Subgroups — C₂×#385×C₁₅₁ #34065618

Order 19328.

1232 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₁₅₁	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂²⋊C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#385×C₁₅₁	C₂	1	8	yes	maximal, direct product factor
6	C₂×#95×C₁₅₁	C₂	1	2	yes	maximal, complemented
7	C₂⁵×C₁₅₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₅₁	—	2	8	no	—
9	C₂³×C₄×C₁₅₁	—	2	4	no	—
10	C₂³×C₄×C₁₅₁	C₂²	1	4	yes	—
11	C₂³×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₅₁	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₅₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₅₁	—	2	8	no	—
16	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	8	yes	complemented
17	C₂×C₂²⋊C₄×C₁₅₁	—	2	4	no	—
18	C₂×C₂²⋊C₄×C₁₅₁	C₄	1	8	yes	complemented
19	C₂×C₂²⋊C₄×C₁₅₁	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₅₁	C₂²	1	4	yes	—
21	#95×C₁₅₁	C₂²	1	8	yes	complemented
22	C₂⁴×C₁₅₁	—	2	4	no	—
23	C₂⁴×C₁₅₁	—	2	4	no	—
24	C₂⁴×C₁₅₁	C₂×C₄	1	4	yes	—
25	C₂⁴×C₁₅₁	D₄	1	4	yes	—
26	C₂⁴×C₁₅₁	Q₈	1	4	yes	—
27	C₂⁴×C₁₅₁	C₂³	1	2	yes	—
28	C₂⁴×C₁₅₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₁	—	2	32	no	—
30	C₂²×C₄×C₁₅₁	—	2	4	no	—
31	C₂²×C₄×C₁₅₁	—	2	8	no	—
32	C₂²×C₄×C₁₅₁	—	2	4	no	—
33	C₂²×C₄×C₁₅₁	—	2	4	no	—
34	C₂²×C₄×C₁₅₁	—	2	4	no	—
35	C₂²×C₄×C₁₅₁	—	2	16	no	—
36	C₂²×C₄×C₁₅₁	—	2	2	no	—
37	C₂²×C₄×C₁₅₁	—	2	2	no	—
38	C₂²×C₄×C₁₅₁	—	2	4	no	—
39	C₂²×C₄×C₁₅₁	—	2	4	no	—
40	C₂²×C₄×C₁₅₁	C₂×C₄	1	8	yes	complemented
41	C₂²×C₄×C₁₅₁	D₄	1	8	yes	—
42	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₅₁	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₅₁	C₂³	1	4	yes	—
46	C₄⋊C₄×C₁₅₁	—	2	16	no	—
47	C₂²⋊C₄×C₁₅₁	—	2	8	no	—
48	C₂²⋊C₄×C₁₅₁	C₂×C₄	1	16	yes	complemented
49	C₂³×C₁₅₁	—	2	16	no	—
50	C₂³×C₁₅₁	—	2	8	no	—
51	C₂³×C₁₅₁	—	2	16	no	—
52	C₂³×C₁₅₁	—	2	8	no	—
53	C₂³×C₁₅₁	—	2	4	no	—
54	C₂³×C₁₅₁	—	2	4	no	—
55	C₂³×C₁₅₁	C₄⋊C₄	1	16	yes	complemented
56	C₂³×C₁₅₁	C₂²×C₄	1	4	yes	—
57	C₂³×C₁₅₁	C₂²×C₄	1	2	yes	—
58	C₂³×C₁₅₁	C₂×D₄	1	4	yes	—
59	C₂³×C₁₅₁	C₂×D₄	1	2	yes	—
60	C₂³×C₁₅₁	C₂×D₄	1	4	yes	—
61	C₂³×C₁₅₁	C₂×Q₈	1	4	yes	—
62	C₂³×C₁₅₁	C₂×Q₈	1	2	yes	—
63	C₂³×C₁₅₁	#41	1	4	yes	—
64	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
65	C₂×C₄×C₁₅₁	—	2	32	no	—
66	C₂×C₄×C₁₅₁	—	2	16	no	—
67	C₂×C₄×C₁₅₁	—	2	16	no	—
68	C₂×C₄×C₁₅₁	—	2	4	no	—
69	C₂×C₄×C₁₅₁	—	2	4	no	—
70	C₂×C₄×C₁₅₁	—	2	4	no	—
71	C₂×C₄×C₁₅₁	—	2	4	no	—
72	C₂×C₄×C₁₅₁	—	2	2	no	—
73	C₂×C₄×C₁₅₁	—	2	2	no	—
74	C₂×C₄×C₁₅₁	—	2	4	no	—
75	C₂×C₄×C₁₅₁	—	2	4	no	—
76	C₂×C₄×C₁₅₁	—	4	4	no	—
77	C₂×C₄×C₁₅₁	—	4	4	no	—
78	C₂×C₄×C₁₅₁	C₂²×C₄	1	8	yes	complemented
79	C₂×C₄×C₁₅₁	C₂×D₄	1	8	yes	—
80	C₂²×C₁₅₁	—	2	8	no	—
