Subgroups — #27173786

Order 12928.

808 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692223	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13108051	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265077	C₂	1	2	yes	maximal, complemented
4	#20265068	C₂	1	2	yes	maximal, complemented
5	#20265067	C₂	1	2	yes	maximal, complemented
6	#20265057	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265056	C₂	1	2	yes	maximal, complemented
8	#20265049	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265032	C₂	1	1	yes	maximal, characteristic, 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	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
14	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	#15692223	C₄	1	8	yes	complemented
16	C₂×#13108053	C₂²	1	2	yes	complemented
17	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	complemented
20	C₂×#13108051	—	2	4	no	—
21	C₂×#13108051	C₂²	1	2	yes	—
22	C₂×#13108051	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	complemented
27	#15692162	C₂²	1	2	yes	complemented
28	#15692162	C₂²	1	2	yes	complemented
29	#15692162	C₂²	1	2	yes	complemented
30	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
33	C₄×D₄×C₁₀₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₁₀₁	—	2	2	no	—
39	C₂³×D₁₀₁	—	2	2	no	—
40	C₂³×D₁₀₁	—	4	2	no	—
41	C₂³×D₁₀₁	—	4	2	no	—
42	C₂³×D₁₀₁	C₂×C₄	1	4	yes	complemented
43	C₂³×D₁₀₁	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₁₀₁	—	4	2	no	—
45	C₂×C₂²⋊D₁₀₁	—	4	2	no	—
46	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	complemented
47	C₂²×Dic₁₀₁	—	2	2	no	—
48	C₂²×Dic₁₀₁	—	4	2	no	—
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	#13108053	—	4	2	no	—
53	C₂²⋊Dic₁₀₁	—	2	2	no	—
54	C₂²⋊Dic₁₀₁	—	4	2	no	—
55	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
56	#13108051	—	2	4	no	—
57	#13108051	—	2	4	no	—
58	#13108051	—	4	2	no	—
59	#13108051	—	4	2	no	—
60	#13108051	C₂×C₄	1	8	yes	complemented
61	C₄×Dic₁₀₁	—	4	2	no	—
62	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₀₁	—	2	2	no	—
65	C₂×D₄×C₁₀₁	—	2	2	no	—
66	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
67	C₂²×C₄×C₁₀₁	—	2	2	no	—
68	C₂²×C₄×C₁₀₁	—	4	2	no	—
69	C₂²×C₄×C₁₀₁	—	4	2	no	—
70	C₂²×C₄×C₁₀₁	D₄	1	4	yes	complemented
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₁₀₁	—	2	2	no	—
77	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
78	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
79	C₄²×C₁₀₁	—	2	2	no	—
80	C₂³×C₁₀₁	—	2	2	no	—
81	C₂³×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	—	2	1	no	—
83	C₂³×C₁₀₁	—	2	1	no	—
84	C₂³×C₁₀₁	—	2	2	no	—
85	C₂³×C₁₀₁	—	2	2	no	—
86	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
87	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
88	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
89	C₂³×C₁₀₁	#41	1	2	yes	—
90	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
91	C₂²×D₁₀₁	—	2	8	no	—
92	C₂²×D₁₀₁	—	2	2	no	—
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	—	4	8	no	—
99	C₂²×D₁₀₁	—	4	4	no	—
100	C₂²×D₁₀₁	—	4	4	no	—
101	C₂²×D₁₀₁	—	4	8	no	—
102	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
103	D₄×C₁₀₁	—	2	8	no	—
104	C₂×C₄×C₁₀₁	—	2	1	no	—
105	C₂×C₄×C₁₀₁	—	2	1	no	—
106	C₂×C₄×C₁₀₁	—	2	2	no	—
107	C₂×C₄×C₁₀₁	—	2	2	no	—
108	C₂×C₄×C₁₀₁	—	2	1	no	—
109	C₂×C₄×C₁₀₁	—	2	1	no	—
110	C₂×C₄×C₁₀₁	—	2	2	no	—
111	C₂×C₄×C₁₀₁	—	2	2	no	—
112	C₂×C₄×C₁₀₁	—	4	4	no	—
113	C₂×C₄×C₁₀₁	—	4	4	no	—
114	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
117	C₂²⋊D₁₀₁	—	4	8	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	—	2	2	no	—
