Subgroups — #27184733

Order 12928.

1242 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265913	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265884	C₂	1	2	yes	maximal, complemented
3	#20265879	C₂	1	4	yes	maximal, complemented
4	#20265858	C₂	1	4	yes	maximal, complemented
5	#20265821	C₂	1	4	yes	maximal, complemented
6	#20265725	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265708	C₂	1	4	yes	maximal, complemented
8	#20265683	C₂	1	4	yes	maximal, complemented
9	C₄×#13108073	C₂	1	2	yes	maximal, complemented
10	C₂×D₄×Dic₁₀₁	C₂	1	4	yes	maximal, complemented
11	D₄²×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108073	C₂²	1	2	yes	—
13	C₂×#13108073	C₂²	1	2	yes	complemented
14	C₂³×Dic₁₀₁	C₂²	1	4	yes	complemented
15	#15692255	C₂²	1	1	yes	characteristic, complemented
16	#15692248	C₂²	1	8	yes	complemented
17	#15692246	C₂²	1	4	yes	complemented
18	#15692242	C₂²	1	4	yes	complemented
19	#15692242	C₂²	1	2	yes	complemented
20	#15692239	C₂²	1	4	yes	complemented
21	#15692236	C₂²	1	8	yes	complemented
22	#15692236	C₂²	1	4	yes	complemented
23	#15692234	C₂²	1	4	yes	complemented
24	#15692232	C₂²	1	8	yes	complemented
25	#15692227	C₂²	1	8	yes	complemented
26	#15692226	C₂²	1	4	yes	complemented
27	#15692224	C₂²	1	4	yes	complemented
28	C₄×Dic₂₀₂	C₂²	1	2	yes	complemented
29	#15692220	C₂²	1	2	yes	complemented
30	D₄×Dic₁₀₁	C₂²	1	4	yes	—
31	D₄×Dic₁₀₁	C₂²	1	2	yes	—
32	D₄×Dic₁₀₁	C₂²	1	8	yes	complemented
33	D₄×Dic₁₀₁	C₂²	1	4	yes	complemented
34	D₄×Dic₁₀₁	C₂²	1	8	yes	complemented
35	D₄×Dic₁₀₁	C₂²	1	4	yes	complemented
36	C₄×C₂²⋊D₁₀₁	C₂²	1	4	yes	complemented
37	#15692215	C₂²	1	4	yes	complemented
38	#15692213	C₂²	1	2	yes	complemented
39	#15692211	C₂²	1	4	yes	complemented
40	#15692211	C₂²	1	4	yes	complemented
41	C₂×C₄⋊Dic₁₀₁	C₂²	1	4	yes	complemented
42	C₂×C₂²⋊Dic₁₀₁	C₂²	1	8	yes	complemented
43	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	complemented
44	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic, complemented
45	C₂²×D₄×C₁₀₁	C₂²	1	4	yes	—
46	#127×C₁₀₁	C₂²	1	1	yes	characteristic
47	#121×C₁₀₁	C₂²	1	4	yes	—
48	#120×C₁₀₁	C₂²	1	4	yes	—
49	C₄×D₄×C₁₀₁	C₂²	1	2	yes	—
50	#13108073	—	2	8	no	—
51	#13108073	—	4	4	no	—
52	C₂×Dic₂₀₂	C₂³	1	2	yes	—
53	C₂×Dic₂₀₂	C₂³	1	2	yes	—
54	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
56	C₂²×Dic₁₀₁	—	4	8	no	—
57	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
58	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
59	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
60	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
61	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
62	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
63	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
65	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
66	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
67	#13108053	C₂³	1	4	yes	—
68	#13108053	C₂³	1	4	yes	—
69	#13108053	C₂³	1	8	yes	—
70	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
71	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
72	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
73	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
74	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
