Subgroups — #27174193

Order 12928.

832 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×#15692251	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692215	C₂	1	2	yes	maximal, complemented
5	#20265078	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265077	C₂	1	2	yes	maximal, complemented
7	#20265054	C₂	1	2	yes	maximal, complemented
8	#20265053	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13108053	C₂	1	1	yes	maximal, characteristic, complemented
11	#390×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
15	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	#15692251	C₄	1	8	yes	complemented
18	C₄×D₄₀₄	—	2	4	no	—
19	D₄×Dic₁₀₁	—	2	4	no	—
20	#15692215	—	2	8	no	—
21	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×#13108053	C₂²	1	1	yes	characteristic
23	C₂×#13108053	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	2	yes	—
26	C₂×#13108051	C₂²	1	2	yes	—
27	C₂×#13108051	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
30	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
31	#15692162	C₂²	1	1	yes	characteristic
32	#15692162	C₂²	1	1	yes	characteristic
33	#15692162	C₂²	1	2	yes	—
34	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
38	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂³×D₁₀₁	C₂³	1	2	yes	—
40	C₂×D₄₀₄	—	2	2	no	—
41	C₂×D₄₀₄	—	2	2	no	—
42	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
43	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
44	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
45	C₂×C₂²⋊D₁₀₁	—	4	2	no	—
46	C₂×C₂²⋊D₁₀₁	—	4	2	no	—
47	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	complemented
48	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	complemented
49	C₂²×Dic₁₀₁	—	2	4	no	—
50	C₂²×Dic₁₀₁	—	2	4	no	—
51	C₂²×Dic₁₀₁	—	4	2	no	—
52	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
55	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₀₁	—	2	4	no	—
58	C₂×C₄×D₁₀₁	—	2	4	no	—
59	C₂×C₄×D₁₀₁	—	4	2	no	—
60	C₄⋊Dic₁₀₁	D₄	1	4	yes	complemented
61	#13108053	—	2	2	no	—
62	#13108053	D₄	1	4	yes	complemented
63	C₂²⋊Dic₁₀₁	—	2	4	no	—
64	#13108051	—	2	4	no	—
65	#13108051	—	2	4	no	—
66	#13108051	—	4	2	no	—
67	C₄×Dic₁₀₁	—	2	2	no	—
68	C₄×Dic₁₀₁	—	2	4	no	—
69	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
70	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
71	C₂×D₄×C₁₀₁	—	2	2	no	—
72	C₂×D₄×C₁₀₁	—	2	2	no	—
73	C₂×D₄×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	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₁	—	2	2	no	—
79	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
80	C₂²⋊C₄×C₁₀₁	—	4	2	no	—
81	C₄²×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	—	2	2	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	C₂²×C₄	1	4	yes	—
89	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
90	C₂²×D₁₀₁	—	2	2	no	—
91	C₂²×D₁₀₁	—	2	2	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	C₂²×C₄	1	4	yes	—
97	D₄×C₁₀₁	—	2	8	no	—
98	C₂×C₄×C₁₀₁	—	2	1	no	—
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	1	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	—	2	1	no	—
104	C₂×C₄×C₁₀₁	—	2	1	no	—
105	C₂×C₄×C₁₀₁	—	2	2	no	—
106	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₀₁	#41	1	2	yes	—
110	C₂²⋊D₁₀₁	—	2	16	no	—
111	C₂²⋊D₁₀₁	—	4	8	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	—	2	1	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	1	no	—
