Subgroups — #27174142

Order 12928.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692231	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20265074	C₂	1	2	yes	maximal, complemented
5	#20265073	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265059	C₂	1	2	yes	maximal, complemented
7	#20265051	C₂	1	2	yes	maximal
8	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13108051	C₂	1	1	yes	maximal, characteristic
10	#20265024	C₂	1	2	yes	maximal
11	#391×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
14	#15692231	C₄	1	8	yes	complemented
15	Q₈×Dic₁₀₁	—	2	4	no	—
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×#13108053	C₂²	1	2	yes	—
20	C₂×#13108053	C₂²	1	2	yes	—
21	C₂×#13108053	C₂²	1	2	yes	—
22	C₄⋊C₄×D₁₀₁	—	2	4	no	—
23	C₂×#13108051	C₂²	1	1	yes	characteristic
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	1	yes	characteristic
31	#15692162	C₂²	1	1	yes	characteristic
32	#15692162	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
36	#95×C₁₀₁	C₂²	1	2	yes	—
37	#95×C₁₀₁	C₂²	1	2	yes	—
38	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₀₁	—	2	2	no	—
45	C₂×C₄×D₁₀₁	—	2	2	no	—
46	C₂×C₄×D₁₀₁	—	2	4	no	—
47	C₂×C₄×D₁₀₁	—	2	4	no	—
48	C₂×C₄×D₁₀₁	—	4	2	no	—
49	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
50	C₄⋊Dic₁₀₁	—	2	4	no	—
51	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
52	C₄⋊Dic₁₀₁	D₄	1	4	yes	—
53	#13108053	—	2	4	no	—
54	#13108053	—	4	2	no	—
55	#13108053	C₂×C₄	1	8	yes	complemented
56	#13108051	—	2	2	no	—
57	#13108051	C₂×C₄	1	8	yes	—
58	#13108051	Q₈	1	4	yes	complemented
59	C₄×Dic₁₀₁	—	2	2	no	—
60	C₄×Dic₁₀₁	—	2	4	no	—
61	C₄×Dic₁₀₁	—	4	2	no	—
62	C₂×Q₈×C₁₀₁	—	2	2	no	—
63	C₂×Q₈×C₁₀₁	—	2	2	no	—
64	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
65	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₁	—	2	2	no	—
71	C₄²×C₁₀₁	—	2	2	no	—
72	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
73	C₂²×D₁₀₁	—	2	1	no	—
74	C₂²×D₁₀₁	—	2	1	no	—
75	C₂²×D₁₀₁	—	2	1	no	—
76	C₂²×D₁₀₁	—	2	1	no	—
77	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
79	C₂²×D₁₀₁	#41	1	2	yes	—
80	Q₈×C₁₀₁	—	2	8	no	—
81	C₂×C₄×C₁₀₁	—	2	2	no	—
82	C₂×C₄×C₁₀₁	—	2	1	no	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	1	no	—
85	C₂×C₄×C₁₀₁	—	2	2	no	—
86	C₂×C₄×C₁₀₁	—	2	1	no	—
87	C₂×C₄×C₁₀₁	—	2	1	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₁₀₁	#41	1	2	yes	—
96	C₂×Dic₁₀₁	—	2	1	no	—
97	C₂×Dic₁₀₁	—	2	1	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	1	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	—	2	1	no	—
104	C₂×Dic₁₀₁	—	2	1	no	—
105	C₂×Dic₁₀₁	—	2	1	no	—
106	C₂×Dic₁₀₁	—	2	1	no	—
107	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
108	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
109	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
