	—
113	C₄×D₁₀₁	—	2	1	no	—
114	Dic₂₀₂	—	2	2	no	—
115	#11717581	—	2	1	no	—
116	#11717581	—	2	2	no	—
117	#11717581	—	2	1	no	—
118	#11717581	—	2	2	no	—
119	#11717581	C₂×D₄	1	2	yes	—
120	#11717581	C₂×D₄	1	2	yes	—
121	C₂²×C₁₀₁	—	2	2	no	—
122	C₂²×C₁₀₁	—	2	1	no	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
125	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×D₁₀₁	—	2	2	no	—
127	C₂×D₁₀₁	—	2	1	no	—
128	C₂×D₁₀₁	—	2	2	no	—
129	C₂×D₁₀₁	—	2	2	no	—
130	C₂×D₁₀₁	—	2	2	no	—
131	C₂×D₁₀₁	—	4	4	no	—
132	C₄×C₁₀₁	—	2	1	no	—
133	C₄×C₁₀₁	—	2	2	no	—
134	C₄×C₁₀₁	—	2	2	no	—
135	Dic₁₀₁	—	2	1	no	—
136	Dic₁₀₁	—	2	2	no	—
137	Dic₁₀₁	C₄×D₄	1	2	yes	—
138	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
139	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₁₀₁	—	2	2	no	—
141	C₂×C₁₀₁	—	4	1	no	—
142	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×C₁₀₁	#568	1	2	yes	—
144	D₁₀₁	—	4	1	no	—
145	D₁₀₁	—	4	2	no	—
146	#2930	—	101	1	no	maximal, Sylow
147	C₁₀₁	#2930	1	1	yes	characteristic, complemented, Sylow
148	C₂×#131	—	101	1	no	—
149	C₂²×M₁₆	—	101	1	no	—
150	#525	—	101	1	no	—
151	#436	—	101	2	no	—
152	#436	—	101	2	no	—
153	#435	—	101	2	no	—
154	#435	—	101	2	no	—
155	#424	—	101	1	no	—
156	#408	—	101	1	no	—
157	#407	—	101	1	no	—
158	#405	—	101	1	no	—
159	C₂×#41	—	101	1	no	—
160	C₂³×C₄	—	101	1	no	—
161	#131	—	202	4	no	—
162	C₂×M₁₆	—	101	1	no	—
163	C₂×M₁₆	—	101	1	no	—
164	C₂×M₁₆	—	101	1	no	—
165	C₂×M₁₆	—	101	2	no	—
166	C₂×M₁₆	—	101	2	no	—
167	C₂×M₁₆	—	101	1	no	—
168	C₂×M₁₆	—	202	4	no	—
169	C₂²×C₈	—	101	1	no	—
170	C₂²×C₈	—	101	1	no	—
171	C₂²×C₈	—	101	1	no	—
172	C₂²×C₈	—	101	1	no	—
173	#123	—	101	1	no	—
174	#123	#87938	1	1	yes	characteristic
175	C₂²⋊Q₈	—	101	2	no	—
176	#121	—	101	2	no	—
177	#120	—	101	1	no	—
178	#120	—	101	1	no	—
179	C₄×D₄	—	101	2	no	—
180	C₄×D₄	—	101	2	no	—
181	#117	—	101	1	no	—
182	C₄⋊C₈	—	101	2	no	—
183	C₄⋊C₈	—	101	2	no	—
184	C₂²⋊C₈	—	101	2	no	—
185	C₂²⋊C₈	—	101	2	no	—
186	C₂²⋊C₈	—	101	2	no	—
187	C₂²⋊C₈	—	101	2	no	—
188	#97	—	101	2	no	—
189	C₄×C₈	—	101	2	no	—
190	C₂⁴	—	101	1	no	—
191	#41	—	202	2	no	—
192	#41	—	202	2	no	—
193	C₂×Q₈	—	101	1	no	—
194	C₂×D₄	—	101	2	no	—
195	C₂×D₄	—	101	1	no	—
196	C₂×D₄	—	101	1	no	—
197	C₂×D₄	—	101	2	no	—
198	C₂×D₄	C₂×#87938	1	1	yes	characteristic
199	C₂²×C₄	—	101	1	no	—
200	C₂²×C₄	—	101	1	no	—
201	C₂²×C₄	—	101	1	no	—
202	C₂²×C₄	—	101	2	no	—
203	C₂²×C₄	—	101	2	no	—
204	C₂²×C₄	—	202	2	no	—
205	C₂²×C₄	—	202	2	no	—
206	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
207	M₁₆	—	202	2	no	—
208	M₁₆	—	202	4	no	—
209	M₁₆	—	202	4	no	—
210	M₁₆	—	202	2	no	—
211	M₁₆	—	202	2	no	—
212	M₁₆	—	202	2	no	—
213	C₂×C₈	—	101	1	no	—
214	C₂×C₈	—	101	1	no	—
215	C₂×C₈	—	101	4	no	—
216	C₂×C₈	—	101	2	no	—
217	C₂×C₈	—	101	2	no	—
218	C₂×C₈	—	101	1	no	—
219	C₂×C₈	—	101	1	no	—
220	C₂×C₈	—	202	2	no	—
