Subgroups — C₁₅₁×#9800 #50314725

Order 38656.

478 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#1997×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#1965×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1566×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1549×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1545×C₁₅₁	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
9	C₂×#127×C₁₅₁	C₂²	1	1	yes	characteristic
10	#486×C₁₅₁	—	2	1	no	—
11	#486×C₁₅₁	—	2	1	no	—
12	#486×C₁₅₁	—	2	2	no	—
13	#431×C₁₅₁	C₂²	1	1	yes	characteristic
14	#423×C₁₅₁	C₂²	1	1	yes	characteristic
15	#423×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×#102×C₁₅₁	C₄	1	4	yes	—
17	C₂×#100×C₁₅₁	—	2	2	no	—
18	C₄×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
20	#343×C₁₅₁	—	2	2	no	—
21	#334×C₁₅₁	—	2	2	no	—
22	#332×C₁₅₁	—	2	2	no	—
23	#331×C₁₅₁	—	2	2	no	—
24	#322×C₁₅₁	—	2	1	no	—
25	C₈²×C₁₅₁	—	2	1	no	—
26	C₂²×Q₈×C₁₅₁	—	2	1	no	—
27	C₂²×D₄×C₁₅₁	—	2	1	no	—
28	C₂²×D₄×C₁₅₁	—	2	1	no	—
29	C₂²×D₄×C₁₅₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₁₅₁	—	2	1	no	—
31	C₂×M₁₆×C₁₅₁	—	2	1	no	—
32	C₂×M₁₆×C₁₅₁	D₄	1	2	yes	—
33	C₂²×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
34	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
35	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
36	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
37	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
38	#127×C₁₅₁	—	2	1	no	—
39	#127×C₁₅₁	—	2	1	no	—
40	#127×C₁₅₁	—	2	1	no	—
41	#127×C₁₅₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₅₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₅₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₁₅₁	—	2	1	no	—
46	C₄⋊C₈×C₁₅₁	—	2	1	no	—
47	C₄⋊C₈×C₁₅₁	—	2	1	no	—
48	C₄⋊C₈×C₁₅₁	—	2	1	no	—
49	C₄⋊C₈×C₁₅₁	D₄	1	2	yes	—
50	#102×C₁₅₁	—	2	4	no	—
51	#102×C₁₅₁	—	2	4	no	—
52	#100×C₁₅₁	—	4	4	no	—
53	C₂²⋊C₈×C₁₅₁	—	2	1	no	—
54	#97×C₁₅₁	—	2	1	no	—
55	C₄×C₈×C₁₅₁	—	2	1	no	—
56	C₄×C₈×C₁₅₁	—	2	1	no	—
57	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₅₁	D₄	1	2	yes	—
60	C₂⁴×C₁₅₁	—	2	2	no	—
61	C₂×Q₈×C₁₅₁	—	4	1	no	—
62	C₂×Q₈×C₁₅₁	—	4	1	no	—
63	C₂×Q₈×C₁₅₁	—	8	2	no	—
64	C₂×D₄×C₁₅₁	—	2	2	no	—
65	C₂×D₄×C₁₅₁	—	2	2	no	—
66	C₂×D₄×C₁₅₁	—	4	4	no	—
67	C₂×D₄×C₁₅₁	—	4	1	no	—
68	C₂×D₄×C₁₅₁	—	4	1	no	—
69	C₂×D₄×C₁₅₁	—	4	1	no	—
70	C₂×D₄×C₁₅₁	—	4	1	no	—
71	C₂×D₄×C₁₅₁	—	4	2	no	—
72	C₂×D₄×C₁₅₁	—	4	2	no	—
73	C₂×D₄×C₁₅₁	—	8	2	no	—
74	C₂²×C₄×C₁₅₁	—	2	2	no	—
75	C₂²×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
78	M₁₆×C₁₅₁	—	2	2	no	—
79	M₁₆×C₁₅₁	—	4	2	no	—
80	M₁₆×C₁₅₁	—	4	2	no	—
81	M₁₆×C₁₅₁	SD₁₆	1	4	yes	—
82	C₂×C₈×C₁₅₁	—	2	2	no	—
83	C₂×C₈×C₁₅₁	—	2	2	no	—
84	C₂×C₈×C₁₅₁	—	4	2	no	—
85	C₂×C₈×C₁₅₁	—	4	1	no	—
86	C₂×C₈×C₁₅₁	—	4	1	no	—
87	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₁₅₁	—	2	2	no	—
90	C₄⋊C₄×C₁₅₁	—	2	2	no	—
91	C₄⋊C₄×C₁₅₁	—	4	1	no	—
92	C₄⋊C₄×C₁₅₁	—	4	1	no	—
