Subgroups — #27174488

Order 12928.

1232 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692229	C₂	1	2	yes	maximal, complemented
3	C₂×#15692223	C₂	1	2	yes	maximal, complemented
4	#20265094	C₂	1	2	yes	maximal, complemented
5	#20265086	C₂	1	2	yes	maximal, complemented
6	#20265079	C₂	1	2	yes	maximal, complemented
7	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265032	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	C₂²	1	2	yes	—
11	C₂²×D₄₀₄	—	2	8	no	—
12	C₂²×D₄₀₄	—	2	4	no	—
13	C₂²×D₄₀₄	C₂²	1	2	yes	—
14	C₂²×D₄₀₄	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	complemented
16	#15692229	—	2	8	no	—
17	#15692223	—	2	8	no	—
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×#13108051	C₂²	1	4	yes	complemented
21	C₂×#13108051	C₂²	1	4	yes	complemented
22	C₂×#13108051	C₂²	1	4	yes	complemented
23	#15692162	C₂²	1	2	yes	complemented
24	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
27	C₄×D₄×C₁₀₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
32	C₂³×D₁₀₁	—	2	4	no	—
33	C₂³×D₁₀₁	—	2	8	no	—
34	C₂³×D₁₀₁	—	2	4	no	—
35	C₂³×D₁₀₁	—	2	4	no	—
36	C₂³×D₁₀₁	—	4	4	no	—
37	C₂³×D₁₀₁	C₂³	1	4	yes	—
38	C₂×D₄₀₄	—	2	32	no	—
39	C₂×D₄₀₄	—	2	8	no	—
40	C₂×D₄₀₄	—	2	4	no	—
41	C₂×D₄₀₄	—	2	4	no	—
42	C₂×D₄₀₄	—	2	4	no	—
43	C₂×D₄₀₄	D₄	1	8	yes	complemented
44	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
45	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
46	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
47	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
48	C₄⋊Dic₁₀₁	—	2	4	no	—
49	C₂²⋊Dic₁₀₁	—	2	4	no	—
50	#13108051	—	2	8	no	—
51	#13108051	—	2	8	no	—
52	#13108051	—	4	4	no	—
53	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
54	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
55	C₂×D₄×C₁₀₁	—	2	2	no	—
56	C₂×D₄×C₁₀₁	—	2	2	no	—
57	C₂×D₄×C₁₀₁	D₄	1	4	yes	complemented
58	C₂²×C₄×C₁₀₁	—	2	4	no	—
59	C₂²×C₄×C₁₀₁	—	2	2	no	—
60	C₂²×C₄×C₁₀₁	—	2	4	no	—
61	C₂²×C₄×C₁₀₁	D₄	1	4	yes	complemented
62	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₀₁	—	2	2	no	—
68	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
69	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
70	C₄²×C₁₀₁	—	2	2	no	—
71	C₂³×C₁₀₁	—	2	2	no	—
72	C₂³×C₁₀₁	—	2	2	no	—
73	C₂³×C₁₀₁	—	2	1	no	—
74	C₂³×C₁₀₁	—	2	2	no	—
75	C₂³×C₁₀₁	—	2	1	no	—
76	C₂³×C₁₀₁	—	2	2	no	—
77	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
78	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
79	C₂³×C₁₀₁	C₂×D₄	1	2	yes	complemented
80	C₂³×C₁₀₁	#41	1	2	yes	—
81	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
82	C₂²×D₁₀₁	—	2	16	no	—
83	C₂²×D₁₀₁	—	2	4	no	—
84	C₂²×D₁₀₁	—	2	4	no	—
85	C₂²×D₁₀₁	—	2	4	no	—
86	C₂²×D₁₀₁	—	2	4	no	—
87	C₂²×D₁₀₁	—	2	4	no	—
88	C₂²×D₁₀₁	—	2	4	no	—
89	C₂²×D₁₀₁	—	4	16	no	—
90	C₂²×D₁₀₁	—	4	16	no	—
91	C₂²×D₁₀₁	—	4	8	no	—
92	C₂²×D₁₀₁	—	4	8	no	—
93	C₂²×D₁₀₁	C₂×D₄	1	8	yes	—
94	D₄×C₁₀₁	—	2	8	no	—
95	C₂×C₄×C₁₀₁	—	2	1	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	1	no	—
98	C₂×C₄×C₁₀₁	—	2	8	no	—
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	1	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	—	4	4	no	—
104	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