124	C₂×Dic₁₀₁	—	2	2	no	—
125	C₂×Dic₁₀₁	—	2	2	no	—
126	C₂×Dic₁₀₁	—	4	4	no	—
127	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
128	C₂²×C₁₀₁	—	2	4	no	—
129	C₂²×C₁₀₁	—	2	2	no	—
130	C₂²×C₁₀₁	—	2	2	no	—
131	C₂²×C₁₀₁	—	2	4	no	—
132	C₂²×C₁₀₁	—	2	2	no	—
133	C₂²×C₁₀₁	—	2	4	no	—
134	C₂²×C₁₀₁	—	4	1	no	—
135	C₂²×C₁₀₁	—	4	1	no	—
136	C₂²×C₁₀₁	—	4	2	no	—
137	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
138	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
139	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
140	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
141	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
142	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
143	C₂×D₁₀₁	—	2	4	no	—
144	C₂×D₁₀₁	—	2	4	no	—
145	C₂×D₁₀₁	—	2	4	no	—
146	C₂×D₁₀₁	—	4	16	no	—
147	C₂×D₁₀₁	—	4	2	no	—
148	C₂×D₁₀₁	—	4	2	no	—
149	C₂×D₁₀₁	—	4	2	no	—
150	C₂×D₁₀₁	—	4	2	no	—
151	C₄×C₁₀₁	—	2	2	no	—
152	C₄×C₁₀₁	—	2	2	no	—
153	C₄×C₁₀₁	—	2	2	no	—
154	C₄×C₁₀₁	—	4	1	no	—
155	C₄×C₁₀₁	—	4	1	no	—
156	Dic₁₀₁	—	2	4	no	—
157	Dic₁₀₁	—	4	2	no	—
158	Dic₁₀₁	—	4	2	no	—
159	C₂×C₁₀₁	—	2	4	no	—
160	C₂×C₁₀₁	—	4	2	no	—
161	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×C₁₀₁	#518	1	2	yes	—
163	C₂×C₁₀₁	#534	1	1	yes	characteristic
164	C₂×C₁₀₁	#535	1	1	yes	characteristic
165	C₂×C₁₀₁	#538	1	2	yes	—
166	D₁₀₁	—	4	4	no	—
167	#2321	—	101	1	no	maximal, Sylow
168	C₁₀₁	#2321	1	1	yes	characteristic, complemented, Sylow
169	C₂×#120	—	101	1	no	—
170	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
171	C₂²×C₂²⋊C₄	—	101	1	no	—
172	#390	—	101	2	no	—
173	#388	—	101	2	no	—
174	#388	—	101	2	no	—
175	#386	—	101	1	no	—
176	#386	—	101	2	no	—
177	#385	—	101	1	no	—
178	#379	—	101	1	no	—
179	C₄×C₂²⋊C₄	—	101	1	no	—
180	C₂⁵	—	101	1	no	—
181	C₂²×D₄	—	101	2	no	—
182	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
183	C₂³×C₄	—	101	1	no	—
184	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
185	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
186	#120	—	101	8	no	—
187	C₄×D₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	101	2	no	—
189	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
190	C₂×C₂²⋊C₄	—	101	1	no	—
191	C₂×C₂²⋊C₄	—	101	2	no	—
192	C₂×C₂²⋊C₄	—	101	2	no	—
193	C₂×C₂²⋊C₄	—	101	2	no	—
194	C₂×C₂²⋊C₄	—	101	2	no	—
195	C₂×C₂²⋊C₄	—	101	1	no	—
196	C₂×C₂²⋊C₄	—	101	2	no	—
197	C₂×C₂²⋊C₄	—	101	2	no	—
198	C₂×C₂²⋊C₄	—	202	4	no	—
199	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
200	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
201	C₂×C₄²	—	101	2	no	—
202	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic, complemented
203	#95	—	101	2	no	—
204	#95	—	101	2	no	—
205	#95	—	101	2	no	—
206	C₂⁴	—	101	4	no	—
207	C₂⁴	—	101	2	no	—
208	C₂⁴	—	404	2	no	—
209	C₂⁴	—	404	2	no	—
210	C₂⁴	—	202	2	no	—
211	C₂⁴	—	202	2	no	—
212	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
213	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	101	8	no	—
217	C₂×D₄	—	404	2	no	—
218	C₂×D₄	—	404	2	no	—
219	C₂×D₄	C₄×D₁₀₁	1	4	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	101	2	no	—
224	C₂²×C₄	—	101	2	no	—
225	C₂²×C₄	—	101	2	no	—
226	C₂²×C₄	—	404	2	no	—
227	C₂²×C₄	—	202	2	no	—
228	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	complemented
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	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
233	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	404	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	101	4	no	—
239	C₂²⋊C₄	—	101	8	no	—
