Subgroups — #27174152

Order 12928.

788 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265080	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265076	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265070	C₂	1	2	yes	maximal, complemented
7	#20265057	C₂	1	2	yes	maximal, complemented
8	#20265052	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#121×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
13	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
14	D₄×Dic₁₀₁	—	2	4	no	—
15	D₄×Dic₁₀₁	—	2	8	no	—
16	C₂×C₄⋊Dic₁₀₁	—	2	4	no	—
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
29	#15692162	C₂²	1	1	yes	characteristic
30	#15692162	C₂²	1	1	yes	characteristic
31	#15692162	C₂²	1	2	yes	—
32	#15692162	C₂²	1	2	yes	—
33	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
34	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	#121×C₁₀₁	C₄	1	8	yes	complemented
37	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
39	C₂²×Dic₁₀₁	—	2	4	no	—
40	C₂²×Dic₁₀₁	—	2	4	no	—
41	C₂²×Dic₁₀₁	—	2	4	no	—
42	C₂²×Dic₁₀₁	—	2	4	no	—
43	C₂²×Dic₁₀₁	—	4	2	no	—
44	C₂²×Dic₁₀₁	—	4	2	no	—
45	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
50	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₁₀₁	—	2	2	no	—
52	C₄⋊Dic₁₀₁	—	2	4	no	—
53	C₄⋊Dic₁₀₁	—	2	4	no	—
54	C₄⋊Dic₁₀₁	D₄	1	4	yes	complemented
55	C₂²⋊Dic₁₀₁	—	2	2	no	—
56	C₂²⋊Dic₁₀₁	—	2	4	no	—
57	C₂²⋊Dic₁₀₁	—	2	4	no	—
58	C₂²⋊Dic₁₀₁	—	4	2	no	—
59	C₂²⋊Dic₁₀₁	D₄	1	4	yes	complemented
60	C₄×Dic₁₀₁	—	2	2	no	—
61	C₄×Dic₁₀₁	—	2	2	no	—
62	C₄×Dic₁₀₁	—	2	4	no	—
63	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
64	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₀₁	—	2	4	no	—
66	C₂×D₄×C₁₀₁	—	2	4	no	—
67	C₂×D₄×C₁₀₁	—	2	2	no	—
68	C₂×D₄×C₁₀₁	—	2	2	no	—
69	C₂×D₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
70	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
71	C₂²×C₄×C₁₀₁	—	2	2	no	—
72	C₂²×C₄×C₁₀₁	—	2	2	no	—
73	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
74	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
76	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
78	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
79	C₂³×C₁₀₁	—	2	1	no	—
80	C₂³×C₁₀₁	—	2	1	no	—
81	C₂³×C₁₀₁	—	2	1	no	—
82	C₂³×C₁₀₁	—	2	1	no	—
83	C₂³×C₁₀₁	—	2	2	no	—
84	C₂³×C₁₀₁	—	2	2	no	—
85	C₂³×C₁₀₁	—	2	2	no	—
86	C₂³×C₁₀₁	—	2	2	no	—
87	C₂³×C₁₀₁	—	2	2	no	—
88	C₂³×C₁₀₁	—	2	2	no	—
89	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
90	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
92	C₂³×C₁₀₁	#41	1	2	yes	—
93	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
94	D₄×C₁₀₁	—	2	16	no	—
95	D₄×C₁₀₁	—	2	8	no	—
96	C₂×C₄×C₁₀₁	—	2	1	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	1	no	—
99	C₂×C₄×C₁₀₁	—	2	2	no	—
100	C₂×C₄×C₁₀₁	—	2	8	no	—
101	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
103	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₀₁	#41	1	2	yes	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	1	no	—
108	C₂×Dic₁₀₁	—	2	1	no	—
109	C₂×Dic₁₀₁	—	2	1	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	2	1	no	—
112	C₂×Dic₁₀₁	—	2	1	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	1	no	—
118	C₂×Dic₁₀₁	—	2	1	no	—
119	C₂×Dic₁₀₁	—	2	2	no	—
120	C₂×Dic₁₀₁	—	2	1	no	—
121	C₂×Dic₁₀₁	—	4	8	no	—
122	C₂×Dic₁₀₁	—	4	8	no	—
123	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
124	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
