Subgroups — #27184772

Order 12928.

1560 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108073	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265826	C₂	1	8	yes	maximal, complemented
3	#20265821	C₂	1	8	yes	maximal, complemented
4	#20265748	C₂	1	2	yes	maximal, complemented
5	#20265736	C₂	1	4	yes	maximal, complemented
6	#20265684	C₂	1	2	yes	maximal, complemented
7	C₂×#15692236	C₂	1	4	yes	maximal, complemented
8	#20265652	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13108073	—	2	8	no	—
11	C₂×#13108073	C₂²	1	8	yes	complemented
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	complemented
14	C₂³×Dic₁₀₁	C₂²	1	2	yes	complemented
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic, complemented
16	#15692253	C₂²	1	4	yes	complemented
17	#15692242	C₂²	1	4	yes	complemented
18	#15692238	C₂²	1	8	yes	complemented
19	#15692236	C₂²	1	8	yes	complemented
20	#15692236	C₂²	1	16	yes	complemented
21	#15692236	C₂²	1	4	yes	complemented
22	#15692234	C₂²	1	8	yes	complemented
23	#15692231	C₂²	1	4	yes	complemented
24	#15692227	C₂²	1	16	yes	complemented
25	#15692226	C₂²	1	4	yes	complemented
26	#15692224	C₂²	1	8	yes	complemented
27	#15692223	C₂²	1	4	yes	complemented
28	D₄×Dic₁₀₁	C₂²	1	8	yes	complemented
29	C₄×C₂²⋊D₁₀₁	C₂²	1	8	yes	complemented
30	#15692214	C₂²	1	2	yes	complemented
31	C₂×#13108053	C₂²	1	8	yes	complemented
32	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	complemented
33	C₂×C₂²⋊Dic₁₀₁	C₂²	1	8	yes	complemented
34	H(5,𝔽₂)×C₁₀₁	—	2	8	no	—
35	C₂×#41×C₁₀₁	C₂²	1	2	yes	—
36	C₂×#41×C₁₀₁	C₂²	1	4	yes	complemented
37	C₂²×D₄×C₁₀₁	C₂²	1	4	yes	—
38	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂²×D₄×C₁₀₁	C₂²	1	4	yes	complemented
40	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
41	#13108073	—	2	32	no	—
42	C₂×Dic₂₀₂	—	2	4	no	—
43	C₂×Dic₂₀₂	C₂³	1	4	yes	—
44	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
45	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
46	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
47	C₂²×Dic₁₀₁	—	2	8	no	—
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
51	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
52	C₂×C₄×D₁₀₁	—	2	4	no	—
53	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
54	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
55	#13108053	C₂³	1	8	yes	—
56	#13108053	C₂³	1	16	yes	—
57	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
58	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
59	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
60	C₂²⋊Dic₁₀₁	C₂³	1	16	yes	—
61	#13108051	C₂³	1	8	yes	—
62	C₄×Dic₁₀₁	C₂³	1	4	yes	—
63	C₂⁴×C₁₀₁	C₂³	1	4	yes	—
64	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
65	#41×C₁₀₁	—	2	16	no	—
66	#41×C₁₀₁	—	2	8	no	—
67	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
68	C₂×D₄×C₁₀₁	—	2	16	no	—
69	C₂×D₄×C₁₀₁	—	2	16	no	—
70	C₂×D₄×C₁₀₁	D₄	1	8	yes	complemented
71	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
72	C₂×D₄×C₁₀₁	C₂³	1	8	yes	—
73	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
74	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
75	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
76	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
77	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
78	C₂³×C₁₀₁	—	2	8	no	—
79	C₂³×C₁₀₁	—	4	4	no	—
80	C₂³×C₁₀₁	—	4	8	no	—
81	C₂³×C₁₀₁	C₂×D₄	1	8	yes	—
82	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
