Subgroups — C₄×#137×C₂₅₁ #44547532

Order 32128.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#41×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#441×C₂₅₁	C₂	1	2	yes	maximal, complemented
5	#440×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	C₄×Q₁₆×C₂₅₁	C₂	1	2	yes	maximal, complemented
7	C₄×SD₁₆×C₂₅₁	C₂	1	2	yes	maximal, complemented
8	#428×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#419×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#417×C₂₅₁	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
14	#137×C₂₅₁	C₄	1	8	yes	direct product factor
15	C₂×Q₁₆×C₂₅₁	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₅₁	—	2	1	no	—
20	C₄×Q₈×C₂₅₁	—	2	1	no	—
21	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	—
22	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	—
24	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₂₅₁	—	2	1	no	—
26	C₄×D₄×C₂₅₁	—	2	1	no	—
27	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
29	#117×C₂₅₁	—	2	1	no	—
30	#117×C₂₅₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₅₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₂₅₁	—	2	1	no	—
34	C₂×C₄²×C₂₅₁	—	2	1	no	—
35	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
36	#107×C₂₅₁	C₂²	1	2	yes	—
37	#106×C₂₅₁	C₂²	1	2	yes	—
38	#103×C₂₅₁	C₂²	1	2	yes	—
39	#103×C₂₅₁	C₂²	1	2	yes	—
40	#103×C₂₅₁	C₂²	1	2	yes	—
41	#102×C₂₅₁	C₂²	1	2	yes	—
42	#97×C₂₅₁	C₂²	1	2	yes	—
43	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
44	#41×C₂₅₁	—	2	2	no	—
45	#41×C₂₅₁	C₂×C₄	1	4	yes	—
46	C₂×Q₈×C₂₅₁	—	2	1	no	—
47	C₂×Q₈×C₂₅₁	—	2	2	no	—
48	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
49	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
50	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₂₅₁	—	2	1	no	—
52	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₅₁	—	2	1	no	—
54	C₂²×C₄×C₂₅₁	—	2	1	no	—
55	C₂²×C₄×C₂₅₁	—	2	1	no	—
56	C₂²×C₄×C₂₅₁	—	2	1	no	—
57	C₂²×C₄×C₂₅₁	—	2	1	no	—
58	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
61	Q₁₆×C₂₅₁	C₂×C₄	1	8	yes	complemented
62	SD₁₆×C₂₅₁	C₂×C₄	1	8	yes	complemented
63	M₁₆×C₂₅₁	—	2	2	no	—
64	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	complemented
65	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
66	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₅₁	—	2	2	no	—
68	C₄⋊C₄×C₂₅₁	—	2	2	no	—
69	C₄⋊C₄×C₂₅₁	—	2	1	no	—
70	C₄⋊C₄×C₂₅₁	—	2	1	no	—
71	C₄⋊C₄×C₂₅₁	—	2	1	no	—
72	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
73	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
76	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
77	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
78	C₄²×C₂₅₁	—	2	1	no	—
79	C₄²×C₂₅₁	—	2	1	no	—
80	C₄²×C₂₅₁	—	2	2	no	—
81	C₄²×C₂₅₁	—	2	1	no	—
82	C₄²×C₂₅₁	—	2	2	no	—
83	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
84	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
85	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
86	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
87	C₂³×C₂₅₁	—	2	1	no	—
88	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₂₅₁	—	2	4	no	—
