Subgroups — #27174041

Order 12928.

704 conjugacy classes of subgroups.

The subgroup lattice has 298 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	C₂×#15692217	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692217	C₂	1	1	yes	maximal, characteristic
4	#20265076	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265051	C₂	1	2	yes	maximal, complemented
6	#20265048	C₂	1	2	yes	maximal, complemented
7	#20265042	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265034	C₂	1	2	yes	maximal, complemented
9	C₄×#13108053	C₂	1	1	yes	maximal, characteristic, complemented
10	#95×D₁₀₁	C₂	1	2	yes	maximal, complemented
11	#389×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	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
16	C₄×D₄₀₄	—	2	4	no	—
17	#15692217	—	2	4	no	—
18	#15692217	C₄	1	8	yes	complemented
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×#13108053	C₂²	1	1	yes	characteristic
22	C₂×#13108053	C₂²	1	1	yes	characteristic
23	C₂×#13108051	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
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	2	yes	—
31	#15692162	C₂²	1	1	yes	characteristic
32	#15692162	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
37	#95×C₁₀₁	C₂²	1	2	yes	—
38	C₂³×D₁₀₁	C₂³	1	2	yes	—
39	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
40	C₂×D₄₀₄	D₄	1	4	yes	—
41	C₂×D₄₀₄	Q₈	1	4	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	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₀₁	—	2	4	no	—
48	C₂×C₄×D₁₀₁	—	2	4	no	—
49	C₂×C₄×D₁₀₁	—	2	4	no	—
50	C₂×C₄×D₁₀₁	—	2	4	no	—
51	C₂×C₄×D₁₀₁	—	4	2	no	—
52	C₂×C₄×D₁₀₁	—	4	2	no	—
53	C₂×C₄×D₁₀₁	—	4	2	no	—
54	C₂×C₄×D₁₀₁	—	4	2	no	—
55	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
56	C₄⋊Dic₁₀₁	—	2	2	no	—
57	C₄⋊Dic₁₀₁	—	4	2	no	—
58	#13108053	—	2	2	no	—
59	#13108053	—	2	2	no	—
60	#13108051	—	2	4	no	—
61	#13108051	—	2	4	no	—
62	#13108051	C₂×C₄	1	8	yes	complemented
63	C₄×Dic₁₀₁	—	2	2	no	—
64	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
65	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₁	—	2	2	no	—
71	C₄⋊C₄×C₁₀₁	—	4	2	no	—
72	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
73	C₄²×C₁₀₁	—	2	2	no	—
74	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
75	C₂²×D₁₀₁	—	2	2	no	—
76	C₂²×D₁₀₁	—	2	2	no	—
77	C₂²×D₁₀₁	—	2	2	no	—
78	C₂²×D₁₀₁	—	2	2	no	—
79	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
80	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
81	C₂²×D₁₀₁	C₂×Q₈	1	4	yes	—
82	C₂×C₄×C₁₀₁	—	2	1	no	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	1	no	—
86	C₂×C₄×C₁₀₁	—	2	1	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	1	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	—	2	2	no	—
92	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₁₀₁	#41	1	2	yes	—
97	C₂×Dic₁₀₁	—	2	1	no	—
98	C₂×Dic₁₀₁	—	2	1	no	—
99	C₂×Dic₁₀₁	—	2	1	no	—
100	C₂×Dic₁₀₁	—	2	1	no	—
101	C₂×Dic₁₀₁	—	2	1	no	—
102	C₂×Dic₁₀₁	—	2	1	no	—
103	C₂×Dic₁₀₁	—	2	1	no	—
104	C₂×Dic₁₀₁	—	2	2	no	—