105	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	complemented
108	C₂²⋊D₁₀₁	—	4	16	no	—
109	C₂×Dic₁₀₁	—	2	4	no	—
110	C₂×Dic₁₀₁	—	2	4	no	—
111	C₂×Dic₁₀₁	—	2	4	no	—
112	D₄₀₄	—	2	16	no	—
113	D₄₀₄	—	2	16	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	2	2	no	—
116	C₂²×C₁₀₁	—	2	4	no	—
117	C₂²×C₁₀₁	—	2	4	no	—
118	C₂²×C₁₀₁	—	2	2	no	—
119	C₂²×C₁₀₁	—	2	4	no	—
120	C₂²×C₁₀₁	—	4	1	no	—
121	C₂²×C₁₀₁	—	4	1	no	—
122	C₂²×C₁₀₁	—	4	2	no	—
123	C₂²×C₁₀₁	#121	1	4	yes	complemented
124	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
125	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
128	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
129	C₂×D₁₀₁	—	2	8	no	—
130	C₂×D₁₀₁	—	2	8	no	—
131	C₂×D₁₀₁	—	2	8	no	—
132	C₂×D₁₀₁	—	4	32	no	—
133	C₂×D₁₀₁	—	4	4	no	—
134	C₂×D₁₀₁	—	4	4	no	—
135	C₂×D₁₀₁	—	4	4	no	—
136	C₂×D₁₀₁	—	4	4	no	—
137	C₄×C₁₀₁	—	2	2	no	—
138	C₄×C₁₀₁	—	2	2	no	—
139	C₄×C₁₀₁	—	4	1	no	—
140	C₄×C₁₀₁	—	4	1	no	—
141	C₄×C₁₀₁	#120	1	4	yes	complemented
142	Dic₁₀₁	—	4	4	no	—
143	C₂×C₁₀₁	—	2	4	no	—
144	C₂×C₁₀₁	—	4	2	no	—
145	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
147	C₂×C₁₀₁	#535	1	1	yes	characteristic
148	C₂×C₁₀₁	D₄²	1	2	yes	—
149	C₂×C₁₀₁	#546	1	2	yes	—
150	D₁₀₁	—	4	8	no	—
151	#2408	—	101	1	no	maximal, Sylow
152	C₁₀₁	#2408	1	1	yes	characteristic, complemented, Sylow
153	C₂³×D₄	—	101	1	no	—
154	C₂×#121	—	101	2	no	—
155	C₂×#120	—	101	2	no	—
156	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
157	#394	—	101	2	no	—
158	#392	—	101	2	no	—
159	#390	—	101	2	no	—
160	#380	—	101	1	no	—
161	#379	—	101	2	no	—
162	C₂⁵	—	101	2	no	—
163	C₂²×D₄	—	101	4	no	—
164	C₂²×D₄	—	101	2	no	—
165	C₂²×D₄	—	101	2	no	—
166	C₂²×D₄	—	202	8	no	—
167	C₂²×D₄	—	202	4	no	—
168	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
169	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
170	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
171	#121	—	202	8	no	—
172	#120	—	202	8	no	—
173	C₄×D₄	—	2	4	no	—
174	C₂×C₄⋊C₄	—	101	2	no	—
175	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
176	C₂×C₂²⋊C₄	—	101	4	no	—
177	C₂×C₂²⋊C₄	—	101	2	no	—
178	C₂×C₂²⋊C₄	—	101	4	no	—
179	C₂×C₂²⋊C₄	—	101	4	no	—
180	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
182	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic, complemented
183	#95	—	101	2	no	—
184	C₂⁴	—	101	4	no	—
185	C₂⁴	—	404	4	no	—
186	C₂⁴	—	202	8	no	—
187	C₂⁴	—	202	4	no	—
188	C₂⁴	—	202	4	no	—
189	C₂⁴	—	202	4	no	—
190	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
191	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	101	8	no	—
195	C₂×D₄	—	404	4	no	—
196	C₂×D₄	—	202	8	no	—
197	C₂×D₄	—	202	32	no	—
198	C₂×D₄	—	202	8	no	—
199	C₂×D₄	—	202	8	no	—
200	C₂×D₄	—	202	4	no	—
201	C₂×D₄	—	202	4	no	—
202	C₂×D₄	—	202	4	no	—
203	C₂×D₄	D₄₀₄	1	4	yes	complemented
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	101	4	no	—
208	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	complemented
209	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	202	4	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	404	4	no	—
219	C₂²⋊C₄	—	202	8	no	—
220	C₂²⋊C₄	—	202	4	no	—
221	C₂²⋊C₄	—	202	8	no	—
222	C₄²	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	101	8	no	—
