Subgroups — #27174063

Order 12928.

660 conjugacy classes of subgroups.

The subgroup lattice has 322 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	C₂×#15692213	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265069	C₂	1	2	yes	maximal, complemented
5	#20265064	C₂	1	2	yes	maximal, complemented
6	#20265054	C₂	1	2	yes	maximal
7	#20265042	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265034	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
11	#380×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₀₁	C₄	1	8	yes	complemented
16	#15692213	—	2	4	no	—
17	C₂×C₄⋊Dic₁₀₁	—	2	4	no	—
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×#13108053	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×#13108051	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	#15692162	C₂²	1	2	yes	—
33	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
37	#95×C₁₀₁	C₂²	1	2	yes	—
38	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
40	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
41	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	4	yes	—
42	C₂²×Dic₁₀₁	—	2	4	no	—
43	C₂²×Dic₁₀₁	—	2	4	no	—
44	C₂²×Dic₁₀₁	—	4	2	no	—
45	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂×C₄×D₁₀₁	—	2	2	no	—
51	C₂×C₄×D₁₀₁	—	2	2	no	—
52	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
53	C₄⋊Dic₁₀₁	—	2	2	no	—
54	C₄⋊Dic₁₀₁	—	2	4	no	—
55	C₄⋊Dic₁₀₁	D₄	1	4	yes	complemented
56	#13108053	C₂×C₄	1	4	yes	complemented
57	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
58	#13108051	—	2	4	no	—
59	#13108051	—	4	2	no	—
60	#13108051	C₂×C₄	1	4	yes	complemented
61	C₄×Dic₁₀₁	—	2	4	no	—
62	C₄×Dic₁₀₁	—	4	2	no	—
63	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
64	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₁	—	2	2	no	—
66	C₂²×C₄×C₁₀₁	—	2	2	no	—
67	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₁	—	2	2	no	—
73	C₄⋊C₄×C₁₀₁	—	2	2	no	—
74	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
75	C₂³×C₁₀₁	—	2	1	no	—
76	C₂³×C₁₀₁	—	2	1	no	—
77	C₂³×C₁₀₁	—	2	1	no	—
78	C₂³×C₁₀₁	—	2	1	no	—
79	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
80	C₂³×C₁₀₁	#41	1	2	yes	—
81	C₂³×C₁₀₁	#41	1	2	yes	—
82	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
83	C₂²×D₁₀₁	—	2	1	no	—
84	C₂²×D₁₀₁	—	2	1	no	—
85	C₂²×D₁₀₁	—	2	1	no	—
86	C₂²×D₁₀₁	—	2	1	no	—
87	C₂²×D₁₀₁	—	2	1	no	—
88	C₂²×D₁₀₁	—	2	1	no	—
89	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	—	2	1	no	—
92	C₂×C₄×C₁₀₁	—	2	8	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	2	no	—
99	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₁	#41	1	2	yes	—
103	C₂²⋊D₁₀₁	—	2	8	no	—
104	C₂×Dic₁₀₁	—	2	1	no	—
105	C₂×Dic₁₀₁	—	2	1	no	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	1	no	—
108	C₂×Dic₁₀₁	—	2	1	no	—
109	C₂×Dic₁₀₁	—	2	1	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	2	1	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	—	2	2	no	—
114	C₂×Dic₁₀₁	—	2	2	no	—
115	C₂×Dic₁₀₁	—	4	8	no	—
116	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
117	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
118	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
119	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
120	C₄×D₁₀₁	—	2	8	no	—
121	C₂²×C₁₀₁	—	2	2	no	—
122	C₂²×C₁₀₁	—	2	2	no	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	—	2	2	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	#117	1	4	yes	complemented
128	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
129	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
131	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
132	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
133	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×D₁₀₁	—	2	2	no	—
136	C₂×D₁₀₁	—	2	2	no	—
137	C₂×D₁₀₁	—	2	2	no	—
138	C₂×D₁₀₁	—	4	1	no	—
139	C₂×D₁₀₁	—	4	1	no	—
140	C₂×D₁₀₁	—	4	1	no	—
141	C₂×D₁₀₁	—	4	1	no	—
142	C₄×C₁₀₁	—	2	2	no	—
143	C₄×C₁₀₁	—	4	2	no	—
144	C₄×C₁₀₁	—	4	2	no	—
145	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
146	Dic₁₀₁	—	2	2	no	—
147	Dic₁₀₁	—	2	2	no	—
148	Dic₁₀₁	—	2	2	no	—
149	Dic₁₀₁	—	2	4	no	—
150	Dic₁₀₁	—	4	2	no	—
151	C₂×C₁₀₁	—	2	4	no	—
152	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
153	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂×C₁₀₁	#520	1	1	yes	characteristic
155	C₂×C₁₀₁	#546	1	1	yes	characteristic
156	C₂×C₁₀₁	#548	1	1	yes	characteristic
157	C₂×C₁₀₁	#552	1	1	yes	characteristic