81	C₂²×C₁₅₁	—	2	4	no	—
82	C₂²×C₁₅₁	—	2	8	no	—
83	C₂²×C₁₅₁	—	2	4	no	—
84	C₂²×C₁₅₁	—	2	32	no	—
85	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	4	yes	—
86	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	8	yes	complemented
87	C₂²×C₁₅₁	C₄×D₄	1	8	yes	—
88	C₂²×C₁₅₁	#120	1	4	yes	—
89	C₂²×C₁₅₁	C₂²⋊Q₈	1	8	yes	—
90	C₂²×C₁₅₁	#123	1	4	yes	—
91	C₂²×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₁₅₁	C₂²×D₄	1	2	yes	—
94	C₂²×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
95	C₂²×C₁₅₁	C₂×#41	1	2	yes	—
96	C₄×C₁₅₁	—	2	8	no	—
97	C₄×C₁₅₁	—	2	8	no	—
98	C₄×C₁₅₁	—	4	2	no	—
99	C₄×C₁₅₁	—	4	2	no	—
100	C₂×C₁₅₁	—	2	8	no	—
101	C₂×C₁₅₁	#385	1	8	yes	direct product factor
102	C₂×C₁₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₁₅₁	C₂×C₄×D₄	1	2	yes	—
104	C₂×C₁₅₁	C₂×#120	1	1	yes	characteristic
105	C₂×C₁₅₁	C₂×C₂²⋊Q₈	1	2	yes	—
106	C₂×C₁₅₁	C₂×#123	1	1	yes	characteristic
107	C₁₅₁	C₂×#385	1	1	yes	characteristic, direct product factor, Sylow
108	C₂×#385	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂⁴×C₄	C₂×C₁₅₁	1	1	yes	characteristic
110	C₂²×C₄⋊C₄	C₂×C₁₅₁	1	2	yes	complemented
111	C₂²×C₂²⋊C₄	C₂×C₁₅₁	1	1	yes	characteristic
112	C₂²×C₂²⋊C₄	C₂×C₁₅₁	1	1	yes	characteristic
113	#385	C₂×C₁₅₁	1	8	yes	direct product factor
114	C₂×#95	C₂×C₁₅₁	1	2	yes	complemented
115	C₂⁵	C₂²×C₁₅₁	1	1	yes	characteristic
116	C₂³×C₄	—	2	4	no	—
117	C₂³×C₄	—	2	8	no	—
118	C₂³×C₄	C₂²×C₁₅₁	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₁₅₁	1	1	yes	characteristic
120	C₂³×C₄	C₂²×C₁₅₁	1	4	yes	—
121	C₂³×C₄	C₂²×C₁₅₁	1	2	yes	—
122	C₂³×C₄	C₂²×C₁₅₁	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	8	no	—
124	C₂×C₄⋊C₄	C₂²×C₁₅₁	1	8	yes	complemented
125	C₂×C₂²⋊C₄	—	2	4	no	—
126	C₂×C₂²⋊C₄	C₄×C₁₅₁	1	8	yes	complemented
127	C₂×C₂²⋊C₄	C₂²×C₁₅₁	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₁₅₁	1	4	yes	—
129	#95	C₂²×C₁₅₁	1	8	yes	complemented
130	C₂⁴	—	2	4	no	—
131	C₂⁴	—	2	4	no	—
132	C₂⁴	C₂×C₄×C₁₅₁	1	4	yes	—
133	C₂⁴	D₄×C₁₅₁	1	4	yes	—
134	C₂⁴	Q₈×C₁₅₁	1	4	yes	—
135	C₂⁴	C₂³×C₁₅₁	1	2	yes	—
136	C₂⁴	C₂³×C₁₅₁	1	1	yes	characteristic
137	C₂²×C₄	—	2	32	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	16	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	C₂×C₄×C₁₅₁	1	8	yes	complemented
149	C₂²×C₄	D₄×C₁₅₁	1	8	yes	—
150	C₂²×C₄	C₂³×C₁₅₁	1	2	yes	—
151	C₂²×C₄	C₂³×C₁₅₁	1	2	yes	—
152	C₂²×C₄	C₂³×C₁₅₁	1	4	yes	—
153	C₂²×C₄	C₂³×C₁₅₁	1	4	yes	—
154	C₄⋊C₄	—	2	16	no	—
155	C₂²⋊C₄	—	2	8	no	—
156	C₂²⋊C₄	C₂×C₄×C₁₅₁	1	16	yes	complemented
157	C₂³	—	2	16	no	—
158	C₂³	—	2	8	no	—
159	C₂³	—	2	16	no	—
160	C₂³	—	2	8	no	—
161	C₂³	—	2	4	no	—
162	C₂³	—	2	4	no	—
163	C₂³	C₄⋊C₄×C₁₅₁	1	16	yes	complemented