90	Q₈×C₂₅₁	—	2	2	no	—
91	Q₈×C₂₅₁	—	2	1	no	—
92	Q₈×C₂₅₁	C₂²×C₄	1	4	yes	—
93	Q₈×C₂₅₁	C₂²×C₄	1	2	yes	—
94	D₄×C₂₅₁	—	2	4	no	—
95	D₄×C₂₅₁	—	2	1	no	—
96	D₄×C₂₅₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₅₁	—	2	4	no	—
98	C₂×C₄×C₂₅₁	—	2	1	no	—
99	C₂×C₄×C₂₅₁	—	2	2	no	—
100	C₂×C₄×C₂₅₁	—	2	4	no	—
101	C₂×C₄×C₂₅₁	—	2	2	no	—
102	C₂×C₄×C₂₅₁	—	2	1	no	—
103	C₂×C₄×C₂₅₁	—	2	1	no	—
104	C₂×C₄×C₂₅₁	—	2	2	no	—
105	C₂×C₄×C₂₅₁	—	4	1	no	—
106	C₂×C₄×C₂₅₁	—	4	2	no	—
107	C₂×C₄×C₂₅₁	—	4	2	no	—
108	C₂×C₄×C₂₅₁	—	4	2	no	—
109	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₅₁	#41	1	2	yes	—
116	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
117	C₈×C₂₅₁	—	2	2	no	—
118	C₈×C₂₅₁	C₂²×C₄	1	4	yes	—
119	C₂²×C₂₅₁	—	2	1	no	—
120	C₂²×C₂₅₁	—	2	1	no	—
121	C₂²×C₂₅₁	—	2	2	no	—
122	C₂²×C₂₅₁	—	4	1	no	—
123	C₂²×C₂₅₁	—	4	1	no	—
124	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
125	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₂₅₁	—	2	2	no	—
127	C₄×C₂₅₁	—	2	4	no	—
128	C₄×C₂₅₁	—	2	1	no	—
129	C₄×C₂₅₁	—	2	1	no	—
130	C₄×C₂₅₁	—	2	1	no	—
131	C₄×C₂₅₁	—	4	1	no	—
132	C₄×C₂₅₁	—	4	1	no	—
133	C₄×C₂₅₁	—	4	2	no	—
134	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
135	C₄×C₂₅₁	#137	1	2	yes	direct product factor
136	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
137	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₂₅₁	—	2	2	no	—
139	C₂×C₂₅₁	—	4	2	no	—
140	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₂₅₁	C₂×#137	1	1	yes	characteristic
142	C₂×C₂₅₁	#575	1	1	yes	characteristic
143	C₂₅₁	C₄×#137	1	1	yes	characteristic, direct product factor, Sylow
144	C₄×#137	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×#137	C₂×C₂₅₁	1	1	yes	characteristic
146	C₄×#41	C₂×C₂₅₁	1	1	yes	characteristic
147	C₂×C₄×Q₈	C₂×C₂₅₁	1	1	yes	characteristic, complemented
148	#441	C₂×C₂₅₁	1	2	yes	complemented
149	#440	C₂×C₂₅₁	1	2	yes	complemented
150	C₄×Q₁₆	C₂×C₂₅₁	1	2	yes	complemented
151	C₄×SD₁₆	C₂×C₂₅₁	1	2	yes	complemented
152	#428	C₂×C₂₅₁	1	1	yes	characteristic, complemented
153	#419	C₂×C₂₅₁	1	1	yes	characteristic, complemented
154	#417	C₂×C₂₅₁	1	1	yes	characteristic
155	C₄×M₁₆	C₂×C₂₅₁	1	1	yes	characteristic, complemented
156	C₂×#41	C₂²×C₂₅₁	1	1	yes	characteristic
157	C₂²×Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
158	#137	C₄×C₂₅₁	1	8	yes	direct product factor
159	C₂×Q₁₆	C₂²×C₂₅₁	1	2	yes	—
160	C₂×SD₁₆	C₂²×C₂₅₁	1	2	yes	—
161	C₂×M₁₆	C₂²×C₂₅₁	1	1	yes	characteristic
162	C₂×M₁₆	C₂²×C₂₅₁	1	1	yes	characteristic
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	—
166	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	—
167	C₄×Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
168	C₄×Q₈	C₂²×C₂₅₁	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
172	C₄×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	C₂²×C₂₅₁	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
180	#107	C₂²×C₂₅₁	1	2	yes	—
181	#106	C₂²×C₂₅₁	1	2	yes	—
182	#103	C₂²×C₂₅₁	1	2	yes	—
183	#103	C₂²×C₂₅₁	1	2	yes	—
184	#103	C₂²×C₂₅₁	1	2	yes	—
185	#102	C₂²×C₂₅₁	1	2	yes	—
186	#97	C₂²×C₂₅₁	1	2	yes	—
187	C₄×C₈	C₂²×C₂₅₁	1	2	yes	—
188	#41	—	2	2	no	—