75	#13108051	C₂³	1	4	yes	—
76	C₄×Dic₁₀₁	C₂³	1	2	yes	—
77	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
78	C₄×Dic₁₀₁	C₂³	1	4	yes	—
79	C₄×Dic₁₀₁	C₂³	1	4	yes	—
80	C₄×Dic₁₀₁	C₂³	1	4	yes	—
81	C₂⁴×C₁₀₁	C₂³	1	4	yes	—
82	C₂×D₄×C₁₀₁	—	2	16	no	—
83	C₂×D₄×C₁₀₁	C₂³	1	8	yes	—
84	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
85	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
86	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
87	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
88	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
89	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
90	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
91	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
92	C₂³×C₁₀₁	—	2	16	no	—
93	C₂³×C₁₀₁	—	4	4	no	—
94	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
95	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
96	C₂²×D₁₀₁	C₂⁴	1	1	yes	characteristic
97	D₄×C₁₀₁	—	2	16	no	—
98	D₄×C₁₀₁	—	2	8	no	—
99	D₄×C₁₀₁	—	4	2	no	—
100	D₄×C₁₀₁	#41	1	8	yes	complemented
101	C₂×C₄×C₁₀₁	—	2	8	no	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
103	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
105	C₂²⋊D₁₀₁	—	2	8	no	—
106	C₂²⋊D₁₀₁	—	4	4	no	—
107	C₂×Dic₁₀₁	—	2	8	no	—
108	C₂×Dic₁₀₁	—	2	8	no	—
109	C₂×Dic₁₀₁	—	2	8	no	—
110	C₂×Dic₁₀₁	—	4	8	no	—
111	C₂×Dic₁₀₁	—	4	4	no	—
112	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
113	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
114	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
115	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
116	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
117	C₄×D₁₀₁	—	2	4	no	—
118	C₄×D₁₀₁	—	2	2	no	—
119	Dic₂₀₂	—	2	4	no	—
120	Dic₂₀₂	—	4	2	no	—
121	C₂²×C₁₀₁	—	2	4	no	—
122	C₂²×C₁₀₁	—	2	4	no	—
123	C₂²×C₁₀₁	—	2	8	no	—
124	C₂²×C₁₀₁	—	2	4	no	—
125	C₂²×C₁₀₁	—	4	16	no	—
126	C₂²×C₁₀₁	C₂×#41	1	8	yes	—
127	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
128	C₂×D₁₀₁	—	2	2	no	—
129	C₂×D₁₀₁	—	2	1	no	—
130	C₄×C₁₀₁	—	2	4	no	—
131	C₄×C₁₀₁	—	2	1	no	—
132	C₄×C₁₀₁	C₂×#41	1	4	yes	—
133	Dic₁₀₁	—	2	4	no	—
134	Dic₁₀₁	—	2	2	no	—
135	Dic₁₀₁	—	2	4	no	—
136	Dic₁₀₁	—	2	4	no	—
137	Dic₁₀₁	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₁₀₁	—	2	8	no	—
139	C₂×C₁₀₁	—	4	4	no	—
140	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
141	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
142	D₁₀₁	—	4	1	no	—
143	#3481	—	101	1	no	maximal, Sylow
144	C₁₀₁	#3481	1	1	yes	characteristic, complemented, Sylow
145	#559	—	101	1	no	—
146	#554	—	101	2	no	—
147	#553	—	101	4	no	—
148	#551	—	101	4	no	—
149	#546	—	101	4	no	—
150	D₄²	D₁₀₁	1	1	yes	characteristic, complemented
151	#532	—	101	1	no	—
152	#529	—	101	4	no	—
153	#525	—	101	4	no	—
154	C₄×#41	—	101	2	no	—
155	C₂×C₄×D₄	—	101	4	no	—
156	C₂×#41	—	101	2	no	—
157	C₂×#41	—	101	2	no	—
158	C₂²×D₄	C₂×D₁₀₁	1	4	yes	—
159	C₂³×C₄	—	101	4	no	—
160	C₄⋊Q₈	—	101	1	no	—
161	#127	C₂×D₁₀₁	1	1	yes	characteristic
162	#126	—	101	8	no	—
163	#125	—	101	4	no	—
164	#124	—	101	4	no	—
165	#124	—	101	2	no	—
166	#123	—	101	4	no	—
167	#123	—	101	8	no	—
168	#123	—	101	4	no	—
169	C₂²⋊Q₈	—	101	4	no	—
170	C₂²⋊Q₈	—	101	8	no	—
171	#121	—	101	8	no	—
