Subgroups — #27174347

Order 12928.

944 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108074	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	2	yes	maximal, complemented
3	#20265097	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265094	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265092	C₂	1	2	yes	maximal, complemented
6	#20265089	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265086	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265085	C₂	1	2	yes	maximal, complemented
9	C₄×#13108051	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108074	—	2	8	no	—
13	C₂×#13108074	—	2	4	no	—
14	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
15	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
19	C₄×C₂²⋊D₁₀₁	—	2	8	no	—
20	C₂×#13108053	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×#13108051	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
27	#15692162	C₂²	1	1	yes	characteristic
28	#15692162	C₂²	1	1	yes	characteristic
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂³×D₁₀₁	C₂³	1	2	yes	—
40	#13108074	—	2	32	no	—
41	C₂×Dic₂₀₂	—	2	2	no	—
42	C₂×Dic₂₀₂	—	2	4	no	—
43	C₂×D₄₀₄	—	2	4	no	—
44	C₂×D₄₀₄	—	2	2	no	—
45	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
46	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
47	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
48	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
49	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
50	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
53	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
54	C₂×C₄×D₁₀₁	—	2	4	no	—
55	C₂×C₄×D₁₀₁	—	2	4	no	—
56	C₂×C₄×D₁₀₁	D₄	1	8	yes	complemented
57	#13108053	—	2	4	no	—
58	C₂²⋊Dic₁₀₁	—	2	4	no	—
59	C₂²⋊Dic₁₀₁	—	2	4	no	—
60	#13108051	—	2	4	no	—
61	#13108051	—	2	4	no	—
62	#13108051	—	2	4	no	—
63	C₄×Dic₁₀₁	—	2	4	no	—
64	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₀₁	—	2	2	no	—
67	C₂×D₄×C₁₀₁	—	2	2	no	—
68	C₂×D₄×C₁₀₁	—	2	2	no	—
69	C₂²×C₄×C₁₀₁	—	2	4	no	—
70	C₂²×C₄×C₁₀₁	—	2	2	no	—
71	C₂²×C₄×C₁₀₁	—	2	4	no	—
72	C₂²×C₄×C₁₀₁	D₄	1	4	yes	complemented
73	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₁	—	2	2	no	—
79	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
80	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
81	C₄²×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	—	2	1	no	—
83	C₂³×C₁₀₁	—	2	2	no	—
84	C₂³×C₁₀₁	—	2	2	no	—
85	C₂³×C₁₀₁	—	2	1	no	—
86	C₂³×C₁₀₁	—	2	1	no	—
87	C₂³×C₁₀₁	—	2	1	no	—
88	C₂³×C₁₀₁	—	2	2	no	—
89	C₂³×C₁₀₁	—	2	2	no	—
90	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
91	C₂³×C₁₀₁	#41	1	2	yes	complemented
92	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
93	C₂²×D₁₀₁	—	2	2	no	—
94	C₂²×D₁₀₁	—	2	2	no	—
95	C₂²×D₁₀₁	—	2	2	no	—
96	C₂²×D₁₀₁	—	2	2	no	—
97	C₂²×D₁₀₁	—	2	2	no	—
98	C₂²×D₁₀₁	—	2	2	no	—
99	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
100	D₄×C₁₀₁	—	2	8	no	—
101	C₂×C₄×C₁₀₁	—	2	1	no	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	—	2	1	no	—
104	C₂×C₄×C₁₀₁	—	2	8	no	—
105	C₂×C₄×C₁₀₁	—	2	1	no	—
106	C₂×C₄×C₁₀₁	—	2	1	no	—
107	C₂×C₄×C₁₀₁	—	2	2	no	—
108	C₂×C₄×C₁₀₁	—	2	2	no	—
109	C₂×C₄×C₁₀₁	—	4	4	no	—
110	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
111	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₁₀₁	#41	1	2	yes	complemented
114	C₂²⋊D₁₀₁	—	2	16	no	—