120	C₂×Dic₁₀₁	—	2	1	no	—
121	C₂×Dic₁₀₁	—	4	8	no	—
122	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
123	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
124	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
125	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
126	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
127	C₂×Dic₁₀₁	#41	1	2	yes	—
128	D₄₀₄	—	2	8	no	—
129	C₄×D₁₀₁	—	4	8	no	—
130	C₂²×C₁₀₁	—	2	4	no	—
131	C₂²×C₁₀₁	—	2	4	no	—
132	C₂²×C₁₀₁	—	2	4	no	—
133	C₂²×C₁₀₁	—	4	2	no	—
134	C₂²×C₁₀₁	—	4	2	no	—
135	C₂²×C₁₀₁	—	4	2	no	—
136	C₂²×C₁₀₁	—	4	2	no	—
137	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
138	C₂²×C₁₀₁	C₂²×D₄	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	C₂×#41	1	1	yes	characteristic
142	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×D₁₀₁	—	2	4	no	—
145	C₂×D₁₀₁	—	2	4	no	—
146	C₂×D₁₀₁	—	2	4	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	—	4	1	no	—
153	C₄×C₁₀₁	—	4	2	no	—
154	C₄×C₁₀₁	C₄×D₄	1	4	yes	complemented
155	Dic₁₀₁	—	2	2	no	—
156	Dic₁₀₁	—	2	4	no	—
157	Dic₁₀₁	—	2	2	no	—
158	Dic₁₀₁	—	4	1	no	—
159	Dic₁₀₁	C₄×D₄	1	4	yes	complemented
160	C₂×C₁₀₁	—	4	4	no	—
161	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂×C₁₀₁	#518	1	1	yes	characteristic
164	C₂×C₁₀₁	D₄²	1	1	yes	characteristic
165	C₂×C₁₀₁	#550	1	1	yes	characteristic
166	C₂×C₁₀₁	#550	1	1	yes	characteristic
167	C₂×C₁₀₁	#559	1	1	yes	characteristic
168	D₁₀₁	—	4	4	no	—
169	#2366	—	101	1	no	maximal, Sylow
170	C₁₀₁	#2366	1	1	yes	characteristic, complemented, Sylow
171	C₂×#127	—	101	1	no	—
172	C₂×C₄×D₄	—	101	1	no	—
173	C₂×C₄×D₄	—	101	2	no	—
174	C₂×C₄×D₄	—	101	1	no	—
175	#390	—	101	1	no	—
176	#390	—	101	2	no	—
177	#390	D₁₀₁	1	1	yes	characteristic, complemented
178	#386	—	101	2	no	—
179	#386	—	101	2	no	—
180	C₄×C₄⋊C₄	—	101	1	no	—
181	C₄×C₄⋊C₄	—	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	—	101	2	no	—
188	#127	—	101	8	no	—
189	C₄×D₄	—	202	4	no	—
190	C₄×D₄	—	202	8	no	—
191	C₄×D₄	—	202	4	no	—
192	C₂×C₄⋊C₄	—	101	1	no	—
193	C₂×C₄⋊C₄	—	101	1	no	—
194	C₂×C₄⋊C₄	—	101	1	no	—
195	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	—	101	2	no	—
197	C₂×C₂²⋊C₄	—	101	2	no	—
198	C₂×C₂²⋊C₄	—	101	2	no	—
199	C₂×C₂²⋊C₄	—	101	2	no	—
200	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
201	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
202	C₂×C₄²	—	101	1	no	—
203	C₂×C₄²	—	101	2	no	—
204	C₂×C₄²	—	101	1	no	—
205	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
206	#95	—	101	2	no	—
207	#95	—	101	1	no	—
208	#95	—	101	1	no	—
209	C₂⁴	—	101	2	no	—
210	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	101	8	no	—
214	C₂×D₄	—	101	8	no	—
215	C₂×D₄	—	101	4	no	—
216	C₂×D₄	—	404	2	no	—
217	C₂×D₄	—	404	2	no	—
218	C₂×D₄	—	202	4	no	—
219	C₂×D₄	—	202	4	no	—
220	C₂×D₄	—	202	2	no	—
221	C₂×D₄	—	202	2	no	—
222	C₂×D₄	C₄×D₁₀₁	1	4	yes	complemented
223	C₂²×C₄	—	101	1	no	—
224	C₂²×C₄	—	101	1	no	—
225	C₂²×C₄	—	101	2	no	—
226	C₂²×C₄	—	101	1	no	—
227	C₂²×C₄	—	101	1	no	—
228	C₂²×C₄	—	404	2	no	—
229	C₂²×C₄	—	404	2	no	—
230	C₂²×C₄	—	202	4	no	—
231	C₂²×C₄	—	202	4	no	—
232	C₂²×C₄	—	202	4	no	—
233	C₂²×C₄	—	202	4	no	—
234	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
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	—	101	4	no	—
240	C₄⋊C₄	—	101	4	no	—
241	C₄⋊C₄	—	202	2	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	404	2	no	—