172	#121	—	101	4	no	—
173	#121	C₂×D₁₀₁	1	4	yes	—
174	#120	—	101	4	no	—
175	#120	C₂×D₁₀₁	1	4	yes	—
176	C₄×Q₈	—	101	2	no	—
177	C₄×Q₈	—	101	2	no	—
178	C₄×D₄	—	101	2	no	—
179	C₄×D₄	—	101	8	no	—
180	C₄×D₄	—	101	4	no	—
181	C₄×D₄	—	101	4	no	—
182	C₄×D₄	—	101	8	no	—
183	C₄×D₄	—	101	4	no	—
184	C₄×D₄	—	101	4	no	—
185	C₄×D₄	—	101	4	no	—
186	C₄×D₄	C₂×D₁₀₁	1	2	yes	—
187	#117	—	101	2	no	—
188	#117	—	101	4	no	—
189	#117	—	101	4	no	—
190	C₂×C₄⋊C₄	—	101	4	no	—
191	C₂×C₂²⋊C₄	—	101	8	no	—
192	C₂×C₄²	—	101	4	no	—
193	C₂×C₄²	—	101	1	no	—
194	C₂⁴	C₂²×D₁₀₁	1	4	yes	—
195	#41	—	404	4	no	—
196	#41	—	202	8	no	—
197	C₂×Q₈	—	101	2	no	—
198	C₂×Q₈	—	101	2	no	—
199	C₂×D₄	—	2	16	no	—
200	C₂×D₄	—	101	4	no	—
201	C₂×D₄	—	101	4	no	—
202	C₂×D₄	C₂²×D₁₀₁	1	8	yes	—
203	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
204	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
205	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
206	C₂²×C₄	—	101	8	no	—
207	C₂²×C₄	—	101	4	no	—
208	C₂²×C₄	—	101	4	no	—
209	C₂²×C₄	—	101	4	no	—
210	C₂²×C₄	—	101	4	no	—
211	C₂²×C₄	—	101	2	no	—
212	C₂²×C₄	—	101	1	no	—
213	C₂²×C₄	—	404	8	no	—
214	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
215	C₄⋊C₄	—	101	4	no	—
216	C₄⋊C₄	—	101	4	no	—
217	C₄⋊C₄	—	101	8	no	—
218	C₄⋊C₄	—	101	2	no	—
219	C₄⋊C₄	—	101	4	no	—
220	C₄⋊C₄	—	101	8	no	—
221	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
222	C₂²⋊C₄	—	101	4	no	—
223	C₂²⋊C₄	—	101	8	no	—
224	C₂²⋊C₄	—	101	8	no	—
225	C₂²⋊C₄	—	101	8	no	—
226	C₂²⋊C₄	—	101	4	no	—
227	C₂²⋊C₄	—	101	4	no	—
228	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
229	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
230	C₄²	—	101	2	no	—
231	C₄²	—	101	1	no	—
232	C₄²	—	101	4	no	—
233	C₄²	—	101	4	no	—
234	C₄²	—	101	4	no	—
235	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
236	C₂³	—	2	16	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	101	1	no	—
239	C₂³	C₂³×D₁₀₁	1	4	yes	—
240	C₂³	C₂³×D₁₀₁	1	4	yes	—
241	Q₈	—	404	2	no	—
242	Q₈	—	202	4	no	—
243	D₄	—	2	16	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	2	no	—
246	D₄	—	404	4	no	—
247	D₄	—	202	8	no	—
248	D₄	#13108073	1	8	yes	complemented
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	101	1	no	—
251	C₂×C₄	—	101	2	no	—
252	C₂×C₄	—	101	4	no	—
253	C₂×C₄	—	101	4	no	—
254	C₂×C₄	—	101	4	no	—
255	C₂×C₄	—	404	4	no	—
256	C₂×C₄	—	404	8	no	—
257	C₂×C₄	—	202	2	no	—
258	C₂×C₄	—	202	8	no	—
259	C₂×C₄	—	202	8	no	—
260	C₂×C₄	—	202	8	no	—
261	C₂×C₄	—	202	4	no	—
262	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
263	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
264	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	4	16	no	—
270	C₂²	—	202	2	no	—
271	C₂²	—	202	1	no	—
272	C₂²	C₂×#13108073	1	8	yes	—
273	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	1	no	—
276	C₄	—	101	2	no	—
277	C₄	—	202	2	no	—
278	C₄	—	202	4	no	—
279	C₄	—	202	4	no	—
280	C₄	—	202	4	no	—
281	C₄	C₂×#13108073	1	4	yes	—
282	C₂	—	2	8	no	—
283	C₂	—	4	4	no	—
284	C₂	—	404	1	no	—
285	C₂	C₂²×#13108073	1	2	yes	—
286	C₂	H(5,𝔽₂)×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Fitting	D₄²×C₁₀₁	C₂
hypercenter	D₄²	D₁₀₁
hypocenter	C₁₀₁	#3481
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ D₄²

