Subgroups — #400×D₁₀₁ #27174382

Order 12928.

604 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#20265119	C₂	1	2	yes	maximal, complemented
4	#20265118	C₂	1	2	yes	maximal, complemented
5	#20265117	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265116	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265115	C₂	1	1	yes	maximal, characteristic, complemented
8	#95×D₁₀₁	C₂	1	1	yes	maximal, characteristic
9	#95×D₁₀₁	C₂	1	2	yes	maximal
10	#95×D₁₀₁	C₂	1	2	yes	maximal
11	#400×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	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×#13108053	C₂²	1	1	yes	characteristic
20	C₂×#13108053	C₂²	1	1	yes	characteristic
21	C₂×#13108053	C₂²	1	2	yes	—
22	C₄⋊C₄×D₁₀₁	—	2	4	no	—
23	C₄⋊C₄×D₁₀₁	—	2	4	no	—
24	#15692162	C₂²	1	1	yes	characteristic
25	#15692162	C₂²	1	2	yes	—
26	#15692162	C₂²	1	2	yes	—
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	#95×C₁₀₁	C₂²	1	1	yes	characteristic
35	#95×C₁₀₁	C₂²	1	2	yes	—
36	#95×C₁₀₁	C₂²	1	2	yes	—
37	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
38	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂×C₄×D₁₀₁	—	2	2	no	—
44	C₂×C₄×D₁₀₁	—	2	2	no	—
45	C₂×C₄×D₁₀₁	—	2	4	no	—
46	C₂×C₄×D₁₀₁	—	2	2	no	—
47	C₂×C₄×D₁₀₁	—	2	4	no	—
48	C₂×C₄×D₁₀₁	—	2	4	no	—
49	C₂×C₄×D₁₀₁	—	2	8	no	—
50	C₂×C₄×D₁₀₁	—	2	4	no	—
51	C₂×C₄×D₁₀₁	—	2	4	no	—
52	C₂×C₄×D₁₀₁	—	2	4	no	—
53	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
54	C₂×C₄×D₁₀₁	Q₈	1	4	yes	—
55	C₄⋊Dic₁₀₁	—	2	2	no	—
56	C₄⋊Dic₁₀₁	—	2	2	no	—
57	#13108053	—	2	2	no	—
58	#13108053	—	2	2	no	—
59	#13108053	—	2	4	no	—
60	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₁	—	2	2	no	—
66	C₄⋊C₄×C₁₀₁	—	2	2	no	—
67	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
68	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
69	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
70	C₂²×D₁₀₁	#41	1	2	yes	—
71	C₂²×D₁₀₁	#41	1	4	yes	—
72	C₂²×D₁₀₁	#41	1	4	yes	—
73	C₂×C₄×C₁₀₁	—	2	1	no	—
74	C₂×C₄×C₁₀₁	—	2	1	no	—
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	—	2	1	no	—
77	C₂×C₄×C₁₀₁	—	2	2	no	—
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	4	no	—
80	C₂×C₄×C₁₀₁	—	2	2	no	—
81	C₂×C₄×C₁₀₁	—	2	2	no	—
82	C₂×C₄×C₁₀₁	—	2	2	no	—
83	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
85	C₂×Dic₁₀₁	—	2	1	no	—
86	C₂×Dic₁₀₁	—	2	1	no	—
87	C₂×Dic₁₀₁	—	2	2	no	—
88	C₂×Dic₁₀₁	—	2	1	no	—
89	C₂×Dic₁₀₁	—	2	2	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	—	2	4	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	2	2	no	—
95	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
96	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
97	C₄×D₁₀₁	—	2	8	no	—
98	C₄×D₁₀₁	—	2	8	no	—
99	C₄×D₁₀₁	—	4	4	no	—
100	C₄×D₁₀₁	—	4	8	no	—
101	C₄×D₁₀₁	—	4	8	no	—
102	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
104	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
105	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
106	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
107	C₂×D₁₀₁	C₂²⋊Q₈	1	4	yes	—
108	C₂×D₁₀₁	#123	1	4	yes	—
109	C₂×D₁₀₁	#124	1	4	yes	—
110	C₂×D₁₀₁	#125	1	8	yes	—
111	C₂×D₁₀₁	#126	1	8	yes	—
112	C₄×C₁₀₁	—	2	2	no	—
113	C₄×C₁₀₁	—	2	2	no	—
114	C₄×C₁₀₁	—	4	1	no	—
115	C₄×C₁₀₁	—	4	2	no	—
116	C₄×C₁₀₁	—	4	2	no	—
117	Dic₁₀₁	—	2	2	no	—
118	Dic₁₀₁	—	2	2	no	—
119	Dic₁₀₁	—	4	1	no	—
120	Dic₁₀₁	—	4	2	no	—
121	Dic₁₀₁	—	4	2	no	—
122	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
124	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
125	C₂×C₁₀₁	C₂×#125	1	2	yes	—
126	C₂×C₁₀₁	C₂×#126	1	2	yes	—
127	D₁₀₁	#400	1	8	yes	direct product factor
128	C₂×#400	—	101	1	no	maximal, Sylow
129	C₁₀₁	C₂×#400	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₄⋊C₄	—	101	1	no	—
131	C₂²×C₄⋊C₄	—	101	1	no	—
