Subgroups — #27179418

Order 12928.

1068 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	C₂³×C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692233	C₂	1	2	yes	maximal, complemented
3	C₂×#15692232	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265051	C₂	1	2	yes	maximal, complemented
5	#20265050	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265048	C₂	1	2	yes	maximal, complemented
7	#20265036	C₂	1	2	yes	maximal, complemented
8	#20265035	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265032	C₂	1	1	yes	maximal, characteristic
10	#20265032	C₂	1	1	yes	maximal, characteristic
11	#385×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic, complemented
14	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	—	2	4	no	—
16	C₂²×C₄×D₁₀₁	—	2	8	no	—
17	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
19	#15692233	—	2	8	no	—
20	#15692232	—	2	4	no	—
21	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
22	C₂×#13108053	C₂²	1	2	yes	complemented
23	C₂×#13108053	C₂²	1	1	yes	characteristic, complemented
24	C₂×#13108053	C₂²	1	1	yes	characteristic, complemented
25	C₂×#13108053	C₂²	1	2	yes	complemented
26	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×#13108051	C₂²	1	2	yes	—
29	C₂×#13108051	C₂²	1	2	yes	—
30	C₂×#13108051	C₂²	1	2	yes	—
31	C₂×#13108051	C₂²	1	2	yes	—
32	#15692162	C₂²	1	2	yes	complemented
33	#15692162	C₂²	1	2	yes	complemented
34	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	complemented
36	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
38	#95×C₁₀₁	C₂²	1	2	yes	complemented
39	C₂³×D₁₀₁	—	2	2	no	—
40	C₂³×D₁₀₁	—	2	2	no	—
41	C₂³×D₁₀₁	—	2	4	no	—
42	C₂³×D₁₀₁	—	2	4	no	—
43	C₂³×D₁₀₁	Q₈	1	4	yes	complemented
44	C₂³×D₁₀₁	C₂³	1	2	yes	—
45	C₂×Dic₂₀₂	—	2	2	no	—
46	C₂×Dic₂₀₂	—	2	4	no	—
47	C₂²×Dic₁₀₁	—	2	4	no	—
48	C₂²×Dic₁₀₁	D₄	1	4	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
50	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
51	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₀₁	—	2	32	no	—
55	C₂×C₄×D₁₀₁	—	2	4	no	—
56	C₂×C₄×D₁₀₁	—	2	4	no	—
57	C₂×C₄×D₁₀₁	—	2	4	no	—
58	C₂×C₄×D₁₀₁	—	2	4	no	—
59	C₂×C₄×D₁₀₁	—	2	4	no	—
60	C₂×C₄×D₁₀₁	D₄	1	8	yes	—
61	C₄⋊Dic₁₀₁	—	2	2	no	—
62	#13108053	—	2	4	no	—
63	#13108053	—	2	2	no	—
64	#13108053	—	2	2	no	—
65	#13108053	—	2	4	no	—
66	C₂²⋊Dic₁₀₁	—	2	2	no	—
67	C₂²⋊Dic₁₀₁	—	2	2	no	—
68	#13108051	—	2	4	no	—
69	#13108051	—	2	4	no	—
70	#13108051	—	2	4	no	—
71	#13108051	—	2	4	no	—
72	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	—	2	2	no	—
74	C₂²×C₄×C₁₀₁	—	2	4	no	—
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₀₁	—	2	4	no	—
80	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
81	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	—	2	2	no	—
83	C₂³×C₁₀₁	—	2	2	no	—
84	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
85	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
86	C₂³×C₁₀₁	#41	1	2	yes	—
87	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
88	C₂²×D₁₀₁	—	2	8	no	—
89	C₂²×D₁₀₁	—	2	16	no	—
90	C₂²×D₁₀₁	—	2	8	no	—
91	C₂²×D₁₀₁	—	2	8	no	—
92	C₂²×D₁₀₁	—	2	2	no	—
93	C₂²×D₁₀₁	—	2	2	no	—
94	C₂²×D₁₀₁	—	2	2	no	—
95	C₂²×D₁₀₁	—	2	2	no	—
96	C₂²×D₁₀₁	—	2	16	no	—
97	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
98	C₂²×D₁₀₁	C₂×Q₈	1	4	yes	—
99	C₂²×D₁₀₁	#41	1	4	yes	—
100	C₂×C₄×C₁₀₁	—	2	8	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	—	2	1	no	—
104	C₂×C₄×C₁₀₁	—	2	1	no	—
105	C₂×C₄×C₁₀₁	—	2	2	no	—
106	C₂×C₄×C₁₀₁	—	2	2	no	—
107	C₂×C₄×C₁₀₁	—	2	2	no	—
108	C₂×C₄×C₁₀₁	—	2	2	no	—
109	C₂×C₄×C₁₀₁	—	2	2	no	—
110	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
111	C₂×Dic₁₀₁	—	2	2	no	—
112	C₂×Dic₁₀₁	—	2	8	no	—
113	C₂×Dic₁₀₁	—	2	2	no	—
114	C₂×Dic₁₀₁	—	2	2	no	—
115	C₂×Dic₁₀₁	—	2	1	no	—
116	C₂×Dic₁₀₁	—	2	1	no	—
117	C₂×Dic₁₀₁	—	2	2	no	—
118	C₂×Dic₁₀₁	—	2	2	no	—
119	C₂×Dic₁₀₁	—	2	2	no	—
120	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
121	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
122	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
123	C₄×D₁₀₁	—	2	16	no	—
124	C₄×D₁₀₁	—	2	16	no	—
125	Dic₂₀₂	—	4	4	no	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	—	2	4	no	—
128	C₂²×C₁₀₁	—	2	2	no	—
129	C₂²×C₁₀₁	—	2	4	no	—
130	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	complemented
131	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
132	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
133	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
134	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
135	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
136	C₂×D₁₀₁	—	2	32	no	—
137	C₂×D₁₀₁	—	2	4	no	—
138	C₂×D₁₀₁	—	2	4	no	—
139	C₂×D₁₀₁	—	2	4	no	—
140	C₂×D₁₀₁	—	2	4	no	—
141	C₂×D₁₀₁	—	2	4	no	—
142	C₂×D₁₀₁	—	2	4	no	—