93	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₅₁	#41	1	1	yes	characteristic
97	C₂³×C₁₅₁	—	4	2	no	—
98	C₂³×C₁₅₁	—	4	4	no	—
99	C₂³×C₁₅₁	—	8	2	no	—
100	C₂³×C₁₅₁	—	8	2	no	—
101	C₂³×C₁₅₁	C₄×D₄	1	1	yes	characteristic
102	Q₈×C₁₅₁	—	8	2	no	—
103	Q₈×C₁₅₁	—	8	2	no	—
104	D₄×C₁₅₁	—	4	4	no	—
105	D₄×C₁₅₁	—	8	2	no	—
106	D₄×C₁₅₁	—	8	2	no	—
107	D₄×C₁₅₁	—	8	2	no	—
108	D₄×C₁₅₁	—	8	2	no	—
109	D₄×C₁₅₁	—	8	1	no	—
110	D₄×C₁₅₁	—	8	1	no	—
111	C₂×C₄×C₁₅₁	—	2	2	no	—
112	C₂×C₄×C₁₅₁	—	2	2	no	—
113	C₂×C₄×C₁₅₁	—	4	2	no	—
114	C₂×C₄×C₁₅₁	—	8	2	no	—
115	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₅₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₅₁	#124	1	1	yes	characteristic
120	C₂×C₄×C₁₅₁	#127	1	1	yes	characteristic
121	C₂×C₄×C₁₅₁	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₁₅₁	#135	1	2	yes	—
123	C₈×C₁₅₁	—	2	2	no	—
124	C₈×C₁₅₁	—	2	2	no	—
125	C₈×C₁₅₁	—	2	4	no	—
126	C₈×C₁₅₁	—	4	2	no	—
127	C₈×C₁₅₁	—	8	1	no	—
128	C₈×C₁₅₁	—	8	1	no	—
129	C₂²×C₁₅₁	—	2	1	no	—
130	C₂²×C₁₅₁	—	2	1	no	—
131	C₂²×C₁₅₁	—	8	2	no	—
132	C₂²×C₁₅₁	—	8	2	no	—
133	C₂²×C₁₅₁	—	8	2	no	—
134	C₂²×C₁₅₁	—	16	4	no	—
135	C₂²×C₁₅₁	#390	1	1	yes	characteristic
136	C₂²×C₁₅₁	C₄×SD₁₆	1	2	yes	—
137	C₄×C₁₅₁	—	2	1	no	—
138	C₄×C₁₅₁	—	2	1	no	—
139	C₄×C₁₅₁	—	8	2	no	—
140	C₄×C₁₅₁	#413	1	2	yes	—
141	C₄×C₁₅₁	#465	1	2	yes	—
142	C₄×C₁₅₁	#472	1	2	yes	—
143	C₄×C₁₅₁	#492	1	1	yes	characteristic
144	C₄×C₁₅₁	#495	1	1	yes	characteristic
145	C₂×C₁₅₁	—	2	2	no	—
146	C₂×C₁₅₁	—	16	2	no	—
147	C₂×C₁₅₁	#1967	1	1	yes	characteristic
148	C₂×C₁₅₁	#1968	1	1	yes	characteristic
149	C₂×C₁₅₁	#1978	1	1	yes	characteristic
150	#9800	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₁₅₁	#9800	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×#486	C₂×C₁₅₁	1	1	yes	characteristic
153	#1997	C₂×C₁₅₁	1	1	yes	characteristic, complemented
154	#1965	C₂×C₁₅₁	1	1	yes	characteristic
155	#1566	C₂×C₁₅₁	1	1	yes	characteristic, complemented
156	#1549	C₂×C₁₅₁	1	1	yes	characteristic, complemented
157	#1545	C₂×C₁₅₁	1	1	yes	characteristic
158	C₈×M₁₆	C₂×C₁₅₁	1	1	yes	characteristic, complemented
159	C₂×C₄⋊Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
160	C₂×#127	C₂²×C₁₅₁	1	1	yes	characteristic
161	#486	—	2	1	no	—
162	#486	—	2	1	no	—
163	#486	—	2	2	no	—
164	#431	C₂²×C₁₅₁	1	1	yes	characteristic
165	#423	C₂²×C₁₅₁	1	1	yes	characteristic
166	#423	C₂²×C₁₅₁	1	1	yes	characteristic
167	C₂×#102	C₄×C₁₅₁	1	4	yes	—
168	C₂×#100	—	2	2	no	—
169	C₄×M₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
171	#343	—	2	2	no	—
172	#334	—	2	2	no	—
173	#332	—	2	2	no	—
174	#331	—	2	2	no	—
175	#322	—	2	1	no	—
176	C₈²	—	2	1	no	—
177	C₂²×Q₈	—	2	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂²×D₄	—	2	1	no	—
180	C₂²×D₄	C₂×C₄×C₁₅₁	1	2	yes	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	D₄×C₁₅₁	1	2	yes	—
184	C₂²×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	2	1	no	—
189	#127	—	2	1	no	—
