Subgroups — #27174065

Order 12928.

588 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×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic
2	#20265066	C₂	1	2	yes	maximal, complemented
3	#20265062	C₂	1	2	yes	maximal, complemented
4	#20265055	C₂	1	2	yes	maximal
5	#20265039	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265036	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265024	C₂	1	2	yes	maximal, complemented
8	#20265024	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265023	C₂	1	1	yes	maximal, characteristic, complemented
10	#95×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#383×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₄×D₄₀₄	C₄	1	8	yes	complemented
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×#13108053	C₂²	1	2	yes	—
19	C₂×#13108051	C₂²	1	2	yes	—
20	C₂×#13108051	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	2	yes	—
22	C₂×#13108051	C₂²	1	1	yes	characteristic
23	C₂×#13108051	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
26	#15692162	C₂²	1	1	yes	characteristic
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	1	yes	characteristic
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
32	#95×C₁₀₁	C₂²	1	1	yes	characteristic
33	#95×C₁₀₁	C₂²	1	1	yes	characteristic
34	#95×C₁₀₁	C₂²	1	2	yes	—
35	#95×C₁₀₁	C₂²	1	2	yes	—
36	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
37	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
38	C₂×D₄₀₄	—	2	2	no	—
39	C₂×D₄₀₄	—	2	2	no	—
40	C₂×D₄₀₄	C₂×C₄	1	4	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	2	yes	—
45	C₂×C₄×D₁₀₁	—	2	4	no	—
46	C₂×C₄×D₁₀₁	—	2	4	no	—
47	C₂×C₄×D₁₀₁	—	4	2	no	—
48	C₂×C₄×D₁₀₁	—	4	2	no	—
49	C₂×C₄×D₁₀₁	—	4	2	no	—
50	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
51	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
52	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
53	#13108053	—	4	2	no	—
54	#13108051	—	2	4	no	—
55	#13108051	—	2	4	no	—
56	#13108051	—	4	2	no	—
57	#13108051	C₂×C₄	1	4	yes	—
58	#13108051	C₂×C₄	1	4	yes	complemented
59	C₄×Dic₁₀₁	—	2	4	no	—
60	C₄×Dic₁₀₁	—	4	2	no	—
61	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
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	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₄²×C₁₀₁	C₂×C₄	1	4	yes	complemented
67	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
68	C₂²×D₁₀₁	—	2	2	no	—
69	C₂²×D₁₀₁	—	2	2	no	—
70	C₂²×D₁₀₁	—	2	2	no	—
71	C₂²×D₁₀₁	—	2	1	no	—
72	C₂²×D₁₀₁	—	2	1	no	—
73	C₂²×D₁₀₁	—	2	2	no	—
74	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
75	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
76	C₂²×D₁₀₁	#41	1	2	yes	—
77	C₂²×D₁₀₁	#41	1	2	yes	—
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	1	no	—
80	C₂×C₄×C₁₀₁	—	2	1	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	—	2	2	no	—
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
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	#41	1	4	yes	—
101	D₄₀₄	—	2	8	no	—
102	C₄×D₁₀₁	—	4	4	no	—
103	C₄×D₁₀₁	—	4	4	no	—
104	C₄×D₁₀₁	—	4	8	no	—
105	C₂²×C₁₀₁	C₂³×C₄	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	2	yes	—
109	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
110	C₂×D₁₀₁	—	2	2	no	—
111	C₂×D₁₀₁	—	2	2	no	—
112	C₂×D₁₀₁	—	2	4	no	—
113	C₂×D₁₀₁	—	2	4	no	—
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	C₂×D₁₀₁	#117	1	4	yes	—
120	C₄×C₁₀₁	—	2	2	no	—
121	C₄×C₁₀₁	—	2	2	no	—
122	C₄×C₁₀₁	—	2	2	no	—
123	C₄×C₁₀₁	—	4	2	no	—
124	C₄×C₁₀₁	—	4	2	no	—
125	Dic₁₀₁	—	2	2	no	—
126	Dic₁₀₁	—	2	2	no	—
127	Dic₁₀₁	—	2	4	no	—
128	Dic₁₀₁	—	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	#551	1	2	yes	—
133	C₂×C₁₀₁	#553	1	2	yes	—
134	D₁₀₁	—	2	4	no	—
135	D₁₀₁	—	4	2	no	—
136	#2353	—	101	1	no	maximal, Sylow
137	C₁₀₁	#2353	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄×D₄	—	101	1	no	—
139	#388	—	101	2	no	—
140	#387	—	101	2	no	—
141	#386	—	101	2	no	—
142	#383	D₁₀₁	1	1	yes	characteristic, complemented
143	#381	—	101	1	no	—
144	#380	—	101	1	no	—
