Subgroups — #27180447

Order 12928.

712 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692286	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692275	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265755	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265339	C₂	1	2	yes	maximal, complemented
5	#20265333	C₂	1	2	yes	maximal, complemented
6	#20265322	C₂	1	2	yes	maximal, complemented
7	#20265316	C₂	1	2	yes	maximal, complemented
8	#20265178	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692179	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265144	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×2₋¹⁺⁴×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108074	C₂²	1	1	yes	characteristic
13	#15692286	—	2	4	no	—
14	#15692275	—	2	4	no	—
15	C₂×#13108064	C₂²	1	1	yes	characteristic
16	C₂×#13108064	C₂²	1	2	yes	—
17	C₂×#13108062	C₂²	1	1	yes	characteristic
18	C₂×#13108061	C₂²	1	1	yes	characteristic
19	C₂×#13108061	C₂²	1	2	yes	—
20	C₂×#13108059	C₂²	1	1	yes	characteristic
21	C₂×#13108058	C₂²	1	1	yes	characteristic
22	C₂²×#11717579	C₂²	1	1	yes	characteristic
23	#15692234	C₂²	1	2	yes	—
24	#15692231	—	2	1	no	—
25	#15692229	C₂²	1	2	yes	—
26	#15692227	—	2	2	no	—
27	#15692226	—	2	1	no	—
28	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
29	#15692179	C₂²	1	2	yes	—
30	#15692179	C₂²	1	2	yes	—
31	#15692179	C₂²	1	2	yes	—
32	#15692176	C₂²	1	2	yes	—
33	C₂²⋊#11717579	C₂²	1	2	yes	—
34	C₂²⋊#11717579	C₂²	1	2	yes	—
35	2₋¹⁺⁴×C₁₀₁	—	2	4	no	—
36	2₋¹⁺⁴×C₁₀₁	—	2	4	no	—
37	C₂×#41×C₁₀₁	—	2	1	no	—
38	C₂×#41×C₁₀₁	—	2	2	no	—
39	C₂×#41×C₁₀₁	C₂²	1	2	yes	—
40	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂²×Q₈×C₁₀₁	—	2	1	no	—
43	C₂²×Q₈×C₁₀₁	C₂²	1	2	yes	—
44	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
45	#13108074	—	2	2	no	—
46	#13108074	—	2	2	no	—
47	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
48	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
50	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
51	C₂²×Dic₁₀₁	—	2	1	no	—
52	C₂×C₄×D₁₀₁	—	2	1	no	—
53	#13108064	—	2	2	no	—
54	#13108064	—	2	4	no	—
55	#13108062	—	2	2	no	—
56	#13108061	—	2	2	no	—
57	#13108061	—	2	4	no	—
58	#13108059	—	2	2	no	—
59	#13108058	—	2	2	no	—
60	C₂×#11717579	—	2	2	no	—
61	C₂×#11717579	C₂³	1	2	yes	—
62	C₂×#11717579	C₂³	1	1	yes	characteristic
63	C₂×#11717579	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₁₀₁	—	2	1	no	—
65	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
66	#13108053	—	2	2	no	—
67	C₂²⋊Dic₁₀₁	—	2	2	no	—
68	#13108051	—	2	2	no	—
69	#41×C₁₀₁	—	2	16	no	—
70	#41×C₁₀₁	—	2	8	no	—
71	#41×C₁₀₁	—	2	4	no	—
72	#41×C₁₀₁	—	2	4	no	—
73	#41×C₁₀₁	—	2	2	no	—
74	#41×C₁₀₁	—	2	2	no	—
75	#41×C₁₀₁	—	2	2	no	—
76	#41×C₁₀₁	D₄	1	4	yes	complemented
77	C₂×Q₈×C₁₀₁	—	2	8	no	—
78	C₂×Q₈×C₁₀₁	—	2	4	no	—
79	C₂×Q₈×C₁₀₁	—	2	4	no	—
80	C₂×Q₈×C₁₀₁	—	2	2	no	—
81	C₂×Q₈×C₁₀₁	—	2	1	no	—
82	C₂×Q₈×C₁₀₁	—	2	2	no	—
83	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
84	C₂×Q₈×C₁₀₁	D₄	1	4	yes	complemented
85	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
87	C₂×D₄×C₁₀₁	—	2	2	no	—
88	C₂×D₄×C₁₀₁	—	2	1	no	—
89	C₂×D₄×C₁₀₁	D₄	1	1	yes	characteristic
90	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
91	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₁	—	2	2	no	—
93	C₂²×C₄×C₁₀₁	—	2	1	no	—
94	C₂²×C₄×C₁₀₁	—	2	2	no	—
95	C₂²×C₄×C₁₀₁	—	2	1	no	—
96	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