143	C₂×D₁₀₁	C₂²⋊Q₈	1	8	yes	complemented
144	C₄×C₁₀₁	—	2	4	no	—
145	C₄×C₁₀₁	—	4	1	no	—
146	C₄×C₁₀₁	—	4	1	no	—
147	C₄×C₁₀₁	—	4	2	no	—
148	Dic₁₀₁	—	2	2	no	—
149	Dic₁₀₁	—	4	1	no	—
150	Dic₁₀₁	—	4	1	no	—
151	Dic₁₀₁	—	4	2	no	—
152	Dic₁₀₁	#120	1	4	yes	—
153	C₂×C₁₀₁	—	2	4	no	—
154	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
155	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	2	yes	—
156	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₁₀₁	#525	1	1	yes	characteristic
158	C₂×C₁₀₁	#525	1	2	yes	—
159	D₁₀₁	—	2	8	no	—
160	#2848	—	101	1	no	maximal, Sylow
161	C₁₀₁	#2848	1	1	yes	characteristic, complemented, Sylow
162	C₂⁴×C₄	—	101	1	no	—
163	C₂×C₂²⋊Q₈	—	101	2	no	—
164	C₂×C₂²⋊Q₈	—	101	1	no	—
165	#385	—	101	1	no	—
166	#385	—	101	2	no	—
167	#385	—	101	2	no	—
168	#385	D₁₀₁	1	1	yes	characteristic, complemented
169	#380	—	101	1	no	—
170	#380	—	101	2	no	—
171	#379	—	101	1	no	—
172	#379	—	101	1	no	—
173	C₂⁵	—	101	1	no	—
174	C₂²×Q₈	—	101	1	no	—
175	C₂³×C₄	—	101	1	no	—
176	C₂³×C₄	—	101	2	no	—
177	C₂³×C₄	—	101	2	no	—
178	C₂³×C₄	—	202	4	no	—
179	C₂³×C₄	—	202	8	no	—
180	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
181	C₂²⋊Q₈	—	202	4	no	—
182	C₂²⋊Q₈	—	202	8	no	—
183	C₂×C₄⋊C₄	—	101	2	no	—
184	C₂×C₄⋊C₄	—	101	1	no	—
185	C₂×C₄⋊C₄	—	101	1	no	—
186	C₂×C₄⋊C₄	—	101	2	no	—
187	C₂×C₄⋊C₄	—	101	1	no	—
188	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	complemented
189	C₂×C₂²⋊C₄	—	101	1	no	—
190	C₂×C₂²⋊C₄	—	101	1	no	—
191	C₂×C₂²⋊C₄	—	101	2	no	—
192	C₂×C₂²⋊C₄	—	101	2	no	—
193	C₂×C₂²⋊C₄	—	101	2	no	—
194	C₂×C₂²⋊C₄	—	101	2	no	—
195	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
197	#95	—	101	2	no	—
198	#95	—	101	2	no	—
199	#95	C₂×D₁₀₁	1	2	yes	complemented
200	C₂⁴	—	101	2	no	—
201	C₂⁴	—	101	4	no	—
202	C₂⁴	—	202	4	no	—
203	C₂⁴	—	202	4	no	—
204	C₂⁴	—	202	2	no	—
205	C₂⁴	—	202	2	no	—
206	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
207	C₂×Q₈	—	202	2	no	—
208	C₂×Q₈	—	202	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	101	1	no	—
212	C₂²×C₄	—	101	4	no	—
213	C₂²×C₄	—	101	1	no	—
214	C₂²×C₄	—	101	8	no	—
215	C₂²×C₄	—	101	1	no	—
216	C₂²×C₄	—	101	2	no	—
217	C₂²×C₄	—	101	1	no	—
218	C₂²×C₄	—	202	4	no	—
219	C₂²×C₄	—	202	4	no	—
220	C₂²×C₄	—	202	4	no	—
221	C₂²×C₄	—	202	4	no	—
222	C₂²×C₄	—	202	32	no	—
223	C₂²×C₄	—	202	4	no	—
224	C₂²×C₄	—	202	4	no	—
225	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
226	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
227	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	202	2	no	—
231	C₄⋊C₄	—	202	2	no	—
232	C₄⋊C₄	—	202	4	no	—
233	C₄⋊C₄	—	202	2	no	—
234	C₄⋊C₄	—	202	4	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	202	4	no	—
238	C₂²⋊C₄	—	202	4	no	—
239	C₂²⋊C₄	—	202	4	no	—
240	C₂²⋊C₄	—	202	4	no	—
241	C₂²⋊C₄	—	202	2	no	—
242	C₂²⋊C₄	—	202	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	101	4	no	—
246	C₂³	—	101	4	no	—
247	C₂³	—	101	4	no	—
248	C₂³	—	202	16	no	—
249	C₂³	—	202	8	no	—
250	C₂³	—	202	8	no	—
251	C₂³	—	202	16	no	—
252	C₂³	—	202	2	no	—
253	C₂³	—	202	2	no	—
254	C₂³	—	202	2	no	—
255	C₂³	—	202	2	no	—
256	C₂³	—	202	8	no	—
257	C₂³	D₄×D₁₀₁	1	2	yes	—
258	C₂³	Q₈×D₁₀₁	1	2	yes	—
259	C₂³	#13108074	1	2	yes	—
260	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
261	Q₈	—	404	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	101	4	no	—
273	C₂×C₄	—	101	2	no	—
274	C₂×C₄	—	101	2	no	—
275	C₂×C₄	—	202	16	no	—
276	C₂×C₄	—	202	16	no	—
277	C₂×C₄	—	202	8	no	—
278	C₂×C₄	—	202	1	no	—
279	C₂×C₄	—	202	2	no	—
280	C₂×C₄	—	202	2	no	—
281	C₂×C₄	—	202	1	no	—
282	C₂×C₄	—	202	2	no	—
283	C₂×C₄	—	202	2	no	—
284	C₂×C₄	—	202	2	no	—
285	C₂×C₄	—	202	2	no	—
286	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	101	8	no	—
292	C₂²	—	202	32	no	—
293	C₂²	—	202	4	no	—
294	C₂²	—	202	4	no	—
295	C₂²	—	202	4	no	—
296	C₂²	—	202	4	no	—
297	C₂²	—	202	4	no	—
298	C₂²	—	202	4	no	—
299	C₂²	#15692233	1	4	yes	complemented
300	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
301	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
302	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
303	C₂²	#41×D₁₀₁	1	2	yes	—
304	C₂²	C₂×#13108074	1	1	yes	characteristic
305	C₄	—	2	4	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	101	4	no	—
310	C₄	—	404	1	no	—
311	C₄	—	404	1	no	—
312	C₄	—	404	2	no	—
313	C₄	—	202	2	no	—
314	C₂	—	2	4	no	—
315	C₂	—	202	8	no	—
316	C₂	#120×D₁₀₁	1	1	yes	characteristic
317	C₂	C₂²⋊Q₈×D₁₀₁	1	2	yes	—
318	C₂	C₂×#15692233	1	1	yes	characteristic
319	C₂	#20265685	1	1	yes	characteristic
320	C₂	#20265686	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	#385×C₁₀₁	C₂
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	#385	D₁₀₁
hypocenter	C₁₀₁	#2848
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #385

