Subgroups — #27174106

Order 12928.

560 conjugacy classes of subgroups.

The subgroup lattice has 272 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	1	yes	maximal, characteristic
2	#20265077	C₂	1	2	yes	maximal
3	#20265071	C₂	1	2	yes	maximal, complemented
4	#20265068	C₂	1	2	yes	maximal, complemented
5	#20265041	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265040	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265039	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13108053	C₂	1	2	yes	maximal, complemented
9	#20265028	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265024	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₁₀₁	C₄	1	8	yes	complemented
15	C₂×#13108053	C₂²	1	2	yes	—
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#13108051	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	2	yes	—
22	C₂×#13108051	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
26	#15692162	C₂²	1	1	yes	characteristic
27	#15692162	C₂²	1	1	yes	characteristic
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
34	#95×C₁₀₁	C₂²	1	1	yes	characteristic
35	#95×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
38	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
39	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	4	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₁₀₁	—	4	2	no	—
46	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
47	#13108053	—	2	4	no	—
48	#13108053	—	2	4	no	—
49	#13108053	C₂×C₄	1	4	yes	—
50	C₂²⋊Dic₁₀₁	—	4	2	no	—
51	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
52	#13108051	—	2	4	no	—
53	#13108051	—	4	2	no	—
54	#13108051	C₂×C₄	1	4	yes	—
55	C₄×Dic₁₀₁	—	2	4	no	—
56	C₄×Dic₁₀₁	—	4	2	no	—
57	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
58	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	—	4	2	no	—
60	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
61	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
66	C₄²×C₁₀₁	—	4	2	no	—
67	C₂³×C₁₀₁	—	2	2	no	—
68	C₂³×C₁₀₁	—	2	1	no	—
69	C₂³×C₁₀₁	—	2	1	no	—
70	C₂³×C₁₀₁	—	2	2	no	—
71	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
72	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
73	C₂²×D₁₀₁	—	2	2	no	—
74	C₂²×D₁₀₁	—	2	1	no	—
75	C₂²×D₁₀₁	—	2	1	no	—
76	C₂²×D₁₀₁	—	2	2	no	—
77	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	2	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	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	4	4	no	—
86	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₀₁	#41	1	4	yes	—
89	C₂²⋊D₁₀₁	—	2	8	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	—	2	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	2	2	no	—
95	C₂×Dic₁₀₁	—	2	2	no	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
101	C₂×Dic₁₀₁	#41	1	2	yes	—
102	C₂×Dic₁₀₁	#41	1	2	yes	—
103	C₄×D₁₀₁	—	4	4	no	—
104	C₂²×C₁₀₁	—	2	2	no	—
105	C₂²×C₁₀₁	—	2	2	no	—
106	C₂²×C₁₀₁	—	2	2	no	—
107	C₂²×C₁₀₁	—	4	2	no	—
108	C₂²×C₁₀₁	—	4	2	no	—
109	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
110	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
111	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
112	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
113	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
114	C₂×D₁₀₁	—	2	2	no	—
115	C₂×D₁₀₁	—	2	2	no	—
116	C₂×D₁₀₁	—	2	2	no	—
117	C₂×D₁₀₁	—	4	2	no	—
118	C₂×D₁₀₁	—	4	2	no	—
119	C₄×C₁₀₁	—	2	2	no	—
120	C₄×C₁₀₁	—	2	2	no	—
121	C₄×C₁₀₁	—	2	4	no	—
122	C₄×C₁₀₁	—	4	2	no	—
123	Dic₁₀₁	—	2	2	no	—
124	Dic₁₀₁	—	2	2	no	—
125	Dic₁₀₁	—	4	2	no	—
126	Dic₁₀₁	—	4	2	no	—
127	Dic₁₀₁	#117	1	4	yes	—
128	C₂×C₁₀₁	—	4	2	no	—
129	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
130	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
131	C₂×C₁₀₁	#518	1	1	yes	characteristic
132	C₂×C₁₀₁	#553	1	2	yes	—
133	C₂×C₁₀₁	#559	1	2	yes	—
134	D₁₀₁	—	4	2	no	—
135	#2357	—	101	1	no	maximal, Sylow
136	C₁₀₁	#2357	1	1	yes	characteristic, complemented, Sylow
137	C₂×C₄×D₄	—	101	1	no	—
138	#390	—	101	2	no	—
139	#388	—	101	2	no	—
140	#388	—	101	2	no	—