Complements

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

N	H ≅ G/N	type
1. #20265913	C₂	semidirect
2. #20265884	C₂	semidirect
3. #20265879	C₂	semidirect
4. #20265858	C₂	semidirect
5. #20265821	C₂	semidirect
6. #20265725	C₂	semidirect
7. #20265708	C₂	semidirect
8. #20265683	C₂	semidirect
9. C₄×#13108073	C₂	semidirect
10. C₂×D₄×Dic₁₀₁	C₂	semidirect
11. D₄²×C₁₀₁	C₂	semidirect
13. C₂×#13108073	C₂²	semidirect
14. C₂³×Dic₁₀₁	C₂²	semidirect
15. #15692255	C₂²	semidirect
16. #15692248	C₂²	semidirect
17. #15692246	C₂²	semidirect
18. #15692242	C₂²	semidirect
19. #15692242	C₂²	semidirect
20. #15692239	C₂²	semidirect
21. #15692236	C₂²	semidirect
22. #15692236	C₂²	semidirect
23. #15692234	C₂²	semidirect
24. #15692232	C₂²	semidirect
25. #15692227	C₂²	semidirect
26. #15692226	C₂²	semidirect
27. #15692224	C₂²	semidirect
28. C₄×Dic₂₀₂	C₂²	semidirect
29. #15692220	C₂²	semidirect
32. D₄×Dic₁₀₁	C₂²	semidirect
33. D₄×Dic₁₀₁	C₂²	semidirect
34. D₄×Dic₁₀₁	C₂²	semidirect
35. D₄×Dic₁₀₁	C₂²	semidirect
36. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
37. #15692215	C₂²	semidirect
38. #15692213	C₂²	semidirect
39. #15692211	C₂²	semidirect
40. #15692211	C₂²	semidirect
41. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
42. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
43. C₂×C₄×Dic₁₀₁	C₂²	semidirect
44. C₄²×D₁₀₁	C₂²	semidirect
100. D₄×C₁₀₁	#41	semidirect
144. C₁₀₁	#3481	semidirect
150. D₄²	D₁₀₁	semidirect
248. D₄	#13108073	semidirect

Maximal subgroups

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

subgroup	index
#20265913	2
#20265884	2
#20265879	2
#20265858	2
#20265821	2
#20265725	2
#20265708	2
#20265683	2
C₄×#13108073	2
C₂×D₄×Dic₁₀₁	2
D₄²×C₁₀₁	2
#3481	101