125	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	—	2	2	no	—
128	C₂²×C₁₀₁	—	2	2	no	—
129	C₂²×C₁₀₁	—	2	2	no	—
130	C₂²×C₁₀₁	—	2	2	no	—
131	C₂²×C₁₀₁	—	2	2	no	—
132	C₂²×C₁₀₁	—	2	4	no	—
133	C₂²×C₁₀₁	—	2	4	no	—
134	C₂²×C₁₀₁	—	2	4	no	—
135	C₂²×C₁₀₁	—	4	2	no	—
136	C₂²×C₁₀₁	—	4	2	no	—
137	C₂²×C₁₀₁	—	4	2	no	—
138	C₂²×C₁₀₁	—	4	2	no	—
139	C₂²×C₁₀₁	C₄×D₄	1	4	yes	complemented
140	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
141	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
143	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
144	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
145	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₁₀₁	—	2	2	no	—
148	C₄×C₁₀₁	—	2	4	no	—
149	C₄×C₁₀₁	C₄×D₄	1	4	yes	complemented
150	Dic₁₀₁	—	2	2	no	—
151	Dic₁₀₁	—	2	4	no	—
152	Dic₁₀₁	—	2	2	no	—
153	Dic₁₀₁	—	4	2	no	—
154	Dic₁₀₁	—	4	1	no	—
155	Dic₁₀₁	—	4	1	no	—
156	C₂×C₁₀₁	—	2	4	no	—
157	C₂×C₁₀₁	—	4	4	no	—
158	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂×C₁₀₁	#520	1	1	yes	characteristic
161	C₂×C₁₀₁	D₄²	1	1	yes	characteristic
162	C₂×C₁₀₁	#547	1	1	yes	characteristic
163	C₂×C₁₀₁	#548	1	1	yes	characteristic
164	C₂×C₁₀₁	#553	1	1	yes	characteristic
165	#2361	—	101	1	no	maximal, Sylow
166	C₁₀₁	#2361	1	1	yes	characteristic, complemented, Sylow
167	C₂×#121	D₁₀₁	1	1	yes	characteristic
168	C₂×C₄×D₄	—	101	1	no	—
169	C₂×C₄×D₄	—	101	2	no	—
170	C₂²×C₄⋊C₄	—	101	1	no	—
171	#390	—	101	1	no	—
172	#389	—	101	1	no	—
173	#388	—	101	2	no	—
174	#386	—	101	2	no	—
175	#385	—	101	2	no	—
176	C₄×C₄⋊C₄	—	101	1	no	—
177	C₄×C₂²⋊C₄	—	101	1	no	—
178	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
179	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
180	C₂³×C₄	—	101	2	no	—
181	C₂³×C₄	—	101	2	no	—
182	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
183	#121	Dic₁₀₁	1	8	yes	complemented
184	C₄×D₄	—	202	8	no	—
185	C₄×D₄	—	202	4	no	—
186	C₂×C₄⋊C₄	—	101	1	no	—
187	C₂×C₄⋊C₄	—	101	2	no	—
188	C₂×C₄⋊C₄	—	101	1	no	—
189	C₂×C₄⋊C₄	—	101	2	no	—
190	C₂×C₄⋊C₄	—	202	4	no	—
191	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	101	1	no	—
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	1	no	—
197	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
198	C₂×C₄²	—	101	1	no	—
199	C₂×C₄²	—	101	2	no	—
200	C₂×C₄²	—	101	1	no	—
201	#95	—	101	1	no	—
202	#95	—	101	1	no	—
203	#95	—	101	2	no	—
204	#95	—	101	2	no	—
205	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
206	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂×Dic₁₀₁	1	8	yes	complemented
212	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	complemented
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	101	1	no	—
216	C₂²×C₄	—	101	1	no	—
217	C₂²×C₄	—	101	2	no	—
218	C₂²×C₄	—	101	1	no	—
219	C₂²×C₄	—	101	1	no	—
220	C₂²×C₄	—	101	2	no	—
221	C₂²×C₄	—	404	2	no	—
222	C₂²×C₄	—	404	2	no	—
223	C₂²×C₄	—	202	4	no	—
224	C₂²×C₄	—	202	4	no	—
225	C₂²×C₄	—	202	4	no	—
226	C₂²×C₄	—	202	4	no	—
227	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	complemented
228	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
229	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
231	C₄⋊C₄	—	101	4	no	—
232	C₄⋊C₄	—	202	4	no	—
233	C₄⋊C₄	—	202	4	no	—
234	C₄⋊C₄	—	202	2	no	—
235	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	complemented
236	C₂²⋊C₄	—	101	4	no	—
237	C₂²⋊C₄	—	404	2	no	—
238	C₂²⋊C₄	—	202	4	no	—
239	C₂²⋊C₄	—	202	4	no	—
240	C₂²⋊C₄	—	202	2	no	—
241	C₂²⋊C₄	C₂×Dic₁₀₁	1	8	yes	complemented
