Subgroups — C₁₅₁×#23196 #50328121

Order 38656.

884 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3147×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#2957×C₁₅₁	C₂	1	4	yes	maximal, complemented
3	#1976×C₁₅₁	C₂	1	4	yes	maximal, complemented
4	#1884×C₁₅₁	C₂	1	4	yes	maximal, complemented
5	#1772×C₁₅₁	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₅₁	C₂²	1	4	yes	—
7	#568×C₁₅₁	C₂²	1	4	yes	—
8	#532×C₁₅₁	C₂²	1	2	yes	—
9	#495×C₁₅₁	C₄	1	4	yes	complemented
10	#494×C₁₅₁	C₄	1	1	yes	characteristic, complemented
11	#493×C₁₅₁	C₄	1	1	yes	characteristic, complemented
12	#492×C₁₅₁	C₄	1	2	yes	complemented
13	#444×C₁₅₁	—	2	8	no	—
14	#443×C₁₅₁	—	2	8	no	—
15	C₂×#108×C₁₅₁	C₂²	1	2	yes	—
16	#427×C₁₅₁	C₂²	1	2	yes	—
17	#421×C₁₅₁	C₂²	1	8	yes	—
18	#410×C₁₅₁	C₂²	1	4	yes	—
19	#405×C₁₅₁	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	#339×C₁₅₁	C₂²	1	4	yes	—
22	C₂×#41×C₁₅₁	C₂³	1	4	yes	—
23	#135×C₁₅₁	—	2	4	no	—
24	#135×C₁₅₁	—	2	4	no	—
25	#135×C₁₅₁	—	2	4	no	—
26	#135×C₁₅₁	—	2	4	no	—
27	C₂×Q₁₆×C₁₅₁	C₂×C₄	1	4	yes	—
28	C₂×SD₁₆×C₁₅₁	C₂×C₄	1	8	yes	—
29	C₂×D₈×C₁₅₁	C₂×C₄	1	4	yes	—
30	#131×C₁₅₁	—	2	16	no	—
31	C₂×M₁₆×C₁₅₁	—	4	4	no	—
32	C₂×M₁₆×C₁₅₁	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₅₁	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	2	yes	—
35	#127×C₁₅₁	C₂×C₄	1	2	yes	—
36	#124×C₁₅₁	C₂×C₄	1	4	yes	—
37	#121×C₁₅₁	—	4	2	no	—
38	C₄×D₄×C₁₅₁	—	4	2	no	—
39	#117×C₁₅₁	C₂³	1	4	yes	—
40	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
41	#108×C₁₅₁	D₄	1	4	yes	complemented
42	#108×C₁₅₁	D₄	1	4	yes	complemented
43	#107×C₁₅₁	—	2	2	no	—
44	#106×C₁₅₁	—	2	2	no	—
45	#104×C₁₅₁	—	2	16	no	—
46	#99×C₁₅₁	—	4	4	no	—
47	#97×C₁₅₁	—	2	4	no	—
48	C₄×C₈×C₁₅₁	—	2	4	no	—
49	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
52	#41×C₁₅₁	—	2	8	no	—
53	#41×C₁₅₁	—	2	8	no	—
54	C₂×Q₈×C₁₅₁	C₂²×C₄	1	4	yes	—
55	C₂×D₄×C₁₅₁	—	2	4	no	—
56	C₂×D₄×C₁₅₁	—	4	2	no	—
57	C₂×D₄×C₁₅₁	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₁₅₁	—	2	1	no	—
59	C₂²×C₄×C₁₅₁	—	2	4	no	—
60	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
61	Q₁₆×C₁₅₁	—	2	4	no	—
62	Q₁₆×C₁₅₁	—	2	4	no	—
63	SD₁₆×C₁₅₁	—	2	8	no	—
64	SD₁₆×C₁₅₁	—	2	8	no	—
65	D₈×C₁₅₁	—	2	4	no	—
66	D₈×C₁₅₁	—	2	4	no	—
67	M₁₆×C₁₅₁	—	2	4	no	—
68	M₁₆×C₁₅₁	—	4	8	no	—
69	M₁₆×C₁₅₁	C₂×D₄	1	8	yes	—
70	C₂×C₈×C₁₅₁	—	2	4	no	—
71	C₂×C₈×C₁₅₁	—	4	8	no	—
72	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₅₁	#41	1	2	yes	—
77	C₂×C₈×C₁₅₁	#41	1	2	yes	—
78	C₄⋊C₄×C₁₅₁	—	2	4	no	—
79	C₄⋊C₄×C₁₅₁	—	4	2	no	—
80	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
81	C₂²⋊C₄×C₁₅₁	—	4	4	no	—
82	C₄²×C₁₅₁	—	2	1	no	—
83	C₄²×C₁₅₁	—	2	4	no	—
84	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
86	C₂³×C₁₅₁	—	2	4	no	—
87	C₂³×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
88	Q₈×C₁₅₁	—	2	8	no	—
89	D₄×C₁₅₁	—	2	8	no	—
90	D₄×C₁₅₁	—	4	8	no	—