230	C₂³	—	404	8	no	—
231	C₂³	—	404	16	no	—
232	C₂³	—	404	16	no	—
233	C₂³	—	404	8	no	—
234	C₂³	—	202	16	no	—
235	C₂³	—	202	4	no	—
236	C₂³	—	202	4	no	—
237	C₂³	—	202	4	no	—
238	C₂³	—	202	4	no	—
239	C₂³	—	202	4	no	—
240	C₂³	—	202	4	no	—
241	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
242	C₂³	C₂×D₄₀₄	1	2	yes	—
243	C₂³	C₂×D₄₀₄	1	2	yes	complemented
244	C₂³	#13108074	1	2	yes	—
245	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
246	D₄	—	2	8	no	—
247	D₄	—	404	16	no	—
248	D₄	—	202	16	no	—
249	D₄	—	202	16	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	202	4	no	—
260	C₂×C₄	—	202	4	no	—
261	C₂×C₄	—	202	4	no	—
262	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
263	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
265	C₂×C₄	C₂×D₄₀₄	1	2	yes	complemented
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	404	32	no	—
276	C₂²	—	404	4	no	—
277	C₂²	—	404	4	no	—
278	C₂²	—	404	4	no	—
279	C₂²	—	404	4	no	—
280	C₂²	—	202	8	no	—
281	C₂²	—	202	8	no	—
282	C₂²	—	202	8	no	—
283	C₂²	#15692229	1	4	yes	complemented
284	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
285	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
286	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
287	C₂²	C₂×#13108074	1	1	yes	characteristic
288	C₂²	#15692301	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	404	4	no	—
294	C₄	#15692223	1	4	yes	complemented
295	C₂	—	2	4	no	—
296	C₂	—	4	2	no	—
297	C₂	—	404	8	no	—
298	C₂	C₂×#15692223	1	1	yes	characteristic
299	C₂	C₂×#15692229	1	1	yes	characteristic
300	C₂	#20265741	1	1	yes	characteristic
301	C₂	D₄×D₄₀₄	1	2	yes	—
302	C₂	#20265823	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	#2408
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄₀₄	C₂	semidirect
2. C₂×#15692229	C₂	semidirect
3. C₂×#15692223	C₂	semidirect
4. #20265094	C₂	semidirect
5. #20265086	C₂	semidirect
6. #20265079	C₂	semidirect
7. #20265037	C₂	semidirect
8. #20265032	C₂	semidirect
9. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
15. C₂²×C₂²⋊D₁₀₁	C₂²	semidirect
18. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
20. C₂×#13108051	C₂²	semidirect
21. C₂×#13108051	C₂²	semidirect
22. C₂×#13108051	C₂²	semidirect
23. #15692162	C₂²	semidirect
26. C₂³×C₄×C₁₀₁	C₂²	semidirect
28. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
30. C₂×C₂²⋊C₄×C₁₀₁	C₂²	semidirect
31. C₂×C₄²×C₁₀₁	C₂²	semidirect
43. C₂×D₄₀₄	D₄	semidirect
57. C₂×D₄×C₁₀₁	D₄	semidirect
61. C₂²×C₄×C₁₀₁	D₄	semidirect
79. C₂³×C₁₀₁	C₂×D₄	semidirect
107. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
123. C₂²×C₁₀₁	#121	semidirect
141. C₄×C₁₀₁	#120	semidirect
152. C₁₀₁	#2408	semidirect
156. C₂×C₄×D₄	D₁₀₁	semidirect
170. C₂³×C₄	C₂×D₁₀₁	semidirect
175. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
181. C₂×C₂²⋊C₄	C₂×D₁₀₁	semidirect
182. C₂×C₄²	C₂×D₁₀₁	semidirect
203. C₂×D₄	D₄₀₄	semidirect
208. C₂²×C₄	C₂²⋊D₁₀₁	semidirect
243. C₂³	C₂×D₄₀₄	semidirect
265. C₂×C₄	C₂×D₄₀₄	semidirect
283. C₂²	#15692229	semidirect
294. C₄	#15692223	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄₀₄	2
C₂×#15692229	2
C₂×#15692223	2
#20265094	2
#20265086	2
#20265079	2
#20265037	2
#20265032	2
C₂×C₄×D₄×C₁₀₁	2
#2408	101