164	C₂³	C₂²×C₄×C₁₅₁	1	4	yes	—
165	C₂³	C₂²×C₄×C₁₅₁	1	2	yes	—
166	C₂³	C₂×D₄×C₁₅₁	1	4	yes	—
167	C₂³	C₂×D₄×C₁₅₁	1	2	yes	—
168	C₂³	C₂×D₄×C₁₅₁	1	4	yes	—
169	C₂³	C₂×Q₈×C₁₅₁	1	4	yes	—
170	C₂³	C₂×Q₈×C₁₅₁	1	2	yes	—
171	C₂³	#41×C₁₅₁	1	4	yes	—
172	C₂³	C₂⁴×C₁₅₁	1	1	yes	characteristic
173	C₂×C₄	—	2	32	no	—
174	C₂×C₄	—	2	16	no	—
175	C₂×C₄	—	2	16	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₂²×C₄×C₁₅₁	1	8	yes	complemented
187	C₂×C₄	C₂×D₄×C₁₅₁	1	8	yes	—
188	C₂²	—	2	32	no	—
189	C₂²	—	2	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	8	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂×C₄⋊C₄×C₁₅₁	1	4	yes	—
194	C₂²	C₂×C₄⋊C₄×C₁₅₁	1	8	yes	complemented
195	C₂²	C₄×D₄×C₁₅₁	1	8	yes	—
196	C₂²	#120×C₁₅₁	1	4	yes	—
197	C₂²	C₂²⋊Q₈×C₁₅₁	1	8	yes	—
198	C₂²	#123×C₁₅₁	1	4	yes	—
199	C₂²	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
200	C₂²	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
201	C₂²	C₂²×D₄×C₁₅₁	1	2	yes	—
202	C₂²	C₂²×Q₈×C₁₅₁	1	1	yes	characteristic
203	C₂²	C₂×#41×C₁₅₁	1	2	yes	—
204	C₄	—	2	8	no	—
205	C₄	—	2	8	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₂	—	2	8	no	—
209	C₂	#385×C₁₅₁	1	8	yes	direct product factor
210	C₂	C₂²×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
211	C₂	C₂×C₄×D₄×C₁₅₁	1	2	yes	—
212	C₂	C₂×#120×C₁₅₁	1	1	yes	characteristic
213	C₂	C₂×C₂²⋊Q₈×C₁₅₁	1	2	yes	—
214	C₂	C₂×#123×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₅₁	C₂³
commutator	C₂²	C₂³×C₄×C₁₅₁
Frattini, cosocle	C₂³	C₂⁴×C₁₅₁
prime radical	C₂⁵×C₁₅₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₄×C₁₅₁	C₂	semidirect
5. #385×C₁₅₁	C₂	direct (N × H)
6. C₂×#95×C₁₅₁	C₂	semidirect
16. C₂×C₄⋊C₄×C₁₅₁	C₂²	semidirect
18. C₂×C₂²⋊C₄×C₁₅₁	C₄	semidirect
21. #95×C₁₅₁	C₂²	semidirect
40. C₂²×C₄×C₁₅₁	C₂×C₄	semidirect
48. C₂²⋊C₄×C₁₅₁	C₂×C₄	semidirect
55. C₂³×C₁₅₁	C₄⋊C₄	semidirect
78. C₂×C₄×C₁₅₁	C₂²×C₄	semidirect
86. C₂²×C₁₅₁	C₂×C₄⋊C₄	semidirect
101. C₂×C₁₅₁	#385	direct (N × H)
107. C₁₅₁	C₂×#385	direct (N × H)
108. C₂×#385	C₁₅₁	direct (N × H)
110. C₂²×C₄⋊C₄	C₂×C₁₅₁	semidirect
113. #385	C₂×C₁₅₁	direct (N × H)
114. C₂×#95	C₂×C₁₅₁	semidirect
124. C₂×C₄⋊C₄	C₂²×C₁₅₁	semidirect
126. C₂×C₂²⋊C₄	C₄×C₁₅₁	semidirect
129. #95	C₂²×C₁₅₁	semidirect
148. C₂²×C₄	C₂×C₄×C₁₅₁	semidirect
156. C₂²⋊C₄	C₂×C₄×C₁₅₁	semidirect
163. C₂³	C₄⋊C₄×C₁₅₁	semidirect
186. C₂×C₄	C₂²×C₄×C₁₅₁	semidirect
194. C₂²	C₂×C₄⋊C₄×C₁₅₁	semidirect
209. C₂	#385×C₁₅₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₁₅₁	2
C₂²×C₄⋊C₄×C₁₅₁	2
C₂²×C₂²⋊C₄×C₁₅₁	2
#385×C₁₅₁	2
C₂×#95×C₁₅₁	2
C₂×#385	151

Subgroups — C2×#385×C151 #34065618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#385×C₁₅₁ #34065618