158	C₂×C₁₀₁	#553	1	1	yes	characteristic
159	D₁₀₁	—	4	2	no	—
160	#2352	—	101	1	no	maximal, Sylow
161	C₁₀₁	#2352	1	1	yes	characteristic, complemented, Sylow
162	C₂×C₄×D₄	—	101	1	no	—
163	C₂×#117	—	101	1	no	—
164	C₂²×C₄⋊C₄	—	101	1	no	—
165	#388	—	101	2	no	—
166	#387	—	101	2	no	—
167	#386	—	101	2	no	—
168	#382	—	101	1	no	—
169	#380	—	101	1	no	—
170	#380	D₁₀₁	1	1	yes	characteristic, complemented
171	C₄×C₄⋊C₄	—	101	1	no	—
172	C₄×C₂²⋊C₄	—	101	2	no	—
173	C₂²×D₄	—	101	1	no	—
174	C₂³×C₄	—	101	1	no	—
175	C₂³×C₄	—	101	2	no	—
176	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
177	C₄×D₄	—	101	8	no	—
178	#117	—	202	4	no	—
179	C₂×C₄⋊C₄	—	101	1	no	—
180	C₂×C₄⋊C₄	—	101	1	no	—
181	C₂×C₄⋊C₄	—	101	1	no	—
182	C₂×C₄⋊C₄	—	101	2	no	—
183	C₂×C₄⋊C₄	—	202	4	no	—
184	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	101	1	no	—
187	C₂×C₂²⋊C₄	—	101	1	no	—
188	C₂×C₂²⋊C₄	—	101	2	no	—
189	C₂×C₂²⋊C₄	—	101	2	no	—
190	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
191	C₂×C₄²	—	101	1	no	—
192	C₂×C₄²	—	101	2	no	—
193	C₂×C₄²	—	101	2	no	—
194	#95	—	101	2	no	—
195	#95	—	101	2	no	—
196	#95	—	101	2	no	—
197	#95	—	101	2	no	—
198	#95	C₂×D₁₀₁	1	2	yes	—
199	C₂⁴	—	101	1	no	—
200	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
201	C₂×D₄	—	101	4	no	—
202	C₂×D₄	—	202	2	no	—
203	C₂×D₄	—	202	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	101	1	no	—
207	C₂²×C₄	—	101	1	no	—
208	C₂²×C₄	—	101	1	no	—
209	C₂²×C₄	—	101	2	no	—
210	C₂²×C₄	—	101	4	no	—
211	C₂²×C₄	—	101	2	no	—
212	C₂²×C₄	—	404	2	no	—
213	C₂²×C₄	—	202	4	no	—
214	C₂²×C₄	—	202	4	no	—
215	C₂²×C₄	—	202	2	no	—
216	C₂²×C₄	—	202	2	no	—
217	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
218	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
219	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
220	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	101	4	no	—
225	C₄⋊C₄	—	101	4	no	—
226	C₄⋊C₄	—	202	4	no	—
227	C₄⋊C₄	—	202	2	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₂²⋊C₄	—	101	4	no	—
230	C₂²⋊C₄	—	101	4	no	—
231	C₂²⋊C₄	—	404	2	no	—
232	C₂²⋊C₄	—	202	4	no	—
233	C₄²	—	101	4	no	—
234	C₄²	—	404	2	no	—
235	C₄²	—	202	4	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	101	2	no	—
241	C₂³	—	202	1	no	—
242	C₂³	—	202	1	no	—
243	C₂³	—	202	1	no	—
244	C₂³	—	202	1	no	—
245	C₂³	—	202	1	no	—
246	C₂³	—	202	1	no	—
247	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
248	C₂³	#13108073	1	2	yes	—
249	C₂³	#13108075	1	2	yes	—
250	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
251	D₄	—	202	8	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	101	2	no	—
262	C₂×C₄	—	101	2	no	—
263	C₂×C₄	—	101	2	no	—
264	C₂×C₄	—	101	4	no	—
265	C₂×C₄	—	404	8	no	—
266	C₂×C₄	—	202	2	no	—
267	C₂×C₄	—	202	2	no	—
268	C₂×C₄	—	202	2	no	—
269	C₂×C₄	—	202	1	no	—
270	C₂×C₄	—	202	1	no	—
271	C₂×C₄	—	202	1	no	—
272	C₂×C₄	—	202	2	no	—
273	C₂×C₄	—	202	1	no	—
274	C₂×C₄	—	202	1	no	—
275	C₂×C₄	—	202	2	no	—
276	C₂×C₄	—	202	1	no	—
277	C₂×C₄	—	202	8	no	—
278	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
279	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
280	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
281	C₂×C₄	#13108073	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	404	1	no	—
289	C₂²	—	404	1	no	—
290	C₂²	—	404	1	no	—
291	C₂²	—	404	1	no	—
292	C₂²	—	202	2	no	—
293	C₂²	—	202	2	no	—
294	C₂²	—	202	2	no	—
295	C₂²	#15692213	1	4	yes	complemented
296	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
297	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
298	C₂²	C₂×#13108073	1	1	yes	characteristic
299	C₂²	C₂×#13108073	1	1	yes	characteristic
300	C₂²	C₂×#13108075	1	1	yes	characteristic
301	C₂²	#15692301	1	1	yes	characteristic
302	C₂²	#15692302	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	404	2	no	—
307	C₄	—	202	4	no	—
308	C₄	—	202	2	no	—
309	C₄	—	202	2	no	—
310	C₄	—	202	2	no	—
311	C₄	#15692215	1	4	yes	—
312	C₂	—	2	4	no	—
313	C₂	—	404	2	no	—
314	C₂	C₂×#15692213	1	1	yes	characteristic
315	C₂	C₂×#15692215	1	1	yes	characteristic
316	C₂	#20265657	1	1	yes	characteristic
317	C₂	#20265818	1	1	yes	characteristic
318	C₂	#20265833	1	1	yes	characteristic
319	C₂	#20265869	1	1	yes	characteristic
320	C₂	#20265879	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#380×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×C₂²⋊D₁₀₁	C₂²
hypercenter	#380	D₁₀₁
hypocenter	C₁₀₁	#2352
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #380

