Subgroups — C₂×#380×C₁₅₁ #34065610

Order 19328.

1232 conjugacy classes of subgroups.

The subgroup lattice has 188 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	2	yes	maximal
4	#380×C₁₅₁	C₂	1	8	yes	maximal, direct product factor
5	C₂×#95×C₁₅₁	C₂	1	2	yes	maximal, complemented
6	C₂⁵×C₁₅₁	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₁₅₁	—	2	8	no	—
8	C₂³×C₄×C₁₅₁	C₄	1	8	yes	complemented
9	C₂³×C₄×C₁₅₁	C₂²	1	4	yes	—
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	C₂×C₄⋊C₄×C₁₅₁	—	2	8	no	—
14	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	8	yes	complemented
15	C₂×C₂²⋊C₄×C₁₅₁	—	2	8	no	—
16	C₂×C₂²⋊C₄×C₁₅₁	C₂²	1	8	yes	—
17	#95×C₁₅₁	C₂²	1	8	yes	complemented
18	C₂⁴×C₁₅₁	—	2	4	no	—
19	C₂⁴×C₁₅₁	—	2	4	no	—
20	C₂⁴×C₁₅₁	C₂×C₄	1	4	yes	—
21	C₂⁴×C₁₅₁	D₄	1	4	yes	—
22	C₂⁴×C₁₅₁	Q₈	1	4	yes	—
23	C₂⁴×C₁₅₁	C₂³	1	2	yes	—
24	C₂⁴×C₁₅₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₅₁	—	2	4	no	—
26	C₂²×C₄×C₁₅₁	—	2	32	no	—
27	C₂²×C₄×C₁₅₁	—	2	16	no	—
28	C₂²×C₄×C₁₅₁	—	2	8	no	—
29	C₂²×C₄×C₁₅₁	—	2	8	no	—
30	C₂²×C₄×C₁₅₁	—	2	8	no	—
31	C₂²×C₄×C₁₅₁	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₁₅₁	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₅₁	C₂×C₄	1	16	yes	complemented
34	C₂²×C₄×C₁₅₁	D₄	1	8	yes	—
35	C₂²×C₄×C₁₅₁	C₂³	1	8	yes	—
36	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₅₁	—	2	16	no	—
39	C₂²⋊C₄×C₁₅₁	—	2	16	no	—
40	C₂³×C₁₅₁	—	2	8	no	—
41	C₂³×C₁₅₁	—	2	8	no	—
42	C₂³×C₁₅₁	—	2	16	no	—
43	C₂³×C₁₅₁	—	2	16	no	—
44	C₂³×C₁₅₁	—	2	4	no	—
45	C₂³×C₁₅₁	—	2	4	no	—
46	C₂³×C₁₅₁	C₄⋊C₄	1	16	yes	complemented
47	C₂³×C₁₅₁	C₂²×C₄	1	4	yes	—
48	C₂³×C₁₅₁	C₂²×C₄	1	2	yes	—
49	C₂³×C₁₅₁	C₂×D₄	1	4	yes	—
50	C₂³×C₁₅₁	C₂×D₄	1	4	yes	—
51	C₂³×C₁₅₁	C₂×D₄	1	2	yes	—
52	C₂³×C₁₅₁	C₂×Q₈	1	4	yes	—
53	C₂³×C₁₅₁	C₂×Q₈	1	2	yes	—
54	C₂³×C₁₅₁	#41	1	4	yes	—
55	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₁₅₁	—	2	8	no	—
57	C₂×C₄×C₁₅₁	—	2	4	no	—
58	C₂×C₄×C₁₅₁	—	2	32	no	—
59	C₂×C₄×C₁₅₁	—	2	8	no	—
60	C₂×C₄×C₁₅₁	—	2	8	no	—
61	C₂×C₄×C₁₅₁	—	2	8	no	—
62	C₂×C₄×C₁₅₁	—	4	16	no	—
63	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	16	yes	complemented
64	C₂×C₄×C₁₅₁	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₁₅₁	C₂²×C₄	1	4	yes	—