91	D₄×C₁₅₁	—	8	4	no	—
92	C₂×C₄×C₁₅₁	—	2	1	no	—
93	C₂×C₄×C₁₅₁	—	2	4	no	—
94	C₂×C₄×C₁₅₁	—	2	1	no	—
95	C₂×C₄×C₁₅₁	—	4	2	no	—
96	C₂×C₄×C₁₅₁	—	4	4	no	—
97	C₂×C₄×C₁₅₁	—	4	4	no	—
98	C₂×C₄×C₁₅₁	—	4	4	no	—
99	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₅₁	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₁₅₁	C₂×#41	1	2	yes	—
102	C₂×C₄×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
103	C₈×C₁₅₁	—	2	8	no	—
104	C₈×C₁₅₁	C₄×D₄	1	2	yes	—
105	C₈×C₁₅₁	C₄×D₄	1	2	yes	—
106	C₈×C₁₅₁	C₄×D₄	1	2	yes	—
107	C₈×C₁₅₁	C₄×D₄	1	2	yes	—
108	C₂²×C₁₅₁	—	4	1	no	—
109	C₂²×C₁₅₁	—	4	4	no	—
110	C₂²×C₁₅₁	—	8	4	no	—
111	C₂²×C₁₅₁	#518	1	1	yes	characteristic
112	C₂²×C₁₅₁	#550	1	2	yes	—
113	C₄×C₁₅₁	—	2	2	no	—
114	C₄×C₁₅₁	—	4	1	no	—
115	C₄×C₁₅₁	—	4	4	no	—
116	C₄×C₁₅₁	—	8	4	no	—
117	C₄×C₁₅₁	C₂×C₄×D₄	1	2	yes	—
118	C₄×C₁₅₁	D₄²	1	1	yes	characteristic
119	C₄×C₁₅₁	#559	1	1	yes	characteristic
120	C₂×C₁₅₁	—	2	1	no	—
121	C₂×C₁₅₁	—	2	2	no	—
122	C₂×C₁₅₁	—	8	4	no	—
123	C₂×C₁₅₁	#2366	1	1	yes	characteristic
124	#23196	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₁₅₁	#23196	1	1	yes	characteristic, direct product factor, Sylow
126	#3147	C₂×C₁₅₁	1	1	yes	characteristic
127	#2957	C₂×C₁₅₁	1	4	yes	complemented
128	#1976	C₂×C₁₅₁	1	4	yes	complemented
129	#1884	C₂×C₁₅₁	1	4	yes	complemented
130	#1772	C₂×C₁₅₁	1	2	yes	complemented
131	C₂×#135	C₂²×C₁₅₁	1	4	yes	—
132	#568	C₂²×C₁₅₁	1	4	yes	—
133	#532	C₂²×C₁₅₁	1	2	yes	—
134	#495	C₄×C₁₅₁	1	4	yes	complemented
135	#494	C₄×C₁₅₁	1	1	yes	characteristic, complemented
136	#493	C₄×C₁₅₁	1	1	yes	characteristic, complemented
137	#492	C₄×C₁₅₁	1	2	yes	complemented
138	#444	—	2	8	no	—
139	#443	—	2	8	no	—
140	C₂×#108	C₂²×C₁₅₁	1	2	yes	—
141	#427	C₂²×C₁₅₁	1	2	yes	—
142	#421	C₂²×C₁₅₁	1	8	yes	—
143	#410	C₂²×C₁₅₁	1	4	yes	—
144	#405	C₂²×C₁₅₁	1	4	yes	—
145	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
146	#339	C₂²×C₁₅₁	1	4	yes	—
147	C₂×#41	C₂³×C₁₅₁	1	4	yes	—
148	#135	—	2	4	no	—
149	#135	—	2	4	no	—
150	#135	—	2	4	no	—
151	#135	—	2	4	no	—
152	C₂×Q₁₆	C₂×C₄×C₁₅₁	1	4	yes	—
153	C₂×SD₁₆	C₂×C₄×C₁₅₁	1	8	yes	—
154	C₂×D₈	C₂×C₄×C₁₅₁	1	4	yes	—
155	#131	—	2	16	no	—
156	C₂×M₁₆	—	4	4	no	—
157	C₂×M₁₆	C₂³×C₁₅₁	1	4	yes	—
158	C₂²×C₈	C₂³×C₁₅₁	1	2	yes	—
159	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	2	yes	—
160	#127	C₂×C₄×C₁₅₁	1	2	yes	—
161	#124	C₂×C₄×C₁₅₁	1	4	yes	—
162	#121	—	4	2	no	—
163	C₄×D₄	—	4	2	no	—
164	#117	C₂³×C₁₅₁	1	4	yes	—
165	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
166	#108	D₄×C₁₅₁	1	4	yes	complemented
167	#108	D₄×C₁₅₁	1	4	yes	complemented
168	#107	—	2	2	no	—
169	#106	—	2	2	no	—
170	#104	—	2	16	no	—
171	#99	—	4	4	no	—
172	#97	—	2	4	no	—
173	C₄×C₈	—	2	4	no	—
174	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
175	C₄×C₈	C₂×C₄×C₁₅₁	1	1	yes	characteristic
176	C₄×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