115	C₂²⋊D₁₀₁	—	2	16	no	—
116	C₂×Dic₁₀₁	—	2	2	no	—
117	C₂×Dic₁₀₁	—	2	2	no	—
118	C₂×Dic₁₀₁	—	2	2	no	—
119	C₂×Dic₁₀₁	—	2	2	no	—
120	C₂×Dic₁₀₁	—	2	2	no	—
121	C₂×Dic₁₀₁	—	2	2	no	—
122	C₂×Dic₁₀₁	—	2	2	no	—
123	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
124	C₂×Dic₁₀₁	#41	1	4	yes	complemented
125	D₄₀₄	—	2	8	no	—
126	C₄×D₁₀₁	—	2	16	no	—
127	Dic₂₀₂	—	2	8	no	—
128	C₂²×C₁₀₁	—	2	2	no	—
129	C₂²×C₁₀₁	—	2	4	no	—
130	C₂²×C₁₀₁	—	2	2	no	—
131	C₂²×C₁₀₁	—	2	2	no	—
132	C₂²×C₁₀₁	—	2	2	no	—
133	C₂²×C₁₀₁	—	4	1	no	—
134	C₂²×C₁₀₁	—	4	2	no	—
135	C₂²×C₁₀₁	—	4	1	no	—
136	C₂²×C₁₀₁	—	4	2	no	—
137	C₂²×C₁₀₁	—	4	2	no	—
138	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
139	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
140	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
141	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
142	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
143	C₂×D₁₀₁	—	2	4	no	—
144	C₂×D₁₀₁	—	2	4	no	—
145	C₂×D₁₀₁	—	2	4	no	—
146	C₂×D₁₀₁	—	4	2	no	—
147	C₂×D₁₀₁	—	4	2	no	—
148	C₂×D₁₀₁	—	4	2	no	—
149	C₂×D₁₀₁	—	4	2	no	—
150	C₄×C₁₀₁	—	2	2	no	—
151	C₄×C₁₀₁	—	2	2	no	—
152	C₄×C₁₀₁	—	4	1	no	—
153	C₄×C₁₀₁	—	4	1	no	—
154	C₄×C₁₀₁	#120	1	4	yes	—
155	Dic₁₀₁	—	2	4	no	—
156	Dic₁₀₁	—	2	4	no	—
157	Dic₁₀₁	—	4	2	no	—
158	C₂×C₁₀₁	—	4	2	no	—
159	C₂×C₁₀₁	—	4	2	no	—
160	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
161	C₂×C₁₀₁	#525	1	2	yes	—
162	C₂×C₁₀₁	#525	1	1	yes	characteristic
163	C₂×C₁₀₁	#532	1	1	yes	characteristic
164	C₂×C₁₀₁	#532	1	2	yes	—
165	D₁₀₁	—	4	4	no	—
166	#2388	—	101	1	no	maximal, Sylow
167	C₁₀₁	#2388	1	1	yes	characteristic, complemented, Sylow
168	C₂²×#41	—	101	1	no	—
169	C₂×C₄×D₄	—	101	2	no	—
170	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
171	#395	—	101	1	no	—
172	#394	—	101	1	no	—
173	#394	—	101	2	no	—
174	#393	—	101	1	no	—
175	#392	—	101	2	no	—
176	#392	—	101	1	no	—
177	C₄×C₂²⋊C₄	—	101	1	no	—
178	C₄×C₂²⋊C₄	—	101	2	no	—
179	C₂×#41	—	202	8	no	—
180	C₂×#41	—	202	4	no	—
181	C₂²×Q₈	—	101	1	no	—
182	C₂²×D₄	—	101	2	no	—
183	C₂²×D₄	—	101	2	no	—
184	C₂²×D₄	—	101	1	no	—
185	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
186	C₂³×C₄	—	101	2	no	—
187	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
188	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
189	C₄×D₄	—	2	4	no	—
190	C₄×D₄	—	202	8	no	—
191	C₂×C₄⋊C₄	—	101	2	no	—
192	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	—	101	2	no	—
194	C₂×C₂²⋊C₄	—	101	2	no	—
195	C₂×C₂²⋊C₄	—	101	2	no	—
196	C₂×C₂²⋊C₄	—	101	2	no	—
197	C₂×C₂²⋊C₄	—	101	2	no	—
198	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
200	C₂×C₄²	—	101	2	no	—
201	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
202	#95	—	101	1	no	—
203	#95	—	101	2	no	—
204	#95	—	101	2	no	—
205	#95	—	101	1	no	—
206	C₂⁴	—	101	2	no	—
207	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
208	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
209	#41	—	202	32	no	—
210	C₂×Q₈	—	202	4	no	—
211	C₂×Q₈	—	202	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	202	4	no	—
216	C₂×D₄	—	202	4	no	—
217	C₂×D₄	—	202	4	no	—
218	C₂×D₄	—	202	4	no	—
219	C₂×D₄	—	202	4	no	—
220	C₂×D₄	—	202	4	no	—
221	C₂×D₄	—	202	4	no	—
222	C₂×D₄	—	202	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	101	2	no	—
227	C₂²×C₄	—	101	2	no	—
228	C₂²×C₄	—	101	8	no	—
229	C₂²×C₄	—	101	2	no	—
