Subgroups — #27180762

Order 12928.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692288	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265688	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692234	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265435	C₂	1	2	yes	maximal, complemented
5	#20265423	C₂	1	2	yes	maximal, complemented
6	#20265418	C₂	1	2	yes	maximal, complemented
7	#20265406	C₂	1	2	yes	maximal, complemented
8	#20265219	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265187	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265179	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×M₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108074	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	#15692288	—	2	4	no	—
15	C₂×Dic₄₀₄	C₂²	1	2	yes	complemented
16	C₂×#13108063	C₂²	1	2	yes	complemented
17	#15692238	—	2	1	no	—
18	#15692234	—	2	1	no	—
19	#15692234	—	2	1	no	—
20	#15692234	C₂²	1	2	yes	—
21	#15692234	C₂²	1	1	yes	characteristic
22	#15692234	C₂²	1	2	yes	—
23	#15692234	C₂²	1	1	yes	characteristic
24	#15692229	C₂²	1	1	yes	characteristic
25	#15692229	C₂²	1	1	yes	characteristic
26	#15692224	—	2	1	no	—
27	C₄×C₂²⋊D₁₀₁	—	2	1	no	—
28	C₄×C₂²⋊D₁₀₁	—	2	1	no	—
29	#15692214	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
31	C₂×#13108053	—	2	1	no	—
32	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
33	#15692189	C₂²	1	2	yes	complemented
34	#15692187	C₂²	1	2	yes	complemented
35	#15692180	C₂²	1	2	yes	complemented
36	#15692180	C₂²	1	2	yes	complemented
37	#15692180	C₂²	1	2	yes	complemented
38	#15692177	C₂²	1	2	yes	complemented
39	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₀₁	—	2	4	no	—
41	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
42	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
44	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
45	#13108074	—	2	2	no	—
46	#13108074	—	2	2	no	—
47	C₂×Dic₂₀₂	—	2	2	no	—
48	C₂×Dic₂₀₂	—	2	2	no	—
49	C₂×Dic₂₀₂	—	2	1	no	—
50	C₂×Dic₂₀₂	C₂³	1	2	yes	—
51	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
52	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
54	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
55	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
56	C₂²×Dic₁₀₁	—	2	1	no	—
57	C₂²×Dic₁₀₁	—	2	1	no	—
58	C₂×C₄×D₁₀₁	—	2	1	no	—
59	Dic₄₀₄	—	2	4	no	—
60	#13108063	—	2	4	no	—
61	C₄⋊Dic₁₀₁	—	2	1	no	—
62	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
63	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
65	#13108053	—	2	2	no	—
66	#13108053	—	2	2	no	—
67	#13108053	—	2	1	no	—
68	#13108053	—	2	1	no	—
69	C₂²⋊Dic₁₀₁	—	2	2	no	—
70	C₂²⋊Dic₁₀₁	—	2	2	no	—
71	C₂²⋊Dic₁₀₁	—	2	1	no	—
72	C₂²⋊Dic₁₀₁	—	2	1	no	—
73	C₂²⋊Dic₁₀₁	—	2	1	no	—
74	#13108051	—	2	1	no	—
75	#13108051	—	2	1	no	—
76	C₄×Dic₁₀₁	—	2	1	no	—
77	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₁	—	2	2	no	—
79	C₂²×C₄×C₁₀₁	—	2	2	no	—
80	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
86	M₁₆×C₁₀₁	—	2	2	no	—
87	M₁₆×C₁₀₁	—	2	4	no	—
88	M₁₆×C₁₀₁	D₄	1	4	yes	complemented
89	C₂×C₈×C₁₀₁	—	2	4	no	—
90	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
91	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
92	C₂³×C₁₀₁	—	2	2	no	—
93	C₂³×C₁₀₁	—	2	2	no	—
94	C₂³×C₁₀₁	—	4	1	no	—
95	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×D₁₀₁	—	2	1	no	—
99	C₂×C₄×C₁₀₁	—	2	2	no	—
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	1	no	—