189	#41	C₂×C₄×C₂₅₁	1	4	yes	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	C₂×C₄×C₂₅₁	1	4	yes	complemented
193	C₂×Q₈	C₂³×C₂₅₁	1	2	yes	—
194	C₂×Q₈	C₂³×C₂₅₁	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂³×C₂₅₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
203	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
205	Q₁₆	C₂×C₄×C₂₅₁	1	8	yes	complemented
206	SD₁₆	C₂×C₄×C₂₅₁	1	8	yes	complemented
207	M₁₆	—	2	2	no	—
208	M₁₆	C₂×C₄×C₂₅₁	1	4	yes	complemented
209	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
210	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
217	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
218	C₄⋊C₄	C₂³×C₂₅₁	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
228	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
229	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
230	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	2	1	no	—
236	Q₈	C₂²×C₄×C₂₅₁	1	4	yes	—
237	Q₈	C₂²×C₄×C₂₅₁	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂²×C₄×C₂₅₁	1	2	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂²×C₄×C₂₅₁	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
259	C₂×C₄	#41×C₂₅₁	1	2	yes	—
260	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂²×C₄×C₂₅₁	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₄×D₄×C₂₅₁	1	2	yes	—
269	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	C₄×D₄×C₂₅₁	1	2	yes	—
279	C₄	#137×C₂₅₁	1	2	yes	direct product factor
280	C₄	C₂³×C₄×C₂₅₁	1	1	yes	characteristic
281	C₄	C₂×#41×C₂₅₁	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₄×D₄×C₂₅₁	1	1	yes	characteristic
285	C₂	C₂×#137×C₂₅₁	1	1	yes	characteristic
286	C₂	#575×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₅₁	C₂×D₄
commutator	C₄	C₂³×C₄×C₂₅₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#41×C₂₅₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Q₈×C₂₅₁	C₂	semidirect
4. #441×C₂₅₁	C₂	semidirect
5. #440×C₂₅₁	C₂	semidirect
6. C₄×Q₁₆×C₂₅₁	C₂	semidirect
7. C₄×SD₁₆×C₂₅₁	C₂	semidirect
8. #428×C₂₅₁	C₂	semidirect
9. #419×C₂₅₁	C₂	semidirect
11. C₄×M₁₆×C₂₅₁	C₂	semidirect
14. #137×C₂₅₁	C₄	direct (N × H)
48. C₂×Q₈×C₂₅₁	C₂×C₄	semidirect
61. Q₁₆×C₂₅₁	C₂×C₄	semidirect
62. SD₁₆×C₂₅₁	C₂×C₄	semidirect
64. M₁₆×C₂₅₁	C₂×C₄	semidirect
135. C₄×C₂₅₁	#137	direct (N × H)
143. C₂₅₁	C₄×#137	direct (N × H)
144. C₄×#137	C₂₅₁	direct (N × H)
147. C₂×C₄×Q₈	C₂×C₂₅₁	semidirect
148. #441	C₂×C₂₅₁	semidirect
149. #440	C₂×C₂₅₁	semidirect
150. C₄×Q₁₆	C₂×C₂₅₁	semidirect
151. C₄×SD₁₆	C₂×C₂₅₁	semidirect
152. #428	C₂×C₂₅₁	semidirect
153. #419	C₂×C₂₅₁	semidirect
155. C₄×M₁₆	C₂×C₂₅₁	semidirect
158. #137	C₄×C₂₅₁	direct (N × H)
192. C₂×Q₈	C₂×C₄×C₂₅₁	semidirect
205. Q₁₆	C₂×C₄×C₂₅₁	semidirect
206. SD₁₆	C₂×C₄×C₂₅₁	semidirect
208. M₁₆	C₂×C₄×C₂₅₁	semidirect
279. C₄	#137×C₂₅₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#137×C₂₅₁	2
C₄×#41×C₂₅₁	2
C₂×C₄×Q₈×C₂₅₁	2
#441×C₂₅₁	2
#440×C₂₅₁	2
C₄×Q₁₆×C₂₅₁	2
C₄×SD₁₆×C₂₅₁	2
#428×C₂₅₁	2
#419×C₂₅₁	2
#417×C₂₅₁	2
C₄×M₁₆×C₂₅₁	2
C₄×#137	251

Subgroups — C4×#137×C251 #44547532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#137×C₂₅₁ #44547532