245	C₂²⋊C₄	—	202	4	no	—
246	C₂²⋊C₄	—	202	4	no	—
247	C₂²⋊C₄	—	202	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	101	4	no	—
250	C₄²	—	202	4	no	—
251	C₄²	—	202	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	101	4	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	C₂×C₄×D₁₀₁	1	4	yes	—
266	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
267	D₄	—	2	8	no	—
268	D₄	—	404	8	no	—
269	D₄	—	202	16	no	—
270	D₄	—	202	8	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	101	2	no	—
280	C₂×C₄	—	101	2	no	—
281	C₂×C₄	—	101	2	no	—
282	C₂×C₄	—	101	2	no	—
283	C₂×C₄	—	101	4	no	—
284	C₂×C₄	—	101	2	no	—
285	C₂×C₄	—	404	8	no	—
286	C₂×C₄	—	404	8	no	—
287	C₂×C₄	—	202	1	no	—
288	C₂×C₄	—	202	1	no	—
289	C₂×C₄	—	202	1	no	—
290	C₂×C₄	—	202	1	no	—
291	C₂×C₄	—	202	2	no	—
292	C₂×C₄	—	202	1	no	—
293	C₂×C₄	—	202	2	no	—
294	C₂×C₄	—	202	1	no	—
295	C₂×C₄	—	202	1	no	—
296	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
297	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
298	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
299	C₂×C₄	#13108073	1	2	yes	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	404	2	no	—
308	C₂²	—	404	2	no	—
309	C₂²	—	404	2	no	—
310	C₂²	—	404	2	no	—
311	C₂²	—	202	4	no	—
312	C₂²	—	202	4	no	—
313	C₂²	—	202	4	no	—
314	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
315	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
316	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
317	C₂²	#41×D₁₀₁	1	1	yes	characteristic
318	C₂²	C₂×#13108073	1	1	yes	characteristic
319	C₂²	#15692300	1	1	yes	characteristic
320	C₂²	#15692301	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	101	4	no	—
325	C₄	—	404	1	no	—
326	C₄	—	202	4	no	—
327	C₄	—	202	2	no	—
328	C₄	—	202	2	no	—
329	C₄	#15692215	1	4	yes	complemented
330	C₂	—	4	4	no	—
331	C₂	—	404	4	no	—
332	C₂	C₄×D₄×D₁₀₁	1	1	yes	characteristic
333	C₂	C₂×#15692215	1	1	yes	characteristic
334	C₂	#20265649	1	1	yes	characteristic
335	C₂	#20265811	1	1	yes	characteristic
336	C₂	#20265845	1	1	yes	characteristic
337	C₂	#20265847	1	1	yes	characteristic
338	C₂	#20265909	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#390×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692251	C₂
hypercenter	#390	D₁₀₁
hypocenter	C₁₀₁	#2366
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #390

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₀₄	C₂	semidirect
3. C₂×D₄×Dic₁₀₁	C₂	semidirect
4. C₂×#15692215	C₂	semidirect
5. #20265078	C₂	semidirect
6. #20265077	C₂	semidirect
7. #20265054	C₂	semidirect
8. #20265053	C₂	semidirect
9. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
10. C₄×#13108053	C₂	semidirect
11. #390×C₁₀₁	C₂	semidirect
17. #15692251	C₄	semidirect
42. C₂×D₄₀₄	C₂×C₄	semidirect
47. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
48. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
60. C₄⋊Dic₁₀₁	D₄	semidirect
62. #13108053	D₄	semidirect
69. C₄×Dic₁₀₁	C₂×C₄	semidirect
73. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
154. C₄×C₁₀₁	C₄×D₄	semidirect
159. Dic₁₀₁	C₄×D₄	semidirect
170. C₁₀₁	#2366	semidirect
177. #390	D₁₀₁	semidirect
222. C₂×D₄	C₄×D₁₀₁	semidirect
329. C₄	#15692215	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692251	2
C₂×C₄×D₄₀₄	2
C₂×D₄×Dic₁₀₁	2
C₂×#15692215	2
#20265078	2
#20265077	2
#20265054	2
#20265053	2
C₄⋊C₄×Dic₁₀₁	2
C₄×#13108053	2
#390×C₁₀₁	2
#2366	101