221	C₂×C₈	—	202	2	no	—
222	C₂×C₈	—	202	2	no	—
223	C₄⋊C₄	—	101	2	no	—
224	C₄⋊C₄	—	101	2	no	—
225	C₄⋊C₄	C₂×#87938	1	2	yes	—
226	C₂²⋊C₄	—	101	1	no	—
227	C₂²⋊C₄	—	101	2	no	—
228	C₂²⋊C₄	—	101	2	no	—
229	C₂²⋊C₄	—	101	2	no	—
230	C₂²⋊C₄	C₂×#87938	1	1	yes	characteristic
231	C₂²⋊C₄	C₂×#87938	1	2	yes	—
232	C₄²	—	101	2	no	—
233	C₂³	—	101	2	no	—
234	C₂³	—	101	1	no	—
235	C₂³	—	404	2	no	—
236	C₂³	—	202	2	no	—
237	C₂³	C₂²×#87938	1	1	yes	characteristic
238	C₂³	C₂²×#87938	1	1	yes	characteristic
239	Q₈	—	202	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	404	2	no	—
242	D₄	—	404	2	no	—
243	D₄	—	202	2	no	—
244	D₄	—	202	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	101	2	no	—
247	C₂×C₄	—	101	2	no	—
248	C₂×C₄	—	101	1	no	—
249	C₂×C₄	—	101	1	no	—
250	C₂×C₄	—	101	2	no	—
251	C₂×C₄	—	101	1	no	—
252	C₂×C₄	—	202	1	no	—
253	C₂×C₄	—	202	2	no	—
254	C₂×C₄	—	202	2	no	—
255	C₂×C₄	—	202	4	no	—
256	C₂×C₄	—	202	1	no	—
257	C₂×C₄	—	202	1	no	—
258	C₂×C₄	—	202	1	no	—
259	C₂×C₄	C₂²×#87938	1	2	yes	—
260	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
261	C₂×C₄	C₂²×#87938	1	2	yes	—
262	C₈	—	101	2	no	—
263	C₈	—	101	2	no	—
264	C₈	—	202	1	no	—
265	C₈	—	202	1	no	—
266	C₈	—	202	2	no	—
267	C₈	—	202	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	404	4	no	—
272	C₂²	—	202	2	no	—
273	C₂²	—	202	1	no	—
274	C₂²	—	202	2	no	—
275	C₂²	—	202	2	no	—
276	C₂²	—	202	2	no	—
277	C₂²	D₄×#87938	1	2	yes	—
278	C₂²	C₂³×#87938	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	101	1	no	—
283	C₄	—	101	1	no	—
284	C₄	—	101	2	no	—
285	C₄	—	202	1	no	—
286	C₄	—	202	2	no	—
287	C₂	—	2	2	no	—
288	C₂	—	4	1	no	—
289	C₂	—	404	2	no	—
290	C₂	—	404	1	no	—
291	C₂	C₂×D₄×#87938	1	1	yes	characteristic
292	C₂	#20266256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#123×C₁₀₁	C₄
prime radical	#20265685	C₂
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	#123	#87938
hypocenter	C₁₀₁	#2930
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. C₂×#15692342	C₂	semidirect
2. C₂²×#13108084	C₂	semidirect
3. #20266211	C₂	semidirect
4. #20266210	C₂	semidirect
5. #20266207	C₂	semidirect
6. #20266206	C₂	semidirect
7. #20266183	C₂	semidirect
8. #20266150	C₂	semidirect
9. #20266149	C₂	semidirect
10. #20266141	C₂	semidirect
16. C₂×#13108084	C₂²	semidirect
19. C₂×#13108083	C₂²	semidirect
20. C₂×D₁₀₁⋊C₈	C₂²	semidirect
23. C₂²×#11717581	C₂²	semidirect
24. #15692322	C₂²	semidirect
25. C₄⋊#11717581	C₂²	semidirect
28. #15692312	C₂²	semidirect
29. C₂²⋊#11717581	C₂²	semidirect
30. #15692310	C₂²	semidirect
31. C₄×#11717581	C₂²	semidirect
52. C₂×#11717581	D₄	semidirect
147. C₁₀₁	#2930	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692342	2
C₂²×#13108084	2
#20266211	2
#20266210	2
#20266207	2
#20266206	2
#20266183	2
#20266150	2
#20266149	2
#20266141	2
#20265685	2
#2930	101