Subgroups — #27173743

Order 12928.

816 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108051	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20265052	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265051	C₂	1	2	yes	maximal, complemented
6	#20265050	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265049	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265048	C₂	1	2	yes	maximal, complemented
9	#20265036	C₂	1	2	yes	maximal, complemented
10	#20265034	C₂	1	2	yes	maximal, complemented
11	#385×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
18	C₂×#13108053	C₂²	1	2	yes	complemented
19	C₂×#13108053	C₂²	1	2	yes	complemented
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×#13108051	—	2	4	no	—
23	C₂×#13108051	C₂²	1	1	yes	characteristic
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	1	yes	characteristic
26	C₂×#13108051	C₂²	1	2	yes	—
27	C₂×#13108051	C₂²	1	2	yes	—
28	C₂²⋊C₄×D₁₀₁	—	2	4	no	—
29	C₂²⋊C₄×D₁₀₁	C₄	1	8	yes	complemented
30	#15692162	C₂²	1	2	yes	complemented
31	#15692162	C₂²	1	2	yes	complemented
32	#15692162	C₂²	1	2	yes	complemented
33	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	complemented
35	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	#95×C₁₀₁	C₂²	1	2	yes	complemented
38	C₂³×D₁₀₁	—	4	2	no	—
39	C₂³×D₁₀₁	—	4	2	no	—
40	C₂³×D₁₀₁	C₂×C₄	1	4	yes	—
41	C₂³×D₁₀₁	D₄	1	4	yes	—
42	C₂³×D₁₀₁	Q₈	1	4	yes	—
43	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
44	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₀₁	—	2	2	no	—
46	C₂²×Dic₁₀₁	—	4	2	no	—
47	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
51	C₂×C₄×D₁₀₁	—	2	2	no	—
52	C₂×C₄×D₁₀₁	—	2	2	no	—
53	C₂×C₄×D₁₀₁	—	4	2	no	—
54	C₂×C₄×D₁₀₁	—	4	2	no	—
55	C₂×C₄×D₁₀₁	—	4	2	no	—
56	C₂×C₄×D₁₀₁	—	4	2	no	—
57	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
58	C₄⋊Dic₁₀₁	—	4	2	no	—
59	#13108053	—	4	2	no	—
60	#13108053	—	4	2	no	—
61	C₂²⋊Dic₁₀₁	—	2	2	no	—
62	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
63	#13108051	—	2	2	no	—
64	#13108051	—	2	2	no	—
65	#13108051	—	2	4	no	—
66	#13108051	—	2	4	no	—
67	#13108051	C₂×C₄	1	8	yes	complemented
68	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	—	2	2	no	—
70	C₂²×C₄×C₁₀₁	—	4	2	no	—
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
75	C₄⋊C₄×C₁₀₁	—	4	2	no	—
76	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
77	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
78	C₂³×C₁₀₁	—	2	2	no	—
79	C₂³×C₁₀₁	—	2	2	no	—
80	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
82	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
83	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₀₁	—	2	2	no	—
85	C₂²×D₁₀₁	—	2	2	no	—
86	C₂²×D₁₀₁	—	2	2	no	—
87	C₂²×D₁₀₁	—	2	2	no	—
88	C₂²×D₁₀₁	—	4	8	no	—
89	C₂²×D₁₀₁	—	4	4	no	—
90	C₂²×D₁₀₁	—	4	4	no	—
91	C₂²×D₁₀₁	—	4	8	no	—
92	C₂²×D₁₀₁	C₄⋊C₄	1	16	yes	complemented
93	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
95	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
96	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
97	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
98	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₁₀₁	—	2	2	no	—
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	—	2	1	no	—
102	C₂×C₄×C₁₀₁	—	2	1	no	—
103	C₂×C₄×C₁₀₁	—	2	2	no	—
104	C₂×C₄×C₁₀₁	—	2	2	no	—
105	C₂×C₄×C₁₀₁	—	2	2	no	—
106	C₂×C₄×C₁₀₁	—	2	2	no	—
107	C₂×C₄×C₁₀₁	—	2	2	no	—
108	C₂×C₄×C₁₀₁	—	4	4	no	—
109	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	2	2	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	—	2	1	no	—
114	C₂×Dic₁₀₁	—	2	1	no	—
115	C₂×Dic₁₀₁	—	2	2	no	—
116	C₂×Dic₁₀₁	—	2	2	no	—
117	C₂×Dic₁₀₁	—	2	2	no	—
118	C₂×Dic₁₀₁	—	2	2	no	—
119	C₂×Dic₁₀₁	—	4	4	no	—
120	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
121	C₄×D₁₀₁	—	4	8	no	—
122	C₄×D₁₀₁	—	4	4	no	—
123	C₄×D₁₀₁	—	4	4	no	—
124	C₂²×C₁₀₁	—	2	4	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	—	2	4	no	—
128	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
129	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
130	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
132	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
133	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
134	C₂×D₁₀₁	—	2	4	no	—
135	C₂×D₁₀₁	—	2	4	no	—
136	C₂×D₁₀₁	—	2	2	no	—
137	C₂×D₁₀₁	—	2	2	no	—
138	C₂×D₁₀₁	—	2	2	no	—
139	C₂×D₁₀₁	—	2	2	no	—
140	C₂×D₁₀₁	—	2	4	no	—
141	C₂×D₁₀₁	—	2	4	no	—
142	C₂×D₁₀₁	—	4	16	no	—
143	C₂×D₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
144	C₂×D₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
145	C₄×C₁₀₁	—	2	4	no	—
146	C₄×C₁₀₁	—	4	1	no	—