103	C₂×C₄×C₁₀₁	—	2	1	no	—
104	C₂×C₄×C₁₀₁	—	2	1	no	—
105	C₂×C₄×C₁₀₁	—	2	1	no	—
106	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₁	#41	1	2	yes	—
111	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₁₀₁	—	4	2	no	—
113	C₂²⋊D₁₀₁	—	4	2	no	—
114	C₂²⋊D₁₀₁	—	4	2	no	—
115	C₂×Dic₁₀₁	—	2	1	no	—
116	C₂×Dic₁₀₁	—	2	1	no	—
117	C₂×Dic₁₀₁	—	2	2	no	—
118	C₂×Dic₁₀₁	—	2	1	no	—
119	C₂×Dic₁₀₁	—	2	2	no	—
120	C₂×Dic₁₀₁	—	4	2	no	—
121	C₂×Dic₁₀₁	—	4	2	no	—
122	D₄₀₄	—	2	1	no	—
123	D₄₀₄	—	2	1	no	—
124	C₄×D₁₀₁	—	4	2	no	—
125	Dic₂₀₂	—	2	2	no	—
126	Dic₂₀₂	—	2	1	no	—
127	Dic₂₀₂	—	2	2	no	—
128	Dic₂₀₂	—	2	1	no	—
129	Dic₂₀₂	—	4	2	no	—
130	C₈×C₁₀₁	—	2	2	no	—
131	C₈×C₁₀₁	C₂×D₄	1	4	yes	—
132	C₂²×C₁₀₁	—	2	1	no	—
133	C₂²×C₁₀₁	—	2	1	no	—
134	C₂²×C₁₀₁	—	2	1	no	—
135	C₂²×C₁₀₁	—	2	1	no	—
136	C₂²×C₁₀₁	—	2	1	no	—
137	C₂²×C₁₀₁	—	2	1	no	—
138	C₂²×C₁₀₁	—	2	1	no	—
139	C₂²×C₁₀₁	—	4	4	no	—
140	C₂²×C₁₀₁	#121	1	2	yes	—
141	C₂²×C₁₀₁	#137	1	2	yes	complemented
142	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₀₁	—	4	1	no	—
144	C₂×D₁₀₁	—	4	1	no	—
145	C₂×D₁₀₁	—	4	1	no	—
146	C₄×C₁₀₁	—	2	1	no	—
147	C₄×C₁₀₁	—	2	1	no	—
148	C₄×C₁₀₁	#121	1	2	yes	—
149	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
150	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
151	Dic₁₀₁	—	4	2	no	—
152	Dic₁₀₁	—	4	1	no	—
153	Dic₁₀₁	—	4	1	no	—
154	Dic₁₀₁	—	4	2	no	—
155	Dic₁₀₁	—	4	1	no	—
156	C₂×C₁₀₁	—	2	2	no	—
157	C₂×C₁₀₁	—	2	2	no	—
158	C₂×C₁₀₁	—	4	1	no	—
159	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
160	C₂×C₁₀₁	C₂×#137	1	1	yes	characteristic
161	C₂×C₁₀₁	#575	1	1	yes	characteristic
162	D₁₀₁	—	8	1	no	—
163	#3056	—	101	1	no	maximal, Sylow
164	C₁₀₁	#3056	1	1	yes	characteristic, complemented, Sylow
165	C₂×#137	—	101	1	no	—
166	C₂²×M₁₆	D₁₀₁	1	1	yes	characteristic, complemented
167	#525	—	101	1	no	—
168	C₂×C₂²⋊Q₈	—	101	1	no	—
169	#470	—	101	2	no	—
170	#468	—	101	2	no	—
171	#467	—	101	2	no	—
172	#465	—	101	2	no	—
173	#428	—	101	1	no	—
174	#419	—	101	1	no	—
175	#417	—	101	1	no	—
176	C₂×#41	—	101	1	no	—
177	C₂²×Q₈	—	101	1	no	—
178	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
179	#137	—	202	4	no	—
180	C₂×Q₁₆	—	101	2	no	—
181	C₂×SD₁₆	—	101	2	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
184	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
185	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
186	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
187	#123	—	202	1	no	—
188	C₂²⋊Q₈	—	101	2	no	—
189	C₂²⋊Q₈	—	101	2	no	—
190	C₂²⋊Q₈	—	101	1	no	—
191	C₂²⋊Q₈	—	101	1	no	—
192	C₂²⋊Q₈	—	202	1	no	—
193	C₂²⋊Q₈	—	202	1	no	—
194	#121	—	101	1	no	—
195	#121	—	101	1	no	—
196	#120	—	202	1	no	—
197	C₄×D₄	—	202	1	no	—
198	C₄×D₄	—	202	1	no	—
199	#117	—	101	1	no	—
200	C₂×C₄⋊C₄	—	101	1	no	—
201	C₂×C₄⋊C₄	—	202	1	no	—
202	C₂×C₂²⋊C₄	—	202	1	no	—
203	#107	—	101	2	no	—
204	#106	—	101	2	no	—
205	#103	—	101	2	no	—
206	#103	—	101	2	no	—
207	#103	—	101	2	no	—
208	#102	—	101	2	no	—
209	C₂⁴	D₄₀₄	1	1	yes	characteristic
210	#41	—	202	2	no	—
211	#41	—	202	2	no	—
212	C₂×Q₈	—	101	2	no	—
213	C₂×Q₈	—	101	1	no	—
214	C₂×Q₈	—	202	2	no	—
215	C₂×Q₈	—	202	2	no	—
216	C₂×Q₈	—	202	1	no	—
217	C₂×D₄	—	101	1	no	—
218	C₂×D₄	—	202	1	no	—
219	C₂×D₄	—	202	1	no	—
220	C₂×D₄	—	202	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	202	1	no	—
224	C₂²×C₄	—	202	1	no	—
225	C₂²×C₄	—	202	1	no	—
226	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