190	#127	—	2	1	no	—
191	#127	—	2	1	no	—
192	#127	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₁₅₁	1	2	yes	—
195	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	D₄×C₁₅₁	1	2	yes	—
201	#102	—	2	4	no	—
202	#102	—	2	4	no	—
203	#100	—	4	4	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
209	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
210	C₄×C₈	D₄×C₁₅₁	1	2	yes	—
211	C₂⁴	—	2	2	no	—
212	C₂×Q₈	—	4	1	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	8	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	4	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	8	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
228	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
229	M₁₆	—	2	2	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	SD₁₆×C₁₅₁	1	4	yes	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	2	yes	—
239	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	2	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
247	C₄²	#41×C₁₅₁	1	1	yes	characteristic
248	C₂³	—	4	2	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	2	no	—
252	C₂³	C₄×D₄×C₁₅₁	1	1	yes	characteristic
253	Q₈	—	8	2	no	—
254	Q₈	—	8	2	no	—
255	D₄	—	4	4	no	—
256	D₄	—	8	2	no	—
257	D₄	—	8	2	no	—
258	D₄	—	8	2	no	—
259	D₄	—	8	2	no	—
260	D₄	—	8	1	no	—
261	D₄	—	8	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
267	C₂×C₄	C₄×D₄×C₁₅₁	1	1	yes	characteristic
268	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
269	C₂×C₄	#121×C₁₅₁	1	1	yes	characteristic
270	C₂×C₄	#124×C₁₅₁	1	1	yes	characteristic
271	C₂×C₄	#127×C₁₅₁	1	1	yes	characteristic
272	C₂×C₄	C₂×SD₁₆×C₁₅₁	1	2	yes	—
273	C₂×C₄	#135×C₁₅₁	1	2	yes	—
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	8	1	no	—
279	C₈	—	8	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	16	4	no	—
286	C₂²	#390×C₁₅₁	1	1	yes	characteristic
287	C₂²	C₄×SD₁₆×C₁₅₁	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	2	no	—
291	C₄	#413×C₁₅₁	1	2	yes	—
292	C₄	#465×C₁₅₁	1	2	yes	—
293	C₄	#472×C₁₅₁	1	2	yes	—
294	C₄	#492×C₁₅₁	1	1	yes	characteristic
295	C₄	#495×C₁₅₁	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	#1967×C₁₅₁	1	1	yes	characteristic
299	C₂	#1968×C₁₅₁	1	1	yes	characteristic
300	C₂	#1978×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₅₁
prime radical	C₂×#127×C₁₅₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1997×C₁₅₁	C₂	semidirect
4. #1566×C₁₅₁	C₂	semidirect
5. #1549×C₁₅₁	C₂	semidirect
7. C₈×M₁₆×C₁₅₁	C₂	semidirect
150. #9800	C₁₅₁	direct (N × H)
151. C₁₅₁	#9800	direct (N × H)
153. #1997	C₂×C₁₅₁	semidirect
155. #1566	C₂×C₁₅₁	semidirect
156. #1549	C₂×C₁₅₁	semidirect
158. C₈×M₁₆	C₂×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₁₅₁	2
#1997×C₁₅₁	2
#1965×C₁₅₁	2
#1566×C₁₅₁	2
#1549×C₁₅₁	2
#1545×C₁₅₁	2
C₈×M₁₆×C₁₅₁	2
#9800	151

Subgroups — C151×#9800 #50314725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#9800 #50314725