105	C₂×Dic₁₀₁	—	2	1	no	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	1	no	—
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
110	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
111	C₂×Dic₁₀₁	#41	1	2	yes	—
112	D₄₀₄	C₄⋊C₄	1	16	yes	complemented
113	C₄×D₁₀₁	—	4	8	no	—
114	C₄×D₁₀₁	—	4	8	no	—
115	C₄×D₁₀₁	—	4	8	no	—
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	1	yes	characteristic
120	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
121	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×D₁₀₁	—	2	4	no	—
124	C₂×D₁₀₁	—	2	4	no	—
125	C₂×D₁₀₁	—	2	4	no	—
126	C₂×D₁₀₁	—	2	4	no	—
127	C₂×D₁₀₁	—	2	4	no	—
128	C₂×D₁₀₁	—	2	4	no	—
129	C₂×D₁₀₁	C₂×C₄⋊C₄	1	8	yes	—
130	C₄×C₁₀₁	—	2	4	no	—
131	C₄×C₁₀₁	—	2	2	no	—
132	C₄×C₁₀₁	—	4	2	no	—
133	C₄×C₁₀₁	—	4	1	no	—
134	C₄×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
135	Dic₁₀₁	—	2	2	no	—
136	Dic₁₀₁	—	2	2	no	—
137	Dic₁₀₁	—	2	2	no	—
138	Dic₁₀₁	—	4	2	no	—
139	Dic₁₀₁	—	4	1	no	—
140	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
142	C₂×C₁₀₁	#518	1	1	yes	characteristic
143	C₂×C₁₀₁	#546	1	1	yes	characteristic
144	C₂×C₁₀₁	#546	1	1	yes	characteristic
145	C₂×C₁₀₁	#554	1	1	yes	characteristic
146	C₂×C₁₀₁	#554	1	1	yes	characteristic
147	D₁₀₁	—	2	8	no	—
148	#2349	—	101	1	no	maximal, Sylow
149	C₁₀₁	#2349	1	1	yes	characteristic, complemented, Sylow
150	C₂×C₄×D₄	—	101	1	no	—
151	C₂×C₄×D₄	—	101	1	no	—
152	C₂×C₄×D₄	—	101	1	no	—
153	#389	—	101	1	no	—
154	#389	D₁₀₁	1	1	yes	characteristic, complemented
155	#385	—	101	2	no	—
156	#385	—	101	2	no	—
157	#382	—	101	1	no	—
158	#380	—	101	2	no	—
159	C₄×C₄⋊C₄	—	101	1	no	—
160	C₂×#95	—	101	2	no	—
161	C₂²×D₄	—	101	1	no	—
162	C₂³×C₄	—	101	2	no	—
163	C₂³×C₄	—	101	2	no	—
164	C₂³×C₄	—	101	2	no	—
165	C₄×D₄	—	101	8	no	—
166	C₄×D₄	—	202	4	no	—
167	C₄×D₄	—	202	4	no	—
168	C₂×C₄⋊C₄	—	101	2	no	—
169	C₂×C₄⋊C₄	—	101	1	no	—
170	C₂×C₄⋊C₄	—	101	1	no	—
171	C₂×C₄⋊C₄	—	101	1	no	—
172	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	—	101	2	no	—
176	C₂×C₂²⋊C₄	—	101	2	no	—
177	C₂×C₂²⋊C₄	—	101	2	no	—
178	C₂×C₄²	—	101	1	no	—
179	C₂×C₄²	—	101	1	no	—
180	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
181	#95	—	101	1	no	—
182	#95	—	101	2	no	—
183	#95	—	101	2	no	—
184	#95	—	101	1	no	—
185	#95	—	101	2	no	—
186	#95	C₂×D₁₀₁	1	2	yes	—
187	C₂⁴	—	101	2	no	—
188	C₂×D₄	—	101	4	no	—
189	C₂×D₄	—	101	4	no	—
190	C₂×D₄	—	101	4	no	—
191	C₂²×C₄	—	101	1	no	—
192	C₂²×C₄	—	101	1	no	—
193	C₂²×C₄	—	101	1	no	—
194	C₂²×C₄	—	101	8	no	—
195	C₂²×C₄	—	101	2	no	—
196	C₂²×C₄	—	101	1	no	—
197	C₂²×C₄	—	404	2	no	—
198	C₂²×C₄	—	404	2	no	—
199	C₂²×C₄	—	404	2	no	—
200	C₂²×C₄	—	404	2	no	—
201	C₂²×C₄	—	202	4	no	—
202	C₂²×C₄	—	202	4	no	—
203	C₂²×C₄	—	202	4	no	—
204	C₂²×C₄	—	202	4	no	—
205	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
206	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	404	2	no	—
213	C₄⋊C₄	—	202	2	no	—
214	C₄⋊C₄	—	202	2	no	—
215	C₄⋊C₄	—	202	2	no	—
216	C₄⋊C₄	C₄×D₁₀₁	1	4	yes	complemented
217	C₂²⋊C₄	—	101	8	no	—
218	C₂²⋊C₄	—	202	4	no	—