227	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
228	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
232	Q₁₆	—	202	4	no	—
233	SD₁₆	—	202	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	C₂²⋊D₁₀₁	1	4	yes	complemented
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
239	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
240	C₄⋊C₄	—	101	2	no	—
241	C₄⋊C₄	—	101	1	no	—
242	C₄⋊C₄	—	101	1	no	—
243	C₄⋊C₄	—	202	2	no	—
244	C₄⋊C₄	—	202	2	no	—
245	C₄⋊C₄	—	202	1	no	—
246	C₄⋊C₄	—	202	1	no	—
247	C₄⋊C₄	—	202	1	no	—
248	C₂²⋊C₄	—	202	2	no	—
249	C₂²⋊C₄	—	202	2	no	—
250	C₂²⋊C₄	—	202	1	no	—
251	C₂²⋊C₄	—	202	1	no	—
252	C₂²⋊C₄	—	202	1	no	—
253	C₂²⋊C₄	—	202	1	no	—
254	C₂²⋊C₄	—	202	1	no	—
255	C₄²	—	202	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	202	1	no	—
260	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
261	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
262	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
263	Q₈	—	404	2	no	—
264	Q₈	—	202	2	no	—
265	Q₈	—	202	1	no	—
266	Q₈	—	202	2	no	—
267	Q₈	—	202	1	no	—
268	D₄	—	404	2	no	—
269	D₄	—	404	2	no	—
270	D₄	—	404	2	no	—
271	D₄	—	202	1	no	—
272	D₄	—	202	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	404	2	no	—
281	C₂×C₄	—	404	2	no	—
282	C₂×C₄	—	404	2	no	—
283	C₂×C₄	—	202	1	no	—
284	C₂×C₄	—	202	1	no	—
285	C₂×C₄	—	202	2	no	—
286	C₂×C₄	—	202	1	no	—
287	C₂×C₄	—	202	2	no	—
288	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
289	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
292	C₂×C₄	#13108074	1	2	yes	—
293	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	C₂×C₂²⋊D₁₀₁	1	4	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	404	1	no	—
305	C₂²	—	404	1	no	—
306	C₂²	—	404	1	no	—
307	C₂²	#15692229	1	2	yes	—
308	C₂²	#15692288	1	2	yes	complemented
309	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	404	1	no	—
313	C₄	—	404	2	no	—
314	C₄	—	404	1	no	—
315	C₄	—	404	2	no	—
316	C₄	—	404	1	no	—
317	C₄	#15692229	1	2	yes	—
318	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
319	C₄	C₂×#13108074	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	808	1	no	—
324	C₂	C₂×#15692229	1	1	yes	characteristic
325	C₂	C₂×#15692288	1	1	yes	characteristic
326	C₂	#20265977	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₀₄
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂²×M₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	#20265688	C₂
hypercenter	C₂²×M₁₆	D₁₀₁
hypocenter	C₁₀₁	#3056
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15692288	C₂	semidirect
3. C₂×#15692234	C₂	semidirect
4. #20265435	C₂	semidirect
5. #20265423	C₂	semidirect
6. #20265418	C₂	semidirect
7. #20265406	C₂	semidirect
8. #20265219	C₂	semidirect
9. #20265187	C₂	semidirect
10. #20265179	C₂	semidirect
11. C₂²×M₁₆×C₁₀₁	C₂	semidirect
15. C₂×Dic₄₀₄	C₂²	semidirect
16. C₂×#13108063	C₂²	semidirect
33. #15692189	C₂²	semidirect
34. #15692187	C₂²	semidirect
35. #15692180	C₂²	semidirect
36. #15692180	C₂²	semidirect
37. #15692180	C₂²	semidirect
38. #15692177	C₂²	semidirect
88. M₁₆×C₁₀₁	D₄	semidirect
141. C₂²×C₁₀₁	#137	semidirect
164. C₁₀₁	#3056	semidirect
166. C₂²×M₁₆	D₁₀₁	semidirect
236. M₁₆	C₂²⋊D₁₀₁	semidirect
308. C₂²	#15692288	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692288	2
#20265688	2
C₂×#15692234	2
#20265435	2
#20265423	2
#20265418	2
#20265406	2
#20265219	2
#20265187	2
#20265179	2
C₂²×M₁₆×C₁₀₁	2
#3056	101