230	C₂²×C₄	—	202	4	no	—
231	C₂²×C₄	—	202	4	no	—
232	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	complemented
233	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
234	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
235	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
237	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	202	4	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	202	4	no	—
243	C₂²⋊C₄	—	202	4	no	—
244	C₂²⋊C₄	—	202	4	no	—
245	C₂²⋊C₄	—	202	4	no	—
246	C₂²⋊C₄	—	202	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	202	4	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	101	4	no	—
258	C₂³	—	202	2	no	—
259	C₂³	—	202	2	no	—
260	C₂³	—	202	2	no	—
261	C₂³	—	202	2	no	—
262	C₂³	—	202	2	no	—
263	C₂³	—	202	2	no	—
264	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
265	C₂³	#13108074	1	2	yes	complemented
266	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
267	Q₈	—	202	8	no	—
268	D₄	—	2	8	no	—
269	D₄	—	202	8	no	—
270	D₄	—	202	16	no	—
271	D₄	—	202	16	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	101	4	no	—
282	C₂×C₄	—	101	4	no	—
283	C₂×C₄	—	202	2	no	—
284	C₂×C₄	—	202	2	no	—
285	C₂×C₄	—	202	2	no	—
286	C₂×C₄	—	202	2	no	—
287	C₂×C₄	—	202	2	no	—
288	C₂×C₄	—	202	2	no	—
289	C₂×C₄	—	202	2	no	—
290	C₂×C₄	—	202	16	no	—
291	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
292	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
293	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
294	C₂×C₄	#13108074	1	2	yes	complemented
295	C₂²	—	2	2	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	404	2	no	—
306	C₂²	—	404	2	no	—
307	C₂²	—	404	2	no	—
308	C₂²	—	404	2	no	—
309	C₂²	—	202	4	no	—
310	C₂²	—	202	4	no	—
311	C₂²	—	202	4	no	—
312	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
313	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
314	C₂²	#41×D₁₀₁	1	2	yes	—
315	C₂²	C₂×#13108074	1	1	yes	characteristic
316	C₂²	C₂×#13108074	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	404	2	no	—
322	C₄	—	202	4	no	—
323	C₄	—	202	4	no	—
324	C₄	#15692223	1	4	yes	—
325	C₂	—	4	2	no	—
326	C₂	—	4	2	no	—
327	C₂	—	404	4	no	—
328	C₂	C₂×#15692223	1	1	yes	characteristic
329	C₂	#20265686	1	2	yes	—
330	C₂	#20265688	1	1	yes	characteristic
331	C₂	#20265723	1	1	yes	characteristic
332	C₂	#20265724	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	#2388
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108074	C₂	semidirect
2. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
3. #20265097	C₂	semidirect
4. #20265094	C₂	semidirect
5. #20265092	C₂	semidirect
6. #20265089	C₂	semidirect
7. #20265086	C₂	semidirect
8. #20265085	C₂	semidirect
9. C₄×#13108051	C₂	semidirect
10. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
11. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
56. C₂×C₄×D₁₀₁	D₄	semidirect
72. C₂²×C₄×C₁₀₁	D₄	semidirect
91. C₂³×C₁₀₁	#41	semidirect
113. C₂×C₄×C₁₀₁	#41	semidirect
124. C₂×Dic₁₀₁	#41	semidirect
167. C₁₀₁	#2388	semidirect
170. C₂×C₄×D₄	D₁₀₁	semidirect
232. C₂²×C₄	C₂²⋊D₁₀₁	semidirect
265. C₂³	#13108074	semidirect
294. C₂×C₄	#13108074	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108074	2
C₂×C₄×C₂²⋊D₁₀₁	2
#20265097	2
#20265094	2
#20265092	2
#20265089	2
#20265086	2
#20265085	2
C₄×#13108051	2
C₄×C₂²⋊Dic₁₀₁	2
C₂×C₄×D₄×C₁₀₁	2
#2388	101