219	C₂²⋊C₄	—	202	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	101	4	no	—
222	C₄²	—	202	2	no	—
223	C₂³	—	101	4	no	—
224	C₂³	—	101	4	no	—
225	C₂³	—	101	4	no	—
226	C₂³	—	202	2	no	—
227	C₂³	—	202	2	no	—
228	C₂³	—	202	2	no	—
229	C₂³	—	202	2	no	—
230	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
231	D₄	—	101	16	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	101	2	no	—
243	C₂×C₄	—	101	2	no	—
244	C₂×C₄	—	101	2	no	—
245	C₂×C₄	—	404	8	no	—
246	C₂×C₄	—	404	8	no	—
247	C₂×C₄	—	404	8	no	—
248	C₂×C₄	—	202	1	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	—	202	1	no	—
251	C₂×C₄	—	202	2	no	—
252	C₂×C₄	—	202	1	no	—
253	C₂×C₄	—	202	2	no	—
254	C₂×C₄	—	202	1	no	—
255	C₂×C₄	—	202	1	no	—
256	C₂×C₄	—	202	1	no	—
257	C₂×C₄	—	202	1	no	—
258	C₂×C₄	—	202	1	no	—
259	C₂×C₄	—	202	1	no	—
260	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
261	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
262	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
263	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
264	C₂×C₄	#13108075	1	2	yes	—
265	C₂²	—	101	8	no	—
266	C₂²	—	202	4	no	—
267	C₂²	—	202	4	no	—
268	C₂²	—	202	4	no	—
269	C₂²	—	202	4	no	—
270	C₂²	—	202	4	no	—
271	C₂²	—	202	4	no	—
272	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
273	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
274	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
275	C₂²	#41×D₁₀₁	1	1	yes	characteristic
276	C₂²	C₂×#13108075	1	1	yes	characteristic
277	C₂²	#15692300	1	1	yes	characteristic
278	C₂²	#15692301	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	4	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	1	no	—
283	C₄	—	404	2	no	—
284	C₄	—	404	1	no	—
285	C₄	—	202	2	no	—
286	C₄	—	202	2	no	—
287	C₄	—	202	2	no	—
288	C₄	C₄⋊C₄×D₁₀₁	1	4	yes	—
289	C₂	—	202	8	no	—
290	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
291	C₂	C₄×#13108075	1	1	yes	characteristic
292	C₂	#20265649	1	1	yes	characteristic
293	C₂	#20265818	1	1	yes	characteristic
294	C₂	#20265824	1	1	yes	characteristic
295	C₂	#20265883	1	1	yes	characteristic
296	C₂	#20265888	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#389×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×D₄₀₄	C₂²
hypercenter	#389	D₁₀₁
hypocenter	C₁₀₁	#2349
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
4. #20265076	C₂	semidirect
5. #20265051	C₂	semidirect
6. #20265048	C₂	semidirect
7. #20265042	C₂	semidirect
8. #20265034	C₂	semidirect
9. C₄×#13108053	C₂	semidirect
10. #95×D₁₀₁	C₂	semidirect
11. #389×C₁₀₁	C₂	semidirect
18. #15692217	C₄	semidirect
55. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
62. #13108051	C₂×C₄	semidirect
64. C₄×Dic₁₀₁	C₂×C₄	semidirect
72. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
112. D₄₀₄	C₄⋊C₄	semidirect
149. C₁₀₁	#2349	semidirect
154. #389	D₁₀₁	semidirect
216. 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×C₄×D₄₀₄	2
C₂×#15692217	2
#20265076	2
#20265051	2
#20265048	2
#20265042	2
#20265034	2
C₄×#13108053	2
#95×D₁₀₁	2
#389×C₁₀₁	2
#2349	101