Subgroups — #27179574

Order 12928.

764 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×#41×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13108057	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×M₁₆×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265145	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265144	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265143	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265142	C₂	1	4	yes	maximal, complemented
8	#20265139	C₂	1	4	yes	maximal, complemented
9	#408×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13108075	C₄	1	1	yes	characteristic
11	C₂×#13108074	C₂²	1	2	yes	—
12	C₂×#13108074	C₂²	1	1	yes	characteristic
13	C₂×#13108073	C₄	1	2	yes	—
14	C₂×#13108073	C₄	1	1	yes	characteristic
15	#41×D₁₀₁	—	2	4	no	—
16	#41×D₁₀₁	—	2	4	no	—
17	C₂×Q₈×D₁₀₁	C₄	1	1	yes	characteristic
18	C₂×D₄×D₁₀₁	C₄	1	2	yes	—
19	C₂×D₄×D₁₀₁	C₄	1	1	yes	characteristic
20	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
21	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×#13108058	C₂²	1	1	yes	characteristic, complemented
23	C₂×#13108057	—	2	4	no	—
24	C₂×#13108057	C₂²	1	2	yes	—
25	C₂×#13108057	C₂²	1	2	yes	—
26	C₂×#13108057	C₂²	1	2	yes	complemented
27	M₁₆×D₁₀₁	—	2	4	no	—
28	C₂²×#11717579	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₈×D₁₀₁	C₂²	1	2	yes	complemented
30	C₂²⋊#11717579	C₂²	1	4	yes	complemented
31	#15692167	C₂²	1	4	yes	complemented
32	#15692167	C₂²	1	4	yes	complemented
33	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
35	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
36	C₂²⋊C₈×C₁₀₁	C₂²	1	4	yes	complemented
37	C₂³×D₁₀₁	C₂×C₄	1	2	yes	—
38	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
39	#13108075	—	2	4	no	—
40	#13108074	—	2	2	no	—
41	#13108074	—	2	2	no	—
42	#13108074	—	2	4	no	—
43	#13108074	—	2	4	no	—
44	#13108073	—	2	8	no	—
45	#13108073	—	2	4	no	—
46	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
48	Q₈×D₁₀₁	—	2	4	no	—
49	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
50	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
54	D₄×D₁₀₁	—	2	8	no	—
55	D₄×D₁₀₁	—	2	4	no	—
56	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
57	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₀₁	—	2	4	no	—
59	C₂×C₄×D₁₀₁	—	2	4	no	—
60	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
61	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
62	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
65	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
66	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
67	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
69	#13108058	—	2	2	no	—
70	#13108057	—	2	4	no	—
71	#13108057	—	2	4	no	—
72	#13108057	—	2	4	no	—
73	C₂×#11717579	—	2	2	no	—
74	C₂×#11717579	C₂³	1	2	yes	—
75	C₂×#11717579	C₂³	1	2	yes	—
76	C₈×D₁₀₁	—	2	4	no	—
77	#41×C₁₀₁	—	2	2	no	—
78	#41×C₁₀₁	—	2	2	no	—
79	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
81	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
83	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
84	M₁₆×C₁₀₁	—	2	2	no	—
85	C₂×C₈×C₁₀₁	—	2	2	no	—
86	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
87	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
88	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
89	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₁₀₁	—	2	4	no	—
91	C₂²×D₁₀₁	—	4	2	no	—
92	C₂²×D₁₀₁	—	4	4	no	—
93	C₂²×D₁₀₁	C₂²⋊C₄	1	4	yes	—
94	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
97	Q₈×C₁₀₁	—	2	2	no	—
98	D₄×C₁₀₁	—	2	4	no	—
99	D₄×C₁₀₁	—	2	2	no	—
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₁₀₁	—	2	4	no	—
