Subgroups — #27174013

Order 12928.

504 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692213	C₂	1	1	yes	maximal, characteristic
2	#20265075	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265073	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265068	C₂	1	2	yes	maximal
5	#20265066	C₂	1	2	yes	maximal
6	#20265041	C₂	1	2	yes	maximal, complemented
7	#20265034	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13108051	C₂	1	1	yes	maximal, characteristic
10	#20265023	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
13	#15692213	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	2	yes	—
18	C₂×#13108051	C₂²	1	1	yes	characteristic
19	C₂×#13108051	C₂²	1	1	yes	characteristic
20	C₂×#13108051	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	#15692162	C₂²	1	1	yes	characteristic
26	#15692162	C₂²	1	2	yes	—
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	1	yes	characteristic
30	#15692162	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
35	#95×C₁₀₁	C₂²	1	2	yes	—
36	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₀₁	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	C₂×C₄×D₁₀₁	—	2	2	no	—
43	C₂×C₄×D₁₀₁	—	2	2	no	—
44	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₁₀₁	—	2	2	no	—
46	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
47	#13108053	—	4	2	no	—
48	#13108053	—	4	2	no	—
49	#13108051	—	2	2	no	—
50	#13108051	—	2	2	no	—
51	#13108051	—	4	2	no	—
52	#13108051	C₂×C₄	1	8	yes	—
53	C₄×Dic₁₀₁	—	2	2	no	—
54	C₄×Dic₁₀₁	—	2	4	no	—
55	C₄×Dic₁₀₁	C₂×C₄	1	8	yes	—
56	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₀₁	—	2	2	no	—
62	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
63	C₄²×C₁₀₁	—	2	4	no	—
64	C₄²×C₁₀₁	—	2	2	no	—
65	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
66	C₂²×D₁₀₁	—	2	1	no	—
67	C₂²×D₁₀₁	—	2	1	no	—
68	C₂²×D₁₀₁	—	2	1	no	—
69	C₂²×D₁₀₁	—	2	1	no	—
70	C₂²×D₁₀₁	—	2	1	no	—
71	C₂²×D₁₀₁	—	2	1	no	—
72	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₀₁	—	2	2	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	1	no	—
79	C₂×C₄×C₁₀₁	—	2	2	no	—
80	C₂×C₄×C₁₀₁	—	2	1	no	—
81	C₂×C₄×C₁₀₁	—	2	2	no	—
82	C₂×C₄×C₁₀₁	—	2	1	no	—
83	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₀₁	#41	1	2	yes	—
86	C₂×C₄×C₁₀₁	#41	1	2	yes	—
87	C₂×C₄×C₁₀₁	#41	1	2	yes	—
88	C₂×Dic₁₀₁	—	2	1	no	—
89	C₂×Dic₁₀₁	—	2	1	no	—
90	C₂×Dic₁₀₁	—	2	1	no	—
91	C₂×Dic₁₀₁	—	2	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	1	no	—
94	C₂×Dic₁₀₁	—	2	2	no	—
95	C₂×Dic₁₀₁	—	2	1	no	—
96	C₂×Dic₁₀₁	—	2	1	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	1	no	—
100	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
101	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
102	C₂×Dic₁₀₁	#41	1	2	yes	—
103	C₄×D₁₀₁	—	2	8	no	—
104	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
106	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
107	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
108	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×D₁₀₁	—	2	2	no	—
112	C₂×D₁₀₁	—	2	2	no	—
113	C₂×D₁₀₁	—	2	2	no	—
114	C₂×D₁₀₁	—	4	1	no	—
115	C₂×D₁₀₁	—	4	1	no	—
116	C₂×D₁₀₁	—	4	1	no	—
117	C₂×D₁₀₁	—	4	1	no	—
118	C₄×C₁₀₁	—	2	4	no	—
119	C₄×C₁₀₁	—	2	2	no	—
120	C₄×C₁₀₁	—	4	1	no	—
121	C₄×C₁₀₁	—	4	2	no	—
122	C₄×C₁₀₁	#117	1	4	yes	—
123	Dic₁₀₁	—	2	2	no	—
124	Dic₁₀₁	—	2	4	no	—
125	Dic₁₀₁	—	2	2	no	—
126	Dic₁₀₁	—	4	2	no	—
127	Dic₁₀₁	—	4	1	no	—
128	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
129	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
130	C₂×C₁₀₁	#520	1	1	yes	characteristic
131	C₂×C₁₀₁	#552	1	1	yes	characteristic
132	C₂×C₁₀₁	#552	1	1	yes	characteristic
133	C₂×C₁₀₁	#555	1	1	yes	characteristic
134	C₂×C₁₀₁	#555	1	1	yes	characteristic
135	D₁₀₁	—	4	2	no	—
136	#2346	—	101	1	no	maximal, Sylow
137	C₁₀₁	#2346	1	1	yes	characteristic, complemented, Sylow
138	C₂×#117	—	101	1	no	—
139	#389	—	101	1	no	—
140	#389	—	101	1	no	—