145	C₄×C₂²⋊C₄	—	101	1	no	—
146	C₄×C₂²⋊C₄	—	101	2	no	—
147	#376	—	101	1	no	—
148	C₂×#95	—	101	1	no	—
149	C₂²×D₄	—	101	1	no	—
150	C₂³×C₄	—	101	1	no	—
151	C₂³×C₄	—	101	1	no	—
152	C₂³×C₄	—	101	2	no	—
153	C₄×D₄	—	101	8	no	—
154	C₂×C₄⋊C₄	—	101	1	no	—
155	C₂×C₄⋊C₄	—	101	2	no	—
156	C₂×C₂²⋊C₄	—	101	1	no	—
157	C₂×C₂²⋊C₄	—	101	1	no	—
158	C₂×C₂²⋊C₄	—	101	2	no	—
159	C₂×C₂²⋊C₄	—	101	2	no	—
160	C₂×C₂²⋊C₄	—	101	2	no	—
161	C₂×C₄²	—	101	2	no	—
162	C₂×C₄²	—	101	2	no	—
163	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
164	#95	—	101	1	no	—
165	#95	—	101	2	no	—
166	#95	—	101	1	no	—
167	#95	—	101	2	no	—
168	#95	—	101	2	no	—
169	#95	C₂×D₁₀₁	1	1	yes	characteristic
170	#95	C₂×D₁₀₁	1	1	yes	characteristic
171	#95	C₂×D₁₀₁	1	2	yes	—
172	#95	C₂×D₁₀₁	1	2	yes	—
173	C₂⁴	—	101	1	no	—
174	C₂⁴	—	101	1	no	—
175	C₂×D₄	—	101	4	no	—
176	C₂×D₄	—	202	2	no	—
177	C₂×D₄	—	202	2	no	—
178	C₂²×C₄	—	101	2	no	—
179	C₂²×C₄	—	101	1	no	—
180	C₂²×C₄	—	101	1	no	—
181	C₂²×C₄	—	101	4	no	—
182	C₂²×C₄	—	101	4	no	—
183	C₂²×C₄	—	101	2	no	—
184	C₂²×C₄	—	404	2	no	—
185	C₂²×C₄	—	404	2	no	—
186	C₂²×C₄	—	404	2	no	—
187	C₂²×C₄	—	202	4	no	—
188	C₂²×C₄	—	202	4	no	—
189	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
190	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
191	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
194	C₄⋊C₄	—	101	4	no	—
195	C₄⋊C₄	—	404	2	no	—
196	C₂²⋊C₄	—	101	4	no	—
197	C₂²⋊C₄	—	101	4	no	—
198	C₂²⋊C₄	—	404	2	no	—
199	C₂²⋊C₄	—	202	4	no	—
200	C₂²⋊C₄	—	202	4	no	—
201	C₄²	—	404	2	no	—
202	C₄²	—	202	4	no	—
203	C₄²	C₄×D₁₀₁	1	4	yes	complemented
204	C₂³	—	101	2	no	—
205	C₂³	—	101	2	no	—
206	C₂³	—	101	2	no	—
207	C₂³	—	101	2	no	—
208	C₂³	—	202	2	no	—
209	C₂³	—	202	1	no	—
210	C₂³	—	202	1	no	—
211	C₂³	—	202	2	no	—
212	C₂³	—	202	2	no	—
213	C₂³	—	202	2	no	—
214	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
215	D₄	—	202	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	101	2	no	—
227	C₂×C₄	—	101	2	no	—
228	C₂×C₄	—	101	4	no	—
229	C₂×C₄	—	404	4	no	—
230	C₂×C₄	—	404	4	no	—
231	C₂×C₄	—	404	8	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	—	202	2	no	—
238	C₂×C₄	—	202	2	no	—
239	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
240	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
242	C₂²	—	101	4	no	—
243	C₂²	—	404	2	no	—
244	C₂²	—	404	2	no	—
245	C₂²	—	202	2	no	—
246	C₂²	—	202	2	no	—
247	C₂²	—	202	2	no	—
248	C₂²	—	202	4	no	—
249	C₂²	—	202	4	no	—
250	C₂²	—	202	2	no	—
251	C₂²	—	202	2	no	—
252	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
253	C₂²	#41×D₁₀₁	1	1	yes	characteristic
254	C₂²	#41×D₁₀₁	1	2	yes	—
255	C₂²	#41×D₁₀₁	1	1	yes	characteristic
256	C₂²	#15692301	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	404	2	no	—
263	C₄	—	202	4	no	—
264	C₄	—	202	2	no	—
265	C₄	—	202	2	no	—
266	C₂	—	404	2	no	—
267	C₂	—	202	4	no	—
268	C₂	#117×D₁₀₁	1	1	yes	characteristic
269	C₂	#20265645	1	1	yes	characteristic
270	C₂	#20265651	1	1	yes	characteristic
271	C₂	#20265852	1	2	yes	—
272	C₂	#20265878	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#383×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×D₄₀₄	C₂²
hypercenter	#383	D₁₀₁
hypocenter	C₁₀₁	#2353
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #383

Complements

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

N	H ≅ G/N	type
2. #20265066	C₂	semidirect
3. #20265062	C₂	semidirect
5. #20265039	C₂	semidirect
6. #20265036	C₂	semidirect
7. #20265024	C₂	semidirect
8. #20265024	C₂	semidirect
9. #20265023	C₂	semidirect
10. #95×D₁₀₁	C₂	semidirect
11. #383×C₁₀₁	C₂	semidirect
16. C₄×D₄₀₄	C₄	semidirect
51. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
52. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
58. #13108051	C₂×C₄	semidirect
66. C₄²×C₁₀₁	C₂×C₄	semidirect
137. C₁₀₁	#2353	semidirect
142. #383	D₁₀₁	semidirect
203. C₄²	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×D₄₀₄	2
#20265066	2
#20265062	2
#20265055	2
#20265039	2
#20265036	2
#20265024	2
#20265023	2
#95×D₁₀₁	2
#383×C₁₀₁	2
#2353	101