108	C₂²⋊D₁₀₁	—	2	4	no	—
109	C₂²⋊D₁₀₁	—	2	4	no	—
110	C₂×Dic₁₀₁	—	2	4	no	—
111	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
112	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
113	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
115	D₄₀₄	—	2	4	no	—
116	D₄₀₄	—	2	2	no	—
117	C₄×D₁₀₁	—	2	4	no	—
118	C₄×D₁₀₁	—	2	2	no	—
119	C₄×D₁₀₁	—	2	2	no	—
120	C₄×D₁₀₁	—	2	4	no	—
121	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
122	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
123	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
124	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
125	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
126	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
127	Dic₂₀₂	—	2	2	no	—
128	Dic₂₀₂	—	2	4	no	—
129	C₈×C₁₀₁	—	2	2	no	—
130	C₈×C₁₀₁	—	2	2	no	—
131	#11717579	—	2	2	no	—
132	#11717579	—	2	2	no	—
133	C₂²×C₁₀₁	—	2	2	no	—
134	C₂²×C₁₀₁	—	2	2	no	—
135	C₂²×C₁₀₁	—	2	4	no	—
136	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
137	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×D₁₀₁	—	2	2	no	—
139	C₂×D₁₀₁	—	2	2	no	—
140	C₂×D₁₀₁	—	2	2	no	—
141	C₂×D₁₀₁	—	2	4	no	—
142	C₂×D₁₀₁	—	4	4	no	—
143	C₂×D₁₀₁	—	4	8	no	—
144	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
145	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
146	C₄×C₁₀₁	—	2	2	no	—
147	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
148	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
150	Dic₁₀₁	—	2	2	no	—
151	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
152	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₁₀₁	—	2	2	no	—
154	C₂×C₁₀₁	—	4	2	no	—
155	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₁₀₁	#568	1	2	yes	—
157	D₁₀₁	—	2	2	no	—
158	D₁₀₁	—	2	2	no	—
159	D₁₀₁	—	4	2	no	—
160	#2879	—	101	1	no	maximal, Sylow
161	C₁₀₁	#2879	1	1	yes	characteristic, complemented, Sylow
162	C₂²×#41	—	101	1	no	—
163	C₂²×M₁₆	—	101	1	no	—
164	C₂²×M₁₆	—	101	1	no	—
165	#408	—	101	1	no	—
166	#408	—	101	1	no	—
167	#408	—	101	1	no	—
168	#408	—	101	4	no	—
169	#408	D₁₀₁	1	1	yes	characteristic, complemented
170	#407	—	101	4	no	—
171	C₂×#41	—	101	1	no	—
172	C₂×#41	—	101	2	no	—
173	C₂×#41	—	101	1	no	—
174	C₂×#41	—	101	2	no	—
175	C₂×#41	—	101	1	no	—
176	C₂×#41	—	202	4	no	—
177	C₂×#41	—	202	4	no	—
178	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
179	C₂²×Q₈	—	101	1	no	—
180	C₂²×D₄	—	101	1	no	—
181	C₂²×D₄	—	101	2	no	—
182	C₂³×C₄	—	101	1	no	—
183	C₂³×C₄	—	101	2	no	—
184	C₂×M₁₆	—	101	1	no	—
185	C₂×M₁₆	—	101	2	no	—
186	C₂×M₁₆	—	101	2	no	—
187	C₂×M₁₆	—	101	2	no	—
188	C₂×M₁₆	—	202	4	no	—
189	C₂×M₁₆	—	202	4	no	—
190	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic, complemented
191	C₂²×C₈	—	101	1	no	—
192	C₂²×C₈	—	101	2	no	—
193	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
194	C₂²⋊C₈	—	101	4	no	—
195	C₂²⋊C₈	—	101	4	no	—
196	C₂²⋊C₈	—	101	4	no	—
197	C₂²⋊C₈	C₂×D₁₀₁	1	4	yes	complemented
198	C₂⁴	—	101	1	no	—
199	C₂⁴	—	101	2	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	—	202	4	no	—
203	#41	—	202	4	no	—
204	#41	—	202	8	no	—
205	#41	—	202	4	no	—
206	#41	—	202	4	no	—
207	#41	—	202	2	no	—
208	#41	—	202	2	no	—
209	C₂×Q₈	—	101	2	no	—
210	C₂×Q₈	—	101	1	no	—
211	C₂×Q₈	—	202	4	no	—
212	C₂×Q₈	C₄×D₁₀₁	1	1	yes	characteristic
213	C₂×D₄	—	101	2	no	—
214	C₂×D₄	—	101	2	no	—
215	C₂×D₄	—	101	2	no	—
216	C₂×D₄	—	101	2	no	—
217	C₂×D₄	—	101	1	no	—
218	C₂×D₄	—	202	4	no	—
219	C₂×D₄	—	202	8	no	—
220	C₂×D₄	C₄×D₁₀₁	1	2	yes	—
221	C₂×D₄	C₄×D₁₀₁	1	1	yes	characteristic
222	C₂²×C₄	—	101	1	no	—
223	C₂²×C₄	—	101	1	no	—
224	C₂²×C₄	—	101	1	no	—
225	C₂²×C₄	—	101	4	no	—