141	#382	—	101	1	no	—
142	#381	—	101	1	no	—
143	#381	—	101	1	no	—
144	C₄×C₄⋊C₄	—	101	2	no	—
145	C₄×C₄⋊C₄	—	101	1	no	—
146	C₄×C₂²⋊C₄	—	101	1	no	—
147	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
148	C₂²×D₄	—	101	1	no	—
149	C₂³×C₄	—	101	1	no	—
150	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
151	C₄×D₄	—	101	8	no	—
152	C₂×C₄⋊C₄	—	101	1	no	—
153	C₂×C₄⋊C₄	—	101	2	no	—
154	C₂×C₄⋊C₄	—	101	2	no	—
155	C₂×C₂²⋊C₄	—	101	1	no	—
156	C₂×C₂²⋊C₄	—	101	1	no	—
157	C₂×C₂²⋊C₄	—	101	2	no	—
158	C₂×C₂²⋊C₄	—	101	2	no	—
159	C₂×C₂²⋊C₄	—	101	2	no	—
160	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
161	C₂×C₄²	—	101	1	no	—
162	C₂×C₄²	—	101	2	no	—
163	C₂×C₄²	—	101	2	no	—
164	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
165	#95	—	101	1	no	—
166	#95	—	101	1	no	—
167	#95	—	101	2	no	—
168	#95	—	101	2	no	—
169	#95	—	101	2	no	—
170	#95	C₂×D₁₀₁	1	1	yes	characteristic
171	#95	C₂×D₁₀₁	1	1	yes	characteristic
172	C₂⁴	—	101	1	no	—
173	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
174	C₂×D₄	—	101	4	no	—
175	C₂×D₄	—	202	2	no	—
176	C₂×D₄	—	202	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	101	1	no	—
179	C₂²×C₄	—	101	1	no	—
180	C₂²×C₄	—	101	1	no	—
181	C₂²×C₄	—	101	4	no	—
182	C₂²×C₄	—	101	2	no	—
183	C₂²×C₄	—	101	2	no	—
184	C₂²×C₄	—	404	2	no	—
185	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
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	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
190	C₄⋊C₄	—	101	4	no	—
191	C₄⋊C₄	—	202	4	no	—
192	C₄⋊C₄	—	202	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	101	4	no	—
195	C₂²⋊C₄	—	101	4	no	—
196	C₂²⋊C₄	—	404	2	no	—
197	C₂²⋊C₄	—	404	2	no	—
198	C₂²⋊C₄	—	202	4	no	—
199	C₄²	—	4	2	no	—
200	C₄²	—	101	4	no	—
201	C₄²	—	404	2	no	—
202	C₄²	—	202	4	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	101	2	no	—
208	C₂³	—	202	1	no	—
209	C₂³	—	202	1	no	—
210	C₂³	—	202	2	no	—
211	C₂³	—	202	2	no	—
212	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
213	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
214	D₄	—	202	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	101	2	no	—
224	C₂×C₄	—	101	2	no	—
225	C₂×C₄	—	101	2	no	—
226	C₂×C₄	—	101	2	no	—
227	C₂×C₄	—	101	2	no	—
228	C₂×C₄	—	404	4	no	—
229	C₂×C₄	—	202	2	no	—
230	C₂×C₄	—	202	2	no	—
231	C₂×C₄	—	202	2	no	—
232	C₂×C₄	—	202	2	no	—
233	C₂×C₄	—	202	2	no	—
234	C₂×C₄	—	202	2	no	—
235	C₂×C₄	—	202	2	no	—
236	C₂×C₄	—	202	2	no	—
237	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
238	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
239	C₂×C₄	#13108074	1	4	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	404	2	no	—
246	C₂²	—	404	2	no	—
247	C₂²	—	202	2	no	—
248	C₂²	—	202	2	no	—
249	C₂²	—	202	2	no	—
250	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
251	C₂²	#41×D₁₀₁	1	1	yes	characteristic
252	C₂²	#41×D₁₀₁	1	1	yes	characteristic
253	C₂²	C₂×#13108074	1	2	yes	—
254	C₂²	#15692300	1	2	yes	—
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	101	4	no	—
260	C₄	—	404	2	no	—
261	C₄	—	404	2	no	—
262	C₄	—	202	2	no	—
263	C₄	—	202	2	no	—
264	C₂	—	4	2	no	—
265	C₂	—	404	2	no	—
266	C₂	#117×D₁₀₁	1	1	yes	characteristic
267	C₂	C₄×#13108074	1	1	yes	characteristic
268	C₂	#20265650	1	1	yes	characteristic
269	C₂	#20265876	1	2	yes	—
270	C₂	#20265910	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×C₂²⋊D₁₀₁	C₂²
hypercenter	C₄×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#2357
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #20265071	C₂	semidirect
4. #20265068	C₂	semidirect
5. #20265041	C₂	semidirect
6. #20265040	C₂	semidirect
7. #20265039	C₂	semidirect
8. C₄×#13108053	C₂	semidirect
9. #20265028	C₂	semidirect
10. #20265024	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₀₁	C₂	semidirect
14. C₄×C₂²⋊D₁₀₁	C₄	semidirect
136. C₁₀₁	#2357	semidirect
147. C₄×C₂²⋊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
#20265077	2
#20265071	2
#20265068	2
#20265041	2
#20265040	2
#20265039	2
C₄×#13108053	2
#20265028	2
#20265024	2
C₄×C₂²⋊C₄×C₁₀₁	2
#2357	101