177	#41	—	2	8	no	—
178	#41	—	2	8	no	—
179	C₂×Q₈	C₂²×C₄×C₁₅₁	1	4	yes	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	C₂²×C₄×C₁₅₁	1	4	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
186	Q₁₆	—	2	4	no	—
187	Q₁₆	—	2	4	no	—
188	SD₁₆	—	2	8	no	—
189	SD₁₆	—	2	8	no	—
190	D₈	—	2	4	no	—
191	D₈	—	2	4	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	—	4	8	no	—
194	M₁₆	C₂×D₄×C₁₅₁	1	8	yes	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	8	no	—
197	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
198	C₂×C₈	C₂²×C₄×C₁₅₁	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
201	C₂×C₈	#41×C₁₅₁	1	2	yes	—
202	C₂×C₈	#41×C₁₅₁	1	2	yes	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	4	4	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	4	no	—
209	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
210	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
211	C₂³	—	2	4	no	—
212	C₂³	H(5,𝔽₂)×C₁₅₁	1	1	yes	characteristic
213	Q₈	—	2	8	no	—
214	D₄	—	2	8	no	—
215	D₄	—	4	8	no	—
216	D₄	—	8	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₁₅₁	1	2	yes	—
226	C₂×C₄	C₂×#41×C₁₅₁	1	2	yes	—
227	C₂×C₄	H(5,𝔽₂)×C₁₅₁	1	1	yes	characteristic
228	C₈	—	2	8	no	—
229	C₈	C₄×D₄×C₁₅₁	1	2	yes	—
230	C₈	C₄×D₄×C₁₅₁	1	2	yes	—
231	C₈	C₄×D₄×C₁₅₁	1	2	yes	—
232	C₈	C₄×D₄×C₁₅₁	1	2	yes	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	#518×C₁₅₁	1	1	yes	characteristic
237	C₂²	#550×C₁₅₁	1	2	yes	—
238	C₄	—	2	2	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	4	no	—
242	C₄	C₂×C₄×D₄×C₁₅₁	1	2	yes	—
243	C₄	D₄²×C₁₅₁	1	1	yes	characteristic
244	C₄	#559×C₁₅₁	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	2	no	—
247	C₂	—	8	4	no	—
248	C₂	#2366×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₅₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₁₅₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₅₁
socle	C₂×C₁₅₁	#2366
prime radical	#3147×C₁₅₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2957×C₁₅₁	C₂	semidirect
3. #1976×C₁₅₁	C₂	semidirect
4. #1884×C₁₅₁	C₂	semidirect
5. #1772×C₁₅₁	C₂	semidirect
9. #495×C₁₅₁	C₄	semidirect
10. #494×C₁₅₁	C₄	semidirect
11. #493×C₁₅₁	C₄	semidirect
12. #492×C₁₅₁	C₄	semidirect
41. #108×C₁₅₁	D₄	semidirect
42. #108×C₁₅₁	D₄	semidirect
124. #23196	C₁₅₁	direct (N × H)
125. C₁₅₁	#23196	direct (N × H)
127. #2957	C₂×C₁₅₁	semidirect
128. #1976	C₂×C₁₅₁	semidirect
129. #1884	C₂×C₁₅₁	semidirect
130. #1772	C₂×C₁₅₁	semidirect
134. #495	C₄×C₁₅₁	semidirect
135. #494	C₄×C₁₅₁	semidirect
136. #493	C₄×C₁₅₁	semidirect
137. #492	C₄×C₁₅₁	semidirect
166. #108	D₄×C₁₅₁	semidirect
167. #108	D₄×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#3147×C₁₅₁	2
#2957×C₁₅₁	2
#1976×C₁₅₁	2
#1884×C₁₅₁	2
#1772×C₁₅₁	2
#23196	151

Subgroups — C151×#23196 #50328121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#23196 #50328121