97	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
98	C₂³×C₁₀₁	—	2	1	no	—
99	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
100	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×D₁₀₁	—	2	1	no	—
102	Q₈×C₁₀₁	—	2	8	no	—
103	Q₈×C₁₀₁	—	2	2	no	—
104	Q₈×C₁₀₁	—	2	4	no	—
105	Q₈×C₁₀₁	—	2	1	no	—
106	Q₈×C₁₀₁	—	2	2	no	—
107	Q₈×C₁₀₁	C₂×D₄	1	2	yes	—
108	Q₈×C₁₀₁	C₂×D₄	1	4	yes	—
109	D₄×C₁₀₁	—	2	8	no	—
110	D₄×C₁₀₁	—	2	4	no	—
111	D₄×C₁₀₁	—	2	4	no	—
112	D₄×C₁₀₁	—	2	2	no	—
113	D₄×C₁₀₁	—	2	1	no	—
114	D₄×C₁₀₁	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₁₀₁	—	2	8	no	—
116	C₂×C₄×C₁₀₁	—	2	8	no	—
117	C₂×C₄×C₁₀₁	—	2	4	no	—
118	C₂×C₄×C₁₀₁	—	2	2	no	—
119	C₂×C₄×C₁₀₁	—	2	1	no	—
120	C₂×C₄×C₁₀₁	—	2	4	no	—
121	C₂×C₄×C₁₀₁	—	2	2	no	—
122	C₂×C₄×C₁₀₁	—	2	2	no	—
123	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
127	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₀₁	—	4	2	no	—
129	C₂²⋊D₁₀₁	—	8	2	no	—
130	C₂×Dic₁₀₁	—	2	2	no	—
131	C₂×Dic₁₀₁	—	2	1	no	—
132	C₂×Dic₁₀₁	—	8	1	no	—
133	D₄₀₄	—	2	1	no	—
134	D₄₀₄	—	2	1	no	—
135	C₄×D₁₀₁	—	4	2	no	—
136	Dic₂₀₂	—	2	1	no	—
137	Dic₂₀₂	—	2	1	no	—
138	#11717579	—	2	1	no	—
139	#11717579	—	2	1	no	—
140	#11717579	—	2	2	no	—
141	C₂²×C₁₀₁	—	2	1	no	—
142	C₂²×C₁₀₁	—	2	1	no	—
143	C₂²×C₁₀₁	—	2	2	no	—
144	C₂²×C₁₀₁	—	2	2	no	—
145	C₂²×C₁₀₁	—	2	2	no	—
146	C₂²×C₁₀₁	—	2	2	no	—
147	C₂²×C₁₀₁	—	4	1	no	—
148	C₂²×C₁₀₁	—	4	1	no	—
149	C₂²×C₁₀₁	#120	1	2	yes	—
150	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₀₁	—	4	1	no	—
152	C₂×D₁₀₁	—	4	1	no	—
153	C₂×D₁₀₁	—	4	1	no	—
154	C₄×C₁₀₁	—	2	4	no	—
155	C₄×C₁₀₁	—	2	2	no	—
156	C₄×C₁₀₁	—	2	2	no	—
157	C₄×C₁₀₁	#120	1	2	yes	—
158	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
160	Dic₁₀₁	—	4	1	no	—
161	Dic₁₀₁	—	4	2	no	—
162	C₂×C₁₀₁	—	2	2	no	—
163	C₂×C₁₀₁	—	4	2	no	—
164	C₂×C₁₀₁	—	4	2	no	—
165	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
166	C₂×C₁₀₁	#577	1	1	yes	characteristic
167	C₂×C₁₀₁	#578	1	1	yes	characteristic
168	D₁₀₁	—	8	1	no	—
169	#3013	—	101	1	no	maximal, Sylow
170	C₁₀₁	#3013	1	1	yes	characteristic, complemented, Sylow
171	C₂×2₋¹⁺⁴	D₁₀₁	1	1	yes	characteristic, complemented
172	C₂×#137	—	101	1	no	—
173	C₂×#135	—	101	1	no	—
174	#537	—	101	1	no	—
175	#452	—	101	2	no	—
176	#451	—	101	2	no	—
177	#449	—	101	2	no	—
178	#448	—	101	2	no	—
179	#417	—	101	1	no	—
180	C₂×#103	—	101	1	no	—
181	#408	—	101	1	no	—
182	2₋¹⁺⁴	—	2	4	no	—
183	2₋¹⁺⁴	—	2	4	no	—
184	C₂×#41	—	2	1	no	—
185	C₂×#41	—	2	2	no	—
186	C₂×#41	—	101	1	no	—
187	C₂×#41	C₂×D₁₀₁	1	2	yes	—
188	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
189	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
190	C₂²×Q₈	—	2	1	no	—
191	C₂²×Q₈	C₂×D₁₀₁	1	2	yes	—
192	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
193	#137	—	202	4	no	—
194	#135	—	202	4	no	—
195	C₂×Q₁₆	—	101	1	no	—
196	C₂×Q₁₆	—	101	2	no	—
197	C₂×SD₁₆	—	101	1	no	—
198	C₂×SD₁₆	—	101	2	no	—
199	C₂×SD₁₆	—	101	1	no	—
200	C₂×D₈	—	101	1	no	—
201	C₂×M₁₆	—	101	1	no	—
202	C₂²×C₈	—	101	1	no	—
203	C₂²⋊Q₈	—	101	2	no	—
204	C₂²⋊Q₈	—	202	1	no	—
205	#121	—	101	2	no	—
206	#121	—	202	1	no	—
207	#121	—	202	2	no	—
208	C₂×C₄⋊C₄	—	101	1	no	—
209	#103	—	101	2	no	—
210	#103	—	101	2	no	—
211	#103	—	101	2	no	—
212	#102	—	101	2	no	—
213	C₂²⋊C₈	—	101	2	no	—
214	C₂²⋊C₈	—	101	2	no	—
215	#41	—	2	16	no	—
216	#41	—	2	8	no	—
217	#41	—	2	4	no	—
218	#41	—	2	4	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	#41	—	202	2	no	—
