Subgroups — #27174039

Order 12928.

704 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	2	yes	maximal
2	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#20265074	C₂	1	2	yes	maximal, complemented
4	#20265051	C₂	1	2	yes	maximal, complemented
5	#20265050	C₂	1	2	yes	maximal, complemented
6	#20265041	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265036	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265028	C₂	1	1	yes	maximal, characteristic, complemented
10	#95×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#95×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₀₁	—	2	8	no	—
16	C₄×C₂²⋊D₁₀₁	C₄	1	8	yes	complemented
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
18	C₂×#13108053	C₂²	1	2	yes	—
19	C₂×#13108053	C₂²	1	2	yes	—
20	C₂×#13108053	C₂²	1	2	yes	—
21	C₂×#13108053	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₀₁	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	#15692162	C₂²	1	1	yes	characteristic
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	1	yes	characteristic
31	#15692162	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
33	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	#95×C₁₀₁	C₂²	1	1	yes	characteristic
35	#95×C₁₀₁	C₂²	1	1	yes	characteristic
36	#95×C₁₀₁	C₂²	1	2	yes	—
37	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊D₁₀₁	D₄	1	4	yes	—
40	C₂×C₂²⋊D₁₀₁	Q₈	1	4	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
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	—	4	2	no	—
50	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
51	C₄⋊Dic₁₀₁	—	4	2	no	—
52	#13108053	—	2	4	no	—
53	#13108053	—	2	4	no	—
54	#13108053	—	4	2	no	—
55	#13108053	C₂×C₄	1	4	yes	complemented
56	C₂²⋊Dic₁₀₁	—	2	4	no	—
57	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
58	#13108051	—	2	4	no	—
59	#13108051	C₂×C₄	1	4	yes	complemented
60	C₄×Dic₁₀₁	—	2	4	no	—
61	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
62	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	—	2	4	no	—
64	C₂²×C₄×C₁₀₁	—	2	4	no	—
65	C₂²×C₄×C₁₀₁	—	4	2	no	—
66	C₂²×C₄×C₁₀₁	—	4	2	no	—
67	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
72	C₂³×C₁₀₁	—	2	2	no	—
73	C₂³×C₁₀₁	—	2	2	no	—
74	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
76	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
77	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
78	C₂²×D₁₀₁	—	2	2	no	—
79	C₂²×D₁₀₁	—	2	2	no	—
80	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
82	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	2	no	—
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	—	4	4	no	—
91	C₂×C₄×C₁₀₁	—	4	8	no	—
92	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₀₁	#41	1	4	yes	—
95	C₂²⋊D₁₀₁	C₄⋊C₄	1	16	yes	complemented
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	2	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	—	2	2	no	—
104	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
105	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
106	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
107	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
108	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
109	C₄×D₁₀₁	—	4	4	no	—
110	C₄×D₁₀₁	—	4	8	no	—
111	C₂²×C₁₀₁	—	2	2	no	—
112	C₂²×C₁₀₁	—	2	2	no	—
113	C₂²×C₁₀₁	—	2	4	no	—
114	C₂²×C₁₀₁	—	2	4	no	—
115	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
116	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
117	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
120	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
121	C₂×D₁₀₁	—	2	2	no	—
122	C₂×D₁₀₁	—	2	2	no	—
123	C₂×D₁₀₁	—	2	4	no	—
124	C₂×D₁₀₁	—	2	4	no	—
125	C₂×D₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
126	C₄×C₁₀₁	—	2	2	no	—
127	C₄×C₁₀₁	—	2	2	no	—
128	C₄×C₁₀₁	—	2	4	no	—
129	C₄×C₁₀₁	—	4	2	no	—
130	Dic₁₀₁	—	2	2	no	—
131	Dic₁₀₁	—	2	2	no	—
132	Dic₁₀₁	—	4	2	no	—
133	Dic₁₀₁	—	4	2	no	—
134	Dic₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₁₀₁	—	2	4	no	—
136	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
138	C₂×C₁₀₁	#518	1	1	yes	characteristic
139	C₂×C₁₀₁	#546	1	2	yes	—
140	C₂×C₁₀₁	#554	1	2	yes	—
141	D₁₀₁	—	2	4	no	—
142	#2349	—	101	1	no	maximal, Sylow
143	C₁₀₁	#2349	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×D₄	—	101	1	no	—
145	C₂×C₄×D₄	—	101	2	no	—
146	#389	—	101	2	no	—
147	#385	—	101	2	no	—
148	#385	—	101	2	no	—
149	#382	—	101	1	no	—
150	#380	—	101	1	no	—
151	#380	—	101	1	no	—