83	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
84	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
85	C₂²×D₁₀₁	—	2	2	no	—
86	C₂²×D₁₀₁	—	4	2	no	—
87	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
88	Q₈×C₁₀₁	—	2	4	no	—
89	D₄×C₁₀₁	—	2	8	no	—
90	D₄×C₁₀₁	—	2	16	no	—
91	D₄×C₁₀₁	—	2	4	no	—
92	D₄×C₁₀₁	C₂×D₄	1	16	yes	—
93	C₂×C₄×C₁₀₁	—	2	8	no	—
94	C₂×C₄×C₁₀₁	—	2	8	no	—
95	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
97	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
98	C₂²⋊D₁₀₁	—	2	16	no	—
99	C₂²⋊D₁₀₁	—	4	8	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	2	4	no	—
102	C₂×Dic₁₀₁	—	2	16	no	—
103	C₂×Dic₁₀₁	—	4	8	no	—
104	C₂×Dic₁₀₁	C₂⁴	1	8	yes	—
105	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
106	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
107	C₄×D₁₀₁	—	2	8	no	—
108	Dic₂₀₂	—	2	8	no	—
109	C₂²×C₁₀₁	—	2	8	no	—
110	C₂²×C₁₀₁	—	2	2	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	—	2	8	no	—
113	C₂²×C₁₀₁	—	4	8	no	—
114	C₂²×C₁₀₁	—	8	8	no	—
115	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
116	C₂²×C₁₀₁	C₂²×D₄	1	8	yes	—
117	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₀₁	—	2	2	no	—
119	C₂×D₁₀₁	—	2	4	no	—
120	C₂×D₁₀₁	—	4	4	no	—
121	C₄×C₁₀₁	—	2	4	no	—
122	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
123	Dic₁₀₁	—	2	4	no	—
124	Dic₁₀₁	—	2	2	no	—
125	Dic₁₀₁	—	2	8	no	—
126	C₂×C₁₀₁	—	2	2	no	—
127	C₂×C₁₀₁	—	2	8	no	—
128	C₂×C₁₀₁	—	4	4	no	—
129	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
130	C₂×C₁₀₁	#585	1	2	yes	—
131	D₁₀₁	—	4	2	no	—
132	#3485	—	101	1	no	maximal, Sylow
133	C₁₀₁	#3485	1	1	yes	characteristic, complemented, Sylow
134	C₂×H(5,𝔽₂)	D₁₀₁	1	1	yes	characteristic, complemented
135	C₂²×#41	—	101	1	no	—
136	#547	—	101	8	no	—
137	#546	—	101	8	no	—
138	#536	—	101	2	no	—
139	#534	—	101	4	no	—
140	#525	—	101	2	no	—
141	C₂×#123	—	101	4	no	—
142	#518	—	101	1	no	—
143	H(5,𝔽₂)	—	2	8	no	—
144	C₂×#41	—	101	8	no	—
145	C₂×#41	—	202	8	no	—
146	C₂×#41	C₂×D₁₀₁	1	2	yes	—
147	C₂×#41	C₂×D₁₀₁	1	4	yes	complemented
148	C₂²×Q₈	—	101	1	no	—
149	C₂²×D₄	—	101	2	no	—
150	C₂²×D₄	C₂×D₁₀₁	1	4	yes	—
151	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
152	C₂²×D₄	C₂×D₁₀₁	1	4	yes	complemented
153	C₂³×C₄	—	101	2	no	—
154	C₂³×C₄	—	101	1	no	—
155	C₄⋊Q₈	—	101	4	no	—
156	#124	—	101	4	no	—
157	#123	—	101	4	no	—
158	#123	—	101	8	no	—
159	#123	—	101	16	no	—
160	#123	—	101	8	no	—
161	C₂²⋊Q₈	—	101	4	no	—
162	C₂²⋊Q₈	—	101	8	no	—
163	#121	—	101	4	no	—
164	#121	—	101	16	no	—
165	#120	—	101	8	no	—
166	#120	—	101	4	no	—
167	C₄×D₄	—	101	8	no	—
168	C₄×D₄	—	101	8	no	—
169	#117	—	101	2	no	—
170	C₂×C₄⋊C₄	—	101	8	no	—
171	C₂×C₂²⋊C₄	—	101	4	no	—
172	C₂×C₂²⋊C₄	—	101	8	no	—
173	C₂⁴	—	101	1	no	—
174	C₂⁴	C₂²×D₁₀₁	1	4	yes	—
175	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
176	#41	—	2	16	no	—
177	#41	—	2	8	no	—
178	#41	—	202	32	no	—
179	C₂×Q₈	—	101	4	no	—
180	C₂×Q₈	—	202	4	no	—
181	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
182	C₂×D₄	—	2	16	no	—
183	C₂×D₄	—	2	16	no	—
184	C₂×D₄	—	101	8	no	—
185	C₂×D₄	—	101	8	no	—
186	C₂×D₄	—	202	8	no	—
187	C₂×D₄	C₂²⋊D₁₀₁	1	8	yes	complemented
188	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
189	C₂×D₄	C₂²×D₁₀₁	1	8	yes	—
190	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
191	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
192	C₂²×C₄	—	101	8	no	—
193	C₂²×C₄	—	101	8	no	—
194	C₂²×C₄	—	101	1	no	—
195	C₂²×C₄	—	101	2	no	—