132	#400	—	101	1	no	—
133	#400	—	101	1	no	—
134	#400	—	101	1	no	—
135	#400	—	101	2	no	—
136	#400	—	101	2	no	—
137	#400	D₁₀₁	1	1	yes	characteristic, direct product factor
138	C₂×#95	—	101	1	no	—
139	C₂×#95	—	101	2	no	—
140	C₂×#95	—	101	2	no	—
141	C₂³×C₄	—	101	1	no	—
142	C₂³×C₄	—	101	1	no	—
143	C₂³×C₄	—	101	1	no	—
144	C₂³×C₄	—	101	2	no	—
145	C₂³×C₄	—	101	2	no	—
146	C₂×C₄⋊C₄	—	101	1	no	—
147	C₂×C₄⋊C₄	—	101	2	no	—
148	C₂×C₄⋊C₄	—	101	1	no	—
149	C₂×C₄⋊C₄	—	101	1	no	—
150	C₂×C₄⋊C₄	—	101	1	no	—
151	C₂×C₄⋊C₄	—	202	4	no	—
152	C₂×C₄⋊C₄	—	202	4	no	—
153	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
155	#95	—	101	2	no	—
156	#95	—	101	2	no	—
157	#95	—	101	2	no	—
158	#95	—	101	1	no	—
159	#95	—	101	2	no	—
160	#95	—	101	2	no	—
161	#95	—	101	2	no	—
162	#95	—	101	2	no	—
163	#95	C₂×D₁₀₁	1	1	yes	characteristic
164	#95	C₂×D₁₀₁	1	2	yes	—
165	#95	C₂×D₁₀₁	1	2	yes	—
166	C₂⁴	—	101	1	no	—
167	C₂²×C₄	—	101	1	no	—
168	C₂²×C₄	—	101	1	no	—
169	C₂²×C₄	—	101	4	no	—
170	C₂²×C₄	—	101	4	no	—
171	C₂²×C₄	—	101	2	no	—
172	C₂²×C₄	—	101	1	no	—
173	C₂²×C₄	—	101	2	no	—
174	C₂²×C₄	—	202	2	no	—
175	C₂²×C₄	—	202	4	no	—
176	C₂²×C₄	—	202	4	no	—
177	C₂²×C₄	—	202	8	no	—
178	C₂²×C₄	—	202	4	no	—
179	C₂²×C₄	—	202	4	no	—
180	C₂²×C₄	—	202	4	no	—
181	C₂²×C₄	—	202	2	no	—
182	C₂²×C₄	—	202	4	no	—
183	C₂²×C₄	—	202	2	no	—
184	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
185	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
187	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	202	4	no	—
192	C₄⋊C₄	—	202	2	no	—
193	C₄⋊C₄	—	202	2	no	—
194	C₄⋊C₄	—	202	2	no	—
195	C₄⋊C₄	—	202	2	no	—
196	C₂³	—	101	2	no	—
197	C₂³	—	101	2	no	—
198	C₂³	—	101	2	no	—
199	C₂³	—	101	4	no	—
200	C₂³	—	101	4	no	—
201	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	101	2	no	—
213	C₂×C₄	—	101	2	no	—
214	C₂×C₄	—	404	4	no	—
215	C₂×C₄	—	404	8	no	—
216	C₂×C₄	—	404	8	no	—
217	C₂×C₄	—	202	8	no	—
218	C₂×C₄	—	202	8	no	—
219	C₂×C₄	—	202	1	no	—
220	C₂×C₄	—	202	2	no	—
221	C₂×C₄	—	202	2	no	—
222	C₂×C₄	—	202	4	no	—
223	C₂×C₄	—	202	2	no	—
224	C₂×C₄	—	202	2	no	—
225	C₂×C₄	—	202	2	no	—
226	C₂×C₄	—	202	1	no	—
227	C₂×C₄	—	202	1	no	—
228	C₂×C₄	—	202	2	no	—
229	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
230	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
231	C₂²	—	101	4	no	—
232	C₂²	—	101	4	no	—
233	C₂²	—	101	4	no	—
234	C₂²	—	101	8	no	—
235	C₂²	—	101	8	no	—
236	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
237	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
238	C₂²	#41×D₁₀₁	1	1	yes	characteristic
239	C₂²	#41×D₁₀₁	1	2	yes	—
240	C₂²	#41×D₁₀₁	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	—	404	1	no	—
247	C₄	—	404	2	no	—
248	C₄	—	404	2	no	—
249	C₄	—	202	2	no	—
250	C₄	—	202	2	no	—
251	C₂	—	101	8	no	—
252	C₂	C₂²⋊Q₈×D₁₀₁	1	1	yes	characteristic
253	C₂	#123×D₁₀₁	1	1	yes	characteristic
254	C₂	#124×D₁₀₁	1	1	yes	characteristic
255	C₂	#125×D₁₀₁	1	2	yes	—
256	C₂	#126×D₁₀₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	#400×C₁₀₁	C₂
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	#400	D₁₀₁
hypocenter	C₁₀₁	C₂×#400
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #400

Complements

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

N	H ≅ G/N	type
3. #20265119	C₂	semidirect
4. #20265118	C₂	semidirect
5. #20265117	C₂	semidirect
6. #20265116	C₂	semidirect
7. #20265115	C₂	semidirect
11. #400×C₁₀₁	C₂	semidirect
127. D₁₀₁	#400	direct (N × H)
129. C₁₀₁	C₂×#400	semidirect
137. #400	D₁₀₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×D₁₀₁	2
#20265119	2
#20265118	2
#20265117	2
#20265116	2
#20265115	2
#95×D₁₀₁	2
#400×C₁₀₁	2
C₂×#400	101

Subgroups — #400×D101 #27174382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #400×D₁₀₁ #27174382