242	C₄²	—	202	4	no	—
243	C₄²	—	202	2	no	—
244	C₄²	—	202	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	C₂²×Dic₁₀₁	1	4	yes	—
256	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
257	C₂³	D₄×D₁₀₁	1	2	yes	—
258	C₂³	#13108073	1	2	yes	—
259	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
260	D₄	—	2	16	no	—
261	D₄	—	2	8	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	101	2	no	—
268	C₂×C₄	—	101	2	no	—
269	C₂×C₄	—	101	4	no	—
270	C₂×C₄	—	404	8	no	—
271	C₂×C₄	—	404	8	no	—
272	C₂×C₄	—	202	1	no	—
273	C₂×C₄	—	202	2	no	—
274	C₂×C₄	—	202	2	no	—
275	C₂×C₄	—	202	1	no	—
276	C₂×C₄	—	202	1	no	—
277	C₂×C₄	—	202	1	no	—
278	C₂×C₄	—	202	1	no	—
279	C₂×C₄	—	202	2	no	—
280	C₂×C₄	—	202	1	no	—
281	C₂×C₄	—	202	2	no	—
282	C₂×C₄	—	202	1	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	C₂²×Dic₁₀₁	1	2	yes	—
288	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
289	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
290	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
291	C₂×C₄	#13108073	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	D₄×Dic₁₀₁	1	4	yes	complemented
306	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
307	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
308	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
309	C₂²	C₂×#13108073	1	1	yes	characteristic
310	C₂²	C₂×#13108073	1	1	yes	characteristic
311	C₂²	#15692301	1	1	yes	characteristic
312	C₂²	#15692302	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	—	404	2	no	—
316	C₄	—	404	1	no	—
317	C₄	—	404	1	no	—
318	C₄	—	202	2	no	—
319	C₄	—	202	4	no	—
320	C₄	—	202	2	no	—
321	C₄	D₄×Dic₁₀₁	1	4	yes	complemented
322	C₂	—	2	4	no	—
323	C₂	—	4	4	no	—
324	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
325	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
326	C₂	#20265660	1	1	yes	characteristic
327	C₂	#20265811	1	1	yes	characteristic
328	C₂	#20265827	1	1	yes	characteristic
329	C₂	#20265833	1	1	yes	characteristic
330	C₂	#20265879	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting, prime radical	C₂×#121×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#121	D₁₀₁
hypocenter	C₁₀₁	#2361
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Dic₁₀₁	C₂	semidirect
2. C₂×D₄×Dic₁₀₁	C₂	semidirect
3. C₂²×C₄⋊Dic₁₀₁	C₂	semidirect
4. #20265080	C₂	semidirect
5. #20265076	C₂	semidirect
6. #20265070	C₂	semidirect
7. #20265057	C₂	semidirect
8. #20265052	C₂	semidirect
9. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
10. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
36. #121×C₁₀₁	C₄	semidirect
54. C₄⋊Dic₁₀₁	D₄	semidirect
59. C₂²⋊Dic₁₀₁	D₄	semidirect
69. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
70. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
73. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
77. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
78. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
139. C₂²×C₁₀₁	C₄×D₄	semidirect
149. C₄×C₁₀₁	C₄×D₄	semidirect
166. C₁₀₁	#2361	semidirect
183. #121	Dic₁₀₁	semidirect
211. C₂×D₄	C₂×Dic₁₀₁	semidirect
212. C₂×D₄	C₂×Dic₁₀₁	semidirect
227. C₂²×C₄	C₂×Dic₁₀₁	semidirect
235. C₄⋊C₄	C₂×Dic₁₀₁	semidirect
241. C₂²⋊C₄	C₂×Dic₁₀₁	semidirect
305. C₂²	D₄×Dic₁₀₁	semidirect
321. C₄	D₄×Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₁₀₁	2
C₂²×C₄⋊Dic₁₀₁	2
#20265080	2
#20265076	2
#20265070	2
#20265057	2
#20265052	2
C₄⋊C₄×Dic₁₀₁	2
C₄×C₂²⋊Dic₁₀₁	2
C₂×#121×C₁₀₁	2
#2361	101