Complements

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

N	H ≅ G/N	type
2. C₂×#15692233	C₂	semidirect
3. C₂×#15692232	C₂	semidirect
4. #20265051	C₂	semidirect
5. #20265050	C₂	semidirect
6. #20265048	C₂	semidirect
7. #20265036	C₂	semidirect
8. #20265035	C₂	semidirect
11. #385×C₁₀₁	C₂	semidirect
13. C₂²×Dic₂₀₂	C₂²	semidirect
21. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
22. C₂×#13108053	C₂²	semidirect
23. C₂×#13108053	C₂²	semidirect
24. C₂×#13108053	C₂²	semidirect
25. C₂×#13108053	C₂²	semidirect
32. #15692162	C₂²	semidirect
33. #15692162	C₂²	semidirect
35. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
38. #95×C₁₀₁	C₂²	semidirect
43. C₂³×D₁₀₁	Q₈	semidirect
130. C₂²×C₁₀₁	C₂²⋊Q₈	semidirect
143. C₂×D₁₀₁	C₂²⋊Q₈	semidirect
161. C₁₀₁	#2848	semidirect
168. #385	D₁₀₁	semidirect
188. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
199. #95	C₂×D₁₀₁	semidirect
299. C₂²	#15692233	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₄×D₁₀₁	2
C₂×#15692233	2
C₂×#15692232	2
#20265051	2
#20265050	2
#20265048	2
#20265036	2
#20265035	2
#20265032	2
#385×C₁₀₁	2
#2848	101