240	C₂²⋊C₄	—	404	2	no	—
241	C₂²⋊C₄	—	404	2	no	—
242	C₂²⋊C₄	—	404	2	no	—
243	C₂²⋊C₄	—	202	2	no	—
244	C₂²⋊C₄	—	202	4	no	—
245	C₂²⋊C₄	—	202	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	404	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	101	4	no	—
255	C₂³	—	404	4	no	—
256	C₂³	—	404	4	no	—
257	C₂³	—	404	8	no	—
258	C₂³	—	404	8	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	—	202	2	no	—
265	C₂³	—	202	8	no	—
266	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
267	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
268	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
269	C₂³	#13108074	1	2	yes	—
270	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
271	D₄	—	2	8	no	—
272	D₄	—	404	8	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	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	—	2	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	101	4	no	—
284	C₂×C₄	—	404	4	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	2	no	—
291	C₂×C₄	—	202	2	no	—
292	C₂×C₄	—	202	2	no	—
293	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
294	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
295	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	404	16	no	—
306	C₂²	—	404	2	no	—
307	C₂²	—	404	2	no	—
308	C₂²	—	404	2	no	—
309	C₂²	—	404	2	no	—
310	C₂²	—	202	4	no	—
311	C₂²	—	202	4	no	—
312	C₂²	—	202	4	no	—
313	C₂²	C₄×C₂²⋊D₁₀₁	1	4	yes	—
314	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
315	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
316	C₂²	C₂×#13108074	1	1	yes	characteristic
317	C₂²	#15692300	1	2	yes	—
318	C₂²	#15692301	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	404	2	no	—
325	C₄	—	404	2	no	—
326	C₄	—	202	4	no	—
327	C₂	—	2	4	no	—
328	C₂	—	4	2	no	—
329	C₂	—	404	4	no	—
330	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
331	C₂	#20265649	1	2	yes	—
332	C₂	#20265736	1	1	yes	characteristic
333	C₂	#20265741	1	1	yes	characteristic
334	C₂	#20265760	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692223	C₂
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	#2321
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#13108051	C₂	semidirect
3. #20265077	C₂	semidirect
4. #20265068	C₂	semidirect
5. #20265067	C₂	semidirect
6. #20265057	C₂	semidirect
7. #20265056	C₂	semidirect
8. #20265049	C₂	semidirect
9. #20265032	C₂	semidirect
10. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
11. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
15. #15692223	C₄	semidirect
16. C₂×#13108053	C₂²	semidirect
19. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
26. C₂×C₄×Dic₁₀₁	C₂²	semidirect
27. #15692162	C₂²	semidirect
28. #15692162	C₂²	semidirect
29. #15692162	C₂²	semidirect
32. C₂³×C₄×C₁₀₁	C₂²	semidirect
34. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
36. C₂×C₂²⋊C₄×C₁₀₁	C₂²	semidirect
37. C₂×C₄²×C₁₀₁	C₂²	semidirect
42. C₂³×D₁₀₁	C₂×C₄	semidirect
46. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
55. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
60. #13108051	C₂×C₄	semidirect
70. C₂²×C₄×C₁₀₁	D₄	semidirect
168. C₁₀₁	#2321	semidirect
170. C₂×C₄×D₄	D₁₀₁	semidirect
185. C₂³×C₄	C₂×D₁₀₁	semidirect
189. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
200. C₂×C₂²⋊C₄	C₂×D₁₀₁	semidirect
202. C₂×C₄²	C₂×D₁₀₁	semidirect
228. C₂²×C₄	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×#15692223	2
C₂²×#13108051	2
#20265077	2
#20265068	2
#20265067	2
#20265057	2
#20265056	2
#20265049	2
#20265032	2
C₄×C₂²⋊Dic₁₀₁	2
C₂×C₄×D₄×C₁₀₁	2
#2321	101