223	#41	—	202	2	no	—
224	#41	C₂²⋊D₁₀₁	1	4	yes	complemented
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	101	1	no	—
232	C₂×Q₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
233	C₂×Q₈	C₂²⋊D₁₀₁	1	4	yes	complemented
234	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	101	1	no	—
239	C₂×D₄	—	202	2	no	—
240	C₂×D₄	—	202	1	no	—
241	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
242	C₂×D₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
243	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	202	1	no	—
249	C₂²×C₄	—	202	1	no	—
250	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
251	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
252	Q₁₆	—	202	2	no	—
253	Q₁₆	—	202	4	no	—
254	SD₁₆	—	202	2	no	—
255	SD₁₆	—	202	4	no	—
256	SD₁₆	—	202	2	no	—
257	D₈	—	202	2	no	—
258	M₁₆	—	202	2	no	—
259	C₂×C₈	—	101	2	no	—
260	C₂×C₈	—	101	1	no	—
261	C₂×C₈	—	101	1	no	—
262	C₂×C₈	—	202	2	no	—
263	C₄⋊C₄	—	101	2	no	—
264	C₄⋊C₄	—	202	2	no	—
265	C₄⋊C₄	—	202	1	no	—
266	C₂²⋊C₄	—	202	2	no	—
267	C₂²⋊C₄	—	202	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	202	1	no	—
270	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
271	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
272	Q₈	—	2	8	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	2	no	—
275	Q₈	—	2	2	no	—
276	Q₈	—	2	4	no	—
277	Q₈	—	202	1	no	—
278	Q₈	—	202	1	no	—
279	Q₈	C₂×C₂²⋊D₁₀₁	1	2	yes	—
280	Q₈	C₂×C₂²⋊D₁₀₁	1	4	yes	—
281	D₄	—	2	8	no	—
282	D₄	—	2	4	no	—
283	D₄	—	2	4	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	2	no	—
286	D₄	—	808	2	no	—
287	D₄	—	404	2	no	—
288	D₄	—	202	1	no	—
289	D₄	—	202	1	no	—
290	D₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	808	1	no	—
300	C₂×C₄	—	404	2	no	—
301	C₂×C₄	—	202	2	no	—
302	C₂×C₄	—	202	1	no	—
303	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
305	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
306	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
308	C₈	—	202	1	no	—
309	C₈	—	202	1	no	—
310	C₈	—	202	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	404	1	no	—
320	C₂²	—	404	1	no	—
321	C₂²	—	404	1	no	—
322	C₂²	#15692224	1	2	yes	—
323	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	404	2	no	—
328	C₄	—	404	1	no	—
329	C₄	#15692224	1	2	yes	—
330	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
331	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	2	no	—
335	C₂	—	808	1	no	—
336	C₂	C₂×#15692224	1	1	yes	characteristic
337	C₂	#20265985	1	1	yes	characteristic
338	C₂	#20265992	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×2₋¹⁺⁴×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	C₂×2₋¹⁺⁴	D₁₀₁
hypocenter	C₁₀₁	#3013
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#15692286	C₂	semidirect
2. C₂×#15692275	C₂	semidirect
3. #20265755	C₂	semidirect
4. #20265339	C₂	semidirect
5. #20265333	C₂	semidirect
6. #20265322	C₂	semidirect
7. #20265316	C₂	semidirect
8. #20265178	C₂	semidirect
9. C₂×#15692179	C₂	semidirect
10. #20265144	C₂	semidirect
11. C₂×2₋¹⁺⁴×C₁₀₁	C₂	semidirect
76. #41×C₁₀₁	D₄	semidirect
84. C₂×Q₈×C₁₀₁	D₄	semidirect
170. C₁₀₁	#3013	semidirect
171. C₂×2₋¹⁺⁴	D₁₀₁	semidirect
224. #41	C₂²⋊D₁₀₁	semidirect
233. C₂×Q₈	C₂²⋊D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692286	2
C₂×#15692275	2
#20265755	2
#20265339	2
#20265333	2
#20265322	2
#20265316	2
#20265178	2
C₂×#15692179	2
#20265144	2
C₂×2₋¹⁺⁴×C₁₀₁	2
#3013	101