141	#388	—	101	2	no	—
142	#388	—	101	2	no	—
143	#382	—	101	2	no	—
144	#380	—	101	1	no	—
145	C₄×C₄⋊C₄	—	101	1	no	—
146	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
147	C₄×C₂²⋊C₄	—	101	1	no	—
148	#376	—	101	2	no	—
149	C₂³×C₄	—	101	1	no	—
150	#117	—	101	8	no	—
151	C₂×C₄⋊C₄	—	101	1	no	—
152	C₂×C₄⋊C₄	—	101	2	no	—
153	C₂×C₄⋊C₄	—	101	2	no	—
154	C₂×C₄⋊C₄	—	101	1	no	—
155	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	—	101	2	no	—
158	C₂×C₂²⋊C₄	—	101	2	no	—
159	C₂×C₂²⋊C₄	—	101	1	no	—
160	C₂×C₂²⋊C₄	—	101	1	no	—
161	C₂×C₄²	—	101	2	no	—
162	C₂×C₄²	—	101	2	no	—
163	C₂×C₄²	—	101	1	no	—
164	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
165	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
166	#95	—	101	1	no	—
167	#95	—	101	2	no	—
168	#95	—	101	2	no	—
169	#95	—	101	2	no	—
170	#95	—	101	1	no	—
171	#95	—	101	2	no	—
172	#95	C₂×D₁₀₁	1	2	yes	—
173	C₂⁴	—	101	1	no	—
174	C₂²×C₄	—	101	1	no	—
175	C₂²×C₄	—	101	1	no	—
176	C₂²×C₄	—	101	2	no	—
177	C₂²×C₄	—	101	4	no	—
178	C₂²×C₄	—	101	2	no	—
179	C₂²×C₄	—	101	1	no	—
180	C₂²×C₄	—	202	2	no	—
181	C₂²×C₄	—	202	2	no	—
182	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
183	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
185	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	101	4	no	—
189	C₄⋊C₄	—	404	2	no	—
190	C₄⋊C₄	—	404	2	no	—
191	C₄⋊C₄	—	202	2	no	—
192	C₄⋊C₄	C₄×D₁₀₁	1	4	yes	—
193	C₂²⋊C₄	—	101	8	no	—
194	C₂²⋊C₄	—	404	2	no	—
195	C₂²⋊C₄	—	202	2	no	—
196	C₂²⋊C₄	—	202	2	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	101	8	no	—
200	C₄²	—	202	4	no	—
201	C₄²	—	202	2	no	—
202	C₂³	—	101	2	no	—
203	C₂³	—	202	1	no	—
204	C₂³	—	202	1	no	—
205	C₂³	—	202	1	no	—
206	C₂³	—	202	1	no	—
207	C₂³	—	202	1	no	—
208	C₂³	—	202	1	no	—
209	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	101	2	no	—
221	C₂×C₄	—	101	4	no	—
222	C₂×C₄	—	101	2	no	—
223	C₂×C₄	—	202	2	no	—
224	C₂×C₄	—	202	1	no	—
225	C₂×C₄	—	202	1	no	—
226	C₂×C₄	—	202	2	no	—
227	C₂×C₄	—	202	2	no	—
228	C₂×C₄	—	202	1	no	—
229	C₂×C₄	—	202	1	no	—
230	C₂×C₄	—	202	1	no	—
231	C₂×C₄	—	202	1	no	—
232	C₂×C₄	—	202	2	no	—
233	C₂×C₄	—	202	1	no	—
234	C₂×C₄	—	202	2	no	—
235	C₂×C₄	—	202	8	no	—
236	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
237	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
238	C₂×C₄	#13108073	1	2	yes	—
239	C₂×C₄	#13108074	1	2	yes	—
240	C₂×C₄	#13108074	1	2	yes	—
241	C₂²	—	404	1	no	—
242	C₂²	—	404	1	no	—
243	C₂²	—	404	1	no	—
244	C₂²	—	404	1	no	—
245	C₂²	—	202	2	no	—
246	C₂²	—	202	2	no	—
247	C₂²	—	202	2	no	—
248	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
249	C₂²	#41×D₁₀₁	1	1	yes	characteristic
250	C₂²	C₂×#13108073	1	1	yes	characteristic
251	C₂²	C₂×#13108074	1	1	yes	characteristic
252	C₂²	C₂×#13108074	1	1	yes	characteristic
253	C₂²	#15692301	1	1	yes	characteristic
254	C₂²	#15692303	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	—	404	2	no	—
260	C₄	—	404	1	no	—
261	C₄	—	202	2	no	—
262	C₄	—	202	2	no	—
263	C₄	—	202	4	no	—
264	C₄	#15692212	1	4	yes	—
265	C₂	—	404	2	no	—
266	C₂	C₂×#15692212	1	1	yes	characteristic
267	C₂	C₄×#13108073	1	1	yes	characteristic
268	C₂	#20265659	1	1	yes	characteristic
269	C₂	#20265862	1	1	yes	characteristic
270	C₂	#20265868	1	1	yes	characteristic
271	C₂	#20265890	1	1	yes	characteristic
272	C₂	#20265892	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#2346
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20265075	C₂	semidirect
3. #20265073	C₂	semidirect
6. #20265041	C₂	semidirect
8. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
10. #20265023	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₀₁	C₂	semidirect
13. #15692213	C₄	semidirect
137. C₁₀₁	#2346	semidirect
146. C₄×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×#15692213	2
#20265075	2
#20265073	2
#20265068	2
#20265066	2
#20265041	2
#20265034	2
C₄⋊C₄×Dic₁₀₁	2
C₄×#13108051	2
#20265023	2
C₄×C₄⋊C₄×C₁₀₁	2
#2346	101