110	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
111	C₄×D₁₀₁	—	2	8	no	—
112	C₄×D₁₀₁	—	4	8	no	—
113	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
114	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
117	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂×D₁₀₁	—	2	2	no	—
121	C₂×D₁₀₁	—	2	2	no	—
122	C₂×D₁₀₁	—	2	2	no	—
123	C₂×D₁₀₁	—	2	2	no	—
124	C₂×D₁₀₁	—	2	2	no	—
125	C₂×D₁₀₁	—	2	2	no	—
126	C₂×D₁₀₁	C₄×Q₈	1	4	yes	complemented
127	C₄×C₁₀₁	—	2	4	no	—
128	C₄×C₁₀₁	—	2	2	no	—
129	C₄×C₁₀₁	—	4	2	no	—
130	C₄×C₁₀₁	—	4	1	no	—
131	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
132	Dic₁₀₁	—	2	2	no	—
133	Dic₁₀₁	—	2	4	no	—
134	Dic₁₀₁	—	2	4	no	—
135	Dic₁₀₁	—	2	2	no	—
136	Dic₁₀₁	—	4	1	no	—
137	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₂×C₁₀₁	#520	1	1	yes	characteristic
140	C₂×C₁₀₁	#546	1	1	yes	characteristic
141	C₂×C₁₀₁	D₄×Q₈	1	1	yes	characteristic
142	C₂×C₁₀₁	#552	1	1	yes	characteristic
143	C₂×C₁₀₁	#554	1	1	yes	characteristic
144	D₁₀₁	—	2	4	no	—
145	#2360	—	101	1	no	maximal, Sylow
146	C₁₀₁	#2360	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₂²⋊Q₈	—	101	1	no	—
148	C₂×C₄×Q₈	—	101	1	no	—
149	C₂²×C₄⋊C₄	—	101	1	no	—
150	#391	D₁₀₁	1	1	yes	characteristic, complemented
151	#389	—	101	1	no	—
152	#389	—	101	2	no	—
153	#387	—	101	2	no	—
154	#385	—	101	2	no	—
155	C₄×C₄⋊C₄	—	101	1	no	—
156	C₄×C₂²⋊C₄	—	101	1	no	—
157	C₄×C₂²⋊C₄	—	101	2	no	—
158	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
159	C₂³×C₄	—	101	1	no	—
160	C₂³×C₄	—	101	2	no	—
161	C₂²⋊Q₈	—	101	8	no	—
162	C₄×Q₈	—	202	4	no	—
163	C₂×C₄⋊C₄	—	101	1	no	—
164	C₂×C₄⋊C₄	—	101	1	no	—
165	C₂×C₄⋊C₄	—	101	2	no	—
166	C₂×C₄⋊C₄	—	101	2	no	—
167	C₂×C₄⋊C₄	—	101	2	no	—
168	C₂×C₄⋊C₄	—	101	2	no	—
169	C₂×C₄⋊C₄	—	202	4	no	—
170	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	—	101	2	no	—
172	C₂×C₂²⋊C₄	—	101	1	no	—
173	C₂×C₂²⋊C₄	—	101	1	no	—
174	C₂×C₄²	—	101	2	no	—
175	C₂×C₄²	—	101	2	no	—
176	C₂×C₄²	—	101	1	no	—
177	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
178	#95	—	101	1	no	—
179	#95	—	101	2	no	—
180	#95	—	101	1	no	—
181	#95	—	101	2	no	—
182	#95	C₂×D₁₀₁	1	2	yes	—
183	#95	C₂×D₁₀₁	1	2	yes	—
184	C₂⁴	—	101	1	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	C₄×D₁₀₁	1	4	yes	complemented
188	C₂²×C₄	—	101	1	no	—
189	C₂²×C₄	—	101	2	no	—
190	C₂²×C₄	—	101	1	no	—
191	C₂²×C₄	—	101	2	no	—
192	C₂²×C₄	—	101	4	no	—
193	C₂²×C₄	—	101	1	no	—
194	C₂²×C₄	—	404	2	no	—
195	C₂²×C₄	—	202	4	no	—
196	C₂²×C₄	—	202	4	no	—
197	C₂²×C₄	—	202	2	no	—
198	C₂²×C₄	—	202	2	no	—
199	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	101	8	no	—
206	C₄⋊C₄	—	101	4	no	—
207	C₄⋊C₄	—	101	4	no	—
208	C₄⋊C₄	—	404	2	no	—
209	C₄⋊C₄	—	202	4	no	—
210	C₄⋊C₄	—	202	4	no	—
211	C₂²⋊C₄	—	101	8	no	—
212	C₂²⋊C₄	—	101	4	no	—
213	C₂²⋊C₄	—	202	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	404	2	no	—