147	C₄×C₁₀₁	—	4	1	no	—
148	C₄×C₁₀₁	—	4	2	no	—
149	Dic₁₀₁	—	2	4	no	—
150	Dic₁₀₁	—	4	1	no	—
151	Dic₁₀₁	—	4	1	no	—
152	Dic₁₀₁	—	4	2	no	—
153	C₂×C₁₀₁	—	2	4	no	—
154	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₁₀₁	#518	1	2	yes	—
156	C₂×C₁₀₁	#534	1	1	yes	characteristic
157	C₂×C₁₀₁	#534	1	1	yes	characteristic
158	C₂×C₁₀₁	#543	1	2	yes	—
159	D₁₀₁	—	2	4	no	—
160	D₁₀₁	—	2	4	no	—
161	#2315	—	101	1	no	maximal, Sylow
162	C₁₀₁	#2315	1	1	yes	characteristic, complemented, Sylow
163	C₂²×C₂²⋊C₄	—	101	1	no	—
164	C₂²×C₂²⋊C₄	—	101	1	no	—
165	C₂²×C₂²⋊C₄	—	101	1	no	—
166	#385	—	101	1	no	—
167	#385	—	101	2	no	—
168	#385	—	101	1	no	—
169	#385	—	101	1	no	—
170	#385	—	101	2	no	—
171	#385	D₁₀₁	1	1	yes	characteristic, complemented
172	#380	—	101	2	no	—
173	#380	—	101	2	no	—
174	C₂⁵	—	101	1	no	—
175	C₂³×C₄	—	101	2	no	—
176	C₂³×C₄	—	101	1	no	—
177	C₂³×C₄	—	101	1	no	—
178	C₂³×C₄	—	101	1	no	—
179	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	101	2	no	—
181	C₂×C₄⋊C₄	—	101	2	no	—
182	C₂×C₄⋊C₄	—	101	2	no	—
183	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	complemented
184	C₂×C₂²⋊C₄	—	101	1	no	—
185	C₂×C₂²⋊C₄	—	101	2	no	—
186	C₂×C₂²⋊C₄	—	101	8	no	—
187	C₂×C₂²⋊C₄	—	101	1	no	—
188	C₂×C₂²⋊C₄	—	101	1	no	—
189	C₂×C₂²⋊C₄	—	101	2	no	—
190	C₂×C₂²⋊C₄	—	101	1	no	—
191	C₂×C₂²⋊C₄	—	101	2	no	—
192	C₂×C₂²⋊C₄	—	202	4	no	—
193	C₂×C₂²⋊C₄	—	202	4	no	—
194	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
196	#95	—	101	2	no	—
197	#95	—	101	2	no	—
198	#95	—	101	2	no	—
199	#95	C₂×D₁₀₁	1	2	yes	complemented
200	C₂⁴	—	101	4	no	—
201	C₂⁴	—	101	4	no	—
202	C₂⁴	—	101	4	no	—
203	C₂⁴	—	101	1	no	—
204	C₂⁴	—	101	1	no	—
205	C₂⁴	—	404	2	no	—
206	C₂⁴	—	404	2	no	—
207	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	101	2	no	—
211	C₂²×C₄	—	101	8	no	—
212	C₂²×C₄	—	101	1	no	—
213	C₂²×C₄	—	101	1	no	—
214	C₂²×C₄	—	101	2	no	—
215	C₂²×C₄	—	404	2	no	—
216	C₂²×C₄	—	404	2	no	—
217	C₂²×C₄	—	404	2	no	—
218	C₂²×C₄	—	404	2	no	—
219	C₂²×C₄	—	404	2	no	—
220	C₂²×C₄	—	202	2	no	—
221	C₂²×C₄	—	202	2	no	—
222	C₂²×C₄	—	202	2	no	—
223	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
225	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	404	2	no	—
229	C₄⋊C₄	—	404	2	no	—
230	C₄⋊C₄	—	404	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	101	4	no	—
233	C₂²⋊C₄	—	101	8	no	—
234	C₂²⋊C₄	—	202	2	no	—
235	C₂²⋊C₄	—	202	2	no	—
236	C₂²⋊C₄	—	202	4	no	—
237	C₂²⋊C₄	—	202	4	no	—
238	C₂²⋊C₄	—	202	2	no	—
239	C₂²⋊C₄	C₄×D₁₀₁	1	4	yes	complemented
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	101	16	no	—
243	C₂³	—	101	2	no	—
244	C₂³	—	101	2	no	—
245	C₂³	—	101	2	no	—
246	C₂³	—	101	2	no	—
247	C₂³	—	101	2	no	—
248	C₂³	—	101	2	no	—
249	C₂³	—	404	4	no	—
250	C₂³	—	404	4	no	—
251	C₂³	—	404	8	no	—
252	C₂³	—	404	8	no	—
253	C₂³	—	202	2	no	—
254	C₂³	—	202	2	no	—
255	C₂³	—	202	2	no	—
256	C₂³	—	202	2	no	—
257	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
258	C₂³	D₄×D₁₀₁	1	2	yes	—
259	C₂³	Q₈×D₁₀₁	1	2	yes	—
260	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
261	C₂×C₄	—	2	2	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	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	101	4	no	—
272	C₂×C₄	—	404	8	no	—
273	C₂×C₄	—	404	4	no	—
274	C₂×C₄	—	404	4	no	—
275	C₂×C₄	—	404	4	no	—
276	C₂×C₄	—	202	1	no	—
277	C₂×C₄	—	202	2	no	—
278	C₂×C₄	—	202	1	no	—
279	C₂×C₄	—	202	2	no	—
280	C₂×C₄	—	202	2	no	—
281	C₂×C₄	—	202	2	no	—
282	C₂×C₄	—	202	2	no	—
283	C₂×C₄	—	202	2	no	—
284	C₂×C₄	—	202	2	no	—
285	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	101	4	no	—
291	C₂²	—	101	4	no	—
292	C₂²	—	404	16	no	—
293	C₂²	—	202	2	no	—
294	C₂²	—	202	2	no	—
295	C₂²	—	202	2	no	—
296	C₂²	—	202	2	no	—
297	C₂²	—	202	4	no	—
298	C₂²	—	202	4	no	—
299	C₂²	—	202	4	no	—
300	C₂²	—	202	4	no	—
301	C₂²	C₄⋊C₄×D₁₀₁	1	4	yes	—
302	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
303	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
304	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
305	C₂²	#15692300	1	2	yes	—
306	C₂²	#15692301	1	2	yes	—
307	C₄	—	2	4	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	404	1	no	—
312	C₄	—	404	1	no	—
313	C₄	—	404	2	no	—
314	C₄	—	202	4	no	—
315	C₂	—	2	4	no	—
316	C₂	—	202	4	no	—
317	C₂	—	202	4	no	—
318	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
319	C₂	#20265649	1	2	yes	—
320	C₂	#20265737	1	1	yes	characteristic
321	C₂	#20265738	1	1	yes	characteristic
322	C₂	#20265802	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#385×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	#385	D₁₀₁
hypocenter	C₁₀₁	#2315
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #385