196	C₂²×C₄	—	101	4	no	—
197	C₂²×C₄	—	202	8	no	—
198	C₂²×C₄	—	202	4	no	—
199	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
200	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
201	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
202	C₄⋊C₄	—	101	8	no	—
203	C₄⋊C₄	—	101	16	no	—
204	C₄⋊C₄	—	101	4	no	—
205	C₂²⋊C₄	—	101	4	no	—
206	C₂²⋊C₄	—	101	8	no	—
207	C₂²⋊C₄	—	101	8	no	—
208	C₂²⋊C₄	—	101	16	no	—
209	C₂²⋊C₄	—	101	8	no	—
210	C₄²	—	101	4	no	—
211	C₂³	—	2	8	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	8	no	—
214	C₂³	—	101	2	no	—
215	C₂³	—	404	2	no	—
216	C₂³	—	202	2	no	—
217	C₂³	C₂×C₂²⋊D₁₀₁	1	8	yes	—
218	C₂³	C₂³×D₁₀₁	1	4	yes	—
219	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
220	C₂³	C₂³×D₁₀₁	1	4	yes	—
221	Q₈	—	2	4	no	—
222	Q₈	—	202	8	no	—
223	D₄	—	2	16	no	—
224	D₄	—	2	8	no	—
225	D₄	—	2	4	no	—
226	D₄	—	404	8	no	—
227	D₄	—	202	16	no	—
228	D₄	C₂×C₂²⋊D₁₀₁	1	16	yes	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	101	8	no	—
232	C₂×C₄	—	101	2	no	—
233	C₂×C₄	—	101	4	no	—
234	C₂×C₄	—	404	8	no	—
235	C₂×C₄	—	202	2	no	—
236	C₂×C₄	—	202	4	no	—
237	C₂×C₄	—	202	16	no	—
238	C₂×C₄	—	202	8	no	—
239	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
240	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
241	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	8	8	no	—
248	C₂²	—	404	4	no	—
249	C₂²	—	202	2	no	—
250	C₂²	—	202	4	no	—
251	C₂²	C₂²×C₂²⋊D₁₀₁	1	8	yes	—
252	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
253	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
254	C₄	—	2	4	no	—
255	C₄	—	202	2	no	—
256	C₄	—	202	4	no	—
257	C₄	—	202	8	no	—
258	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	2	8	no	—
261	C₂	—	4	4	no	—
262	C₂	—	404	2	no	—
263	C₂	C₂³×C₂²⋊D₁₀₁	1	1	yes	characteristic
264	C₂	#20266012	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Fitting	C₂×H(5,𝔽₂)×C₁₀₁	C₂
hypercenter	C₂×H(5,𝔽₂)	D₁₀₁
hypocenter	C₁₀₁	#3485
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108073	C₂	semidirect
2. #20265826	C₂	semidirect
3. #20265821	C₂	semidirect
4. #20265748	C₂	semidirect
5. #20265736	C₂	semidirect
6. #20265684	C₂	semidirect
7. C₂×#15692236	C₂	semidirect
8. #20265652	C₂	semidirect
9. C₂×H(5,𝔽₂)×C₁₀₁	C₂	semidirect
11. C₂×#13108073	C₂²	semidirect
12. C₂²×Dic₂₀₂	C₂²	semidirect
13. C₂²×C₂²⋊D₁₀₁	C₂²	semidirect
14. C₂³×Dic₁₀₁	C₂²	semidirect
15. C₂²×C₄×D₁₀₁	C₂²	semidirect
16. #15692253	C₂²	semidirect
17. #15692242	C₂²	semidirect
18. #15692238	C₂²	semidirect
19. #15692236	C₂²	semidirect
20. #15692236	C₂²	semidirect
21. #15692236	C₂²	semidirect
22. #15692234	C₂²	semidirect
23. #15692231	C₂²	semidirect
24. #15692227	C₂²	semidirect
25. #15692226	C₂²	semidirect
26. #15692224	C₂²	semidirect
27. #15692223	C₂²	semidirect
28. D₄×Dic₁₀₁	C₂²	semidirect
29. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
30. #15692214	C₂²	semidirect
31. C₂×#13108053	C₂²	semidirect
32. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
33. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
36. C₂×#41×C₁₀₁	C₂²	semidirect
39. C₂²×D₄×C₁₀₁	C₂²	semidirect
70. C₂×D₄×C₁₀₁	D₄	semidirect
133. C₁₀₁	#3485	semidirect
134. C₂×H(5,𝔽₂)	D₁₀₁	semidirect
147. C₂×#41	C₂×D₁₀₁	semidirect
152. C₂²×D₄	C₂×D₁₀₁	semidirect
187. C₂×D₄	C₂²⋊D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108073	2
#20265826	2
#20265821	2
#20265748	2
#20265736	2
#20265684	2
C₂×#15692236	2
#20265652	2
C₂×H(5,𝔽₂)×C₁₀₁	2
#3485	101