152	C₄×C₄⋊C₄	—	101	1	no	—
153	C₂×#95	—	101	1	no	—
154	C₂×#95	D₁₀₁	1	1	yes	characteristic, complemented
155	C₂²×D₄	—	101	1	no	—
156	C₂³×C₄	—	101	1	no	—
157	C₂³×C₄	—	101	2	no	—
158	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
159	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
160	C₄×D₄	—	101	8	no	—
161	C₄×D₄	—	202	8	no	—
162	C₂×C₄⋊C₄	—	101	1	no	—
163	C₂×C₄⋊C₄	—	101	2	no	—
164	C₂×C₄⋊C₄	—	101	2	no	—
165	C₂×C₄⋊C₄	—	101	2	no	—
166	C₂×C₄⋊C₄	—	101	2	no	—
167	C₂×C₂²⋊C₄	—	101	1	no	—
168	C₂×C₂²⋊C₄	—	101	1	no	—
169	C₂×C₂²⋊C₄	—	101	2	no	—
170	C₂×C₂²⋊C₄	—	101	2	no	—
171	C₂×C₄²	—	101	1	no	—
172	C₂×C₄²	—	101	2	no	—
173	#95	—	101	1	no	—
174	#95	—	101	2	no	—
175	#95	—	101	1	no	—
176	#95	—	101	2	no	—
177	#95	C₂×D₁₀₁	1	1	yes	characteristic
178	#95	C₂×D₁₀₁	1	1	yes	characteristic
179	#95	C₂×D₁₀₁	1	2	yes	—
180	C₂⁴	—	101	1	no	—
181	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
182	C₂×D₄	—	101	4	no	—
183	C₂×D₄	—	101	4	no	—
184	C₂×D₄	—	101	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	101	1	no	—
190	C₂²×C₄	—	101	1	no	—
191	C₂²×C₄	—	101	1	no	—
192	C₂²×C₄	—	101	4	no	—
193	C₂²×C₄	—	101	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	404	2	no	—
196	C₂²×C₄	—	404	2	no	—
197	C₂²×C₄	—	202	4	no	—
198	C₂²×C₄	—	202	4	no	—
199	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
200	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
204	C₄⋊C₄	—	101	4	no	—
205	C₄⋊C₄	—	404	2	no	—
206	C₄⋊C₄	—	404	2	no	—
207	C₄⋊C₄	—	202	4	no	—
208	C₄⋊C₄	—	202	4	no	—
209	C₂²⋊C₄	—	101	4	no	—
210	C₂²⋊C₄	—	101	4	no	—
211	C₂²⋊C₄	—	202	4	no	—
212	C₂²⋊C₄	—	202	4	no	—
213	C₄²	—	101	4	no	—
214	C₄²	—	202	4	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	101	2	no	—
218	C₂³	—	101	2	no	—
219	C₂³	—	101	2	no	—
220	C₂³	—	202	2	no	—
221	C₂³	—	202	2	no	—
222	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
223	C₂³	D₄×D₁₀₁	1	2	yes	—
224	C₂³	Q₈×D₁₀₁	1	2	yes	—
225	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
226	D₄	—	101	16	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	101	2	no	—
237	C₂×C₄	—	101	2	no	—
238	C₂×C₄	—	101	2	no	—
239	C₂×C₄	—	101	2	no	—
240	C₂×C₄	—	101	2	no	—
241	C₂×C₄	—	404	8	no	—
242	C₂×C₄	—	404	4	no	—
243	C₂×C₄	—	202	2	no	—
244	C₂×C₄	—	202	2	no	—
245	C₂×C₄	—	202	2	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	—	202	2	no	—
248	C₂×C₄	—	202	2	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	2	no	—
251	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
252	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
253	C₂×C₄	#13108074	1	4	yes	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	101	4	no	—
259	C₂²	—	202	4	no	—
260	C₂²	—	202	4	no	—
261	C₂²	—	202	2	no	—
262	C₂²	—	202	2	no	—
263	C₂²	C₄⋊C₄×D₁₀₁	1	4	yes	—
264	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
265	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
266	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
267	C₂²	C₂×#13108074	1	2	yes	—
268	C₂²	#15692300	1	2	yes	—
269	C₄	—	2	4	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	101	4	no	—
274	C₄	—	404	2	no	—
275	C₄	—	404	2	no	—
276	C₄	—	202	2	no	—
277	C₄	—	202	2	no	—
278	C₂	—	2	4	no	—
279	C₂	—	202	4	no	—
280	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
281	C₂	C₄×#13108074	1	1	yes	characteristic
282	C₂	#20265650	1	1	yes	characteristic
283	C₂	#20265817	1	2	yes	—
284	C₂	#20265882	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×#95×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×C₂²⋊D₁₀₁	C₂²
hypercenter	C₂×#95	D₁₀₁
hypocenter	C₁₀₁	#2349
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
3. #20265074	C₂	semidirect
4. #20265051	C₂	semidirect
5. #20265050	C₂	semidirect
6. #20265041	C₂	semidirect
7. #20265037	C₂	semidirect
8. #20265036	C₂	semidirect
9. #20265028	C₂	semidirect
10. #95×D₁₀₁	C₂	semidirect
11. C₂×#95×C₁₀₁	C₂	semidirect
16. C₄×C₂²⋊D₁₀₁	C₄	semidirect
50. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
55. #13108053	C₂×C₄	semidirect
57. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
59. #13108051	C₂×C₄	semidirect
61. C₄×Dic₁₀₁	C₂×C₄	semidirect
67. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
95. C₂²⋊D₁₀₁	C₄⋊C₄	semidirect
143. C₁₀₁	#2349	semidirect
154. C₂×#95	D₁₀₁	semidirect
199. C₂²×C₄	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×C₄×C₂²⋊D₁₀₁	2
#20265074	2
#20265051	2
#20265050	2
#20265041	2
#20265037	2
#20265036	2
#20265028	2
#95×D₁₀₁	2
C₂×#95×C₁₀₁	2
#2349	101