Complements

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

N	H ≅ G/N	type
4. #20265052	C₂	semidirect
5. #20265051	C₂	semidirect
6. #20265050	C₂	semidirect
7. #20265049	C₂	semidirect
8. #20265048	C₂	semidirect
9. #20265036	C₂	semidirect
10. #20265034	C₂	semidirect
11. #385×C₁₀₁	C₂	semidirect
17. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
18. C₂×#13108053	C₂²	semidirect
19. C₂×#13108053	C₂²	semidirect
29. C₂²⋊C₄×D₁₀₁	C₄	semidirect
30. #15692162	C₂²	semidirect
31. #15692162	C₂²	semidirect
32. #15692162	C₂²	semidirect
34. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
37. #95×C₁₀₁	C₂²	semidirect
57. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
62. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
67. #13108051	C₂×C₄	semidirect
77. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
92. C₂²×D₁₀₁	C₄⋊C₄	semidirect
162. C₁₀₁	#2315	semidirect
171. #385	D₁₀₁	semidirect
183. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
199. #95	C₂×D₁₀₁	semidirect
239. C₂²⋊C₄	C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108051	2
C₂×C₂²⋊C₄×D₁₀₁	2
#20265052	2
#20265051	2
#20265050	2
#20265049	2
#20265048	2
#20265036	2
#20265034	2
#385×C₁₀₁	2
#2315	101