226	C₂²×C₄	—	101	4	no	—
227	C₂²×C₄	—	101	1	no	—
228	C₂²×C₄	—	101	1	no	—
229	C₂²×C₄	—	101	2	no	—
230	C₂²×C₄	—	101	2	no	—
231	C₂²×C₄	—	101	2	no	—
232	C₂²×C₄	—	101	2	no	—
233	C₂²×C₄	—	202	4	no	—
234	C₂²×C₄	—	202	4	no	—
235	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
236	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
237	M₁₆	—	2	2	no	—
238	M₁₆	—	202	4	no	—
239	M₁₆	—	202	4	no	—
240	M₁₆	—	202	4	no	—
241	M₁₆	—	202	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	101	2	no	—
244	C₂×C₈	—	101	2	no	—
245	C₂×C₈	—	202	4	no	—
246	C₂×C₈	—	202	2	no	—
247	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
248	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
249	C₂³	—	101	4	no	—
250	C₂³	—	101	1	no	—
251	C₂³	—	101	1	no	—
252	C₂³	—	101	2	no	—
253	C₂³	—	404	2	no	—
254	C₂³	—	404	4	no	—
255	C₂³	—	202	4	no	—
256	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
257	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	Q₈	—	202	4	no	—
260	Q₈	—	202	2	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	2	no	—
263	D₄	—	202	4	no	—
264	D₄	—	202	4	no	—
265	D₄	—	202	4	no	—
266	D₄	—	202	4	no	—
267	D₄	—	202	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	101	1	no	—
271	C₂×C₄	—	101	1	no	—
272	C₂×C₄	—	101	4	no	—
273	C₂×C₄	—	101	2	no	—
274	C₂×C₄	—	101	2	no	—
275	C₂×C₄	—	101	2	no	—
276	C₂×C₄	—	101	2	no	—
277	C₂×C₄	—	101	4	no	—
278	C₂×C₄	—	101	4	no	—
279	C₂×C₄	—	101	2	no	—
280	C₂×C₄	—	202	4	no	—
281	C₂×C₄	—	202	4	no	—
282	C₂×C₄	—	202	4	no	—
283	C₂×C₄	—	202	2	no	—
284	C₂×C₄	—	202	2	no	—
285	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
286	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
287	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
288	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
289	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	202	2	no	—
293	C₈	—	202	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	101	2	no	—
298	C₂²	—	101	2	no	—
299	C₂²	—	404	4	no	—
300	C₂²	—	404	8	no	—
301	C₂²	—	202	2	no	—
302	C₂²	—	202	4	no	—
303	C₂²	—	202	2	no	—
304	C₂²	—	202	2	no	—
305	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
306	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	101	2	no	—
309	C₄	—	101	2	no	—
310	C₄	—	202	2	no	—
311	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
312	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
313	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	—	404	2	no	—
317	C₂	—	202	2	no	—
318	C₂	—	202	2	no	—
319	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
320	C₂	#20265942	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#408×C₁₀₁	C₂
prime radical	C₂×#41×D₁₀₁	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#408	D₁₀₁
hypocenter	C₁₀₁	#2879
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. C₂²×#13108057	C₂	semidirect
3. C₂×M₁₆×D₁₀₁	C₂	semidirect
4. #20265145	C₂	semidirect
5. #20265144	C₂	semidirect
6. #20265143	C₂	semidirect
7. #20265142	C₂	semidirect
8. #20265139	C₂	semidirect
9. #408×C₁₀₁	C₂	semidirect
22. C₂×#13108058	C₂²	semidirect
26. C₂×#13108057	C₂²	semidirect
28. C₂²×#11717579	C₂²	semidirect
29. C₂×C₈×D₁₀₁	C₂²	semidirect
30. C₂²⋊#11717579	C₂²	semidirect
31. #15692167	C₂²	semidirect
32. #15692167	C₂²	semidirect
34. C₂×M₁₆×C₁₀₁	C₂²	semidirect
35. C₂²×C₈×C₁₀₁	C₂²	semidirect
36. C₂²⋊C₈×C₁₀₁	C₂²	semidirect
161. C₁₀₁	#2879	semidirect
169. #408	D₁₀₁	semidirect
190. C₂×M₁₆	C₂×D₁₀₁	semidirect
193. C₂²×C₈	C₂×D₁₀₁	semidirect
197. C₂²⋊C₈	C₂×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₀₁	2
C₂²×#13108057	2
C₂×M₁₆×D₁₀₁	2
#20265145	2
#20265144	2
#20265143	2
#20265142	2
#20265139	2
#408×C₁₀₁	2
#2879	101