216	C₄²	—	202	4	no	—
217	C₄²	—	202	2	no	—
218	C₂³	—	101	2	no	—
219	C₂³	—	101	2	no	—
220	C₂³	—	101	2	no	—
221	C₂³	—	202	1	no	—
222	C₂³	—	202	1	no	—
223	C₂³	—	202	1	no	—
224	C₂³	—	202	1	no	—
225	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	101	4	no	—
238	C₂×C₄	—	101	2	no	—
239	C₂×C₄	—	101	4	no	—
240	C₂×C₄	—	101	2	no	—
241	C₂×C₄	—	404	8	no	—
242	C₂×C₄	—	202	1	no	—
243	C₂×C₄	—	202	1	no	—
244	C₂×C₄	—	202	1	no	—
245	C₂×C₄	—	202	1	no	—
246	C₂×C₄	—	202	1	no	—
247	C₂×C₄	—	202	2	no	—
248	C₂×C₄	—	202	1	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	2	no	—
251	C₂×C₄	—	202	1	no	—
252	C₂×C₄	—	202	2	no	—
253	C₂×C₄	—	202	8	no	—
254	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
255	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
256	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
257	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
258	C₂×C₄	#13108073	1	2	yes	—
259	C₂²	—	101	4	no	—
260	C₂²	—	202	2	no	—
261	C₂²	—	202	2	no	—
262	C₂²	—	202	2	no	—
263	C₂²	—	202	2	no	—
264	C₂²	—	202	2	no	—
265	C₂²	—	202	2	no	—
266	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
267	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
268	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
269	C₂²	#41×D₁₀₁	1	1	yes	characteristic
270	C₂²	C₂×#13108073	1	1	yes	characteristic
271	C₂²	#15692301	1	1	yes	characteristic
272	C₂²	#15692303	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	404	1	no	—
278	C₄	—	202	4	no	—
279	C₄	—	202	2	no	—
280	C₄	—	202	2	no	—
281	C₄	—	202	4	no	—
282	C₄	#15692215	1	4	yes	—
283	C₂	—	202	4	no	—
284	C₂	C₂×#15692215	1	1	yes	characteristic
285	C₂	C₄×Q₈×D₁₀₁	1	1	yes	characteristic
286	C₂	#20265659	1	1	yes	characteristic
287	C₂	#20265820	1	1	yes	characteristic
288	C₂	#20265842	1	1	yes	characteristic
289	C₂	#20265861	1	1	yes	characteristic
290	C₂	#20265885	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#391×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	#391	D₁₀₁
hypocenter	C₁₀₁	#2360
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₁₀₁	C₂	semidirect
4. #20265074	C₂	semidirect
5. #20265073	C₂	semidirect
6. #20265059	C₂	semidirect
8. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
11. #391×C₁₀₁	C₂	semidirect
14. #15692231	C₄	semidirect
51. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
55. #13108053	C₂×C₄	semidirect
58. #13108051	Q₈	semidirect
64. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
126. C₂×D₁₀₁	C₄×Q₈	semidirect
146. C₁₀₁	#2360	semidirect
150. #391	D₁₀₁	semidirect
187. C₂×Q₈	C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692231	2
C₂×Q₈×Dic₁₀₁	2
C₂×C₄⋊C₄×D₁₀₁	2
#20265074	2
#20265073	2
#20265059	2
#20265051	2
C₄×C₄⋊Dic₁₀₁	2
C₄×#13108051	2
#20265024	2
#391×C₁₀₁	2
#2360	101