Complements

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

N	H ≅ G/N	type
2. C₂×#15692213	C₂	semidirect
3. C₂²×C₄⋊Dic₁₀₁	C₂	semidirect
4. #20265069	C₂	semidirect
5. #20265064	C₂	semidirect
7. #20265042	C₂	semidirect
8. #20265034	C₂	semidirect
9. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
10. C₂²⋊C₄×Dic₁₀₁	C₂	semidirect
11. #380×C₁₀₁	C₂	semidirect
15. C₄×C₂²⋊D₁₀₁	C₄	semidirect
52. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
55. C₄⋊Dic₁₀₁	D₄	semidirect
56. #13108053	C₂×C₄	semidirect
60. #13108051	C₂×C₄	semidirect
63. C₄×Dic₁₀₁	C₂×C₄	semidirect
127. C₂²×C₁₀₁	#117	semidirect
161. C₁₀₁	#2352	semidirect
170. #380	D₁₀₁	semidirect
295. C₂²	#15692213	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
C₂×#15692213	2
C₂²×C₄⋊Dic₁₀₁	2
#20265069	2
#20265064	2
#20265054	2
#20265042	2
#20265034	2
C₄⋊C₄×Dic₁₀₁	2
C₂²⋊C₄×Dic₁₀₁	2
#380×C₁₀₁	2
#2352	101