66	C₂×C₄×C₁₅₁	C₂×D₄	1	8	yes	—
67	C₂²×C₁₅₁	—	2	4	no	—
68	C₂²×C₁₅₁	—	2	4	no	—
69	C₂²×C₁₅₁	—	2	8	no	—
70	C₂²×C₁₅₁	—	2	8	no	—
71	C₂²×C₁₅₁	—	2	32	no	—
72	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	4	yes	—
73	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	8	yes	complemented
75	C₂²×C₁₅₁	#117	1	4	yes	—
76	C₂²×C₁₅₁	#121	1	8	yes	—
77	C₂²×C₁₅₁	C₂²⋊Q₈	1	8	yes	—
78	C₂²×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₁₅₁	C₂²×D₄	1	2	yes	—
80	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₁₅₁	C₂×#41	1	2	yes	—
83	C₄×C₁₅₁	—	2	4	no	—
84	C₄×C₁₅₁	—	4	8	no	—
85	C₄×C₁₅₁	C₂×C₂²⋊C₄	1	8	yes	complemented
86	C₂×C₁₅₁	—	2	8	no	—
87	C₂×C₁₅₁	#380	1	8	yes	direct product factor
88	C₂×C₁₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₁₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₁₅₁	C₂×#117	1	1	yes	characteristic
91	C₂×C₁₅₁	C₂×#121	1	2	yes	—
92	C₂×C₁₅₁	C₂×C₂²⋊Q₈	1	2	yes	—
93	C₁₅₁	C₂×#380	1	1	yes	characteristic, direct product factor, Sylow
94	C₂×#380	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂⁴×C₄	C₂×C₁₅₁	1	1	yes	characteristic
96	C₂²×C₄⋊C₄	C₂×C₁₅₁	1	2	yes	complemented
97	C₂²×C₂²⋊C₄	C₂×C₁₅₁	1	2	yes	—
98	#380	C₂×C₁₅₁	1	8	yes	direct product factor
99	C₂×#95	C₂×C₁₅₁	1	2	yes	complemented
100	C₂⁵	C₂²×C₁₅₁	1	1	yes	characteristic
101	C₂³×C₄	—	2	8	no	—
102	C₂³×C₄	C₄×C₁₅₁	1	8	yes	complemented
103	C₂³×C₄	C₂²×C₁₅₁	1	4	yes	—
104	C₂³×C₄	C₂²×C₁₅₁	1	1	yes	characteristic
105	C₂³×C₄	C₂²×C₁₅₁	1	4	yes	—
106	C₂³×C₄	C₂²×C₁₅₁	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	8	no	—
108	C₂×C₄⋊C₄	C₂²×C₁₅₁	1	8	yes	complemented
109	C₂×C₂²⋊C₄	—	2	8	no	—
110	C₂×C₂²⋊C₄	C₂²×C₁₅₁	1	8	yes	—
111	#95	C₂²×C₁₅₁	1	8	yes	complemented
112	C₂⁴	—	2	4	no	—
113	C₂⁴	—	2	4	no	—
114	C₂⁴	C₂×C₄×C₁₅₁	1	4	yes	—
115	C₂⁴	D₄×C₁₅₁	1	4	yes	—
116	C₂⁴	Q₈×C₁₅₁	1	4	yes	—
117	C₂⁴	C₂³×C₁₅₁	1	2	yes	—
118	C₂⁴	C₂³×C₁₅₁	1	1	yes	characteristic
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	—	2	32	no	—
121	C₂²×C₄	—	2	16	no	—
122	C₂²×C₄	—	2	8	no	—
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	2	8	no	—
125	C₂²×C₄	C₂×C₄×C₁₅₁	1	4	yes	—
126	C₂²×C₄	C₂×C₄×C₁₅₁	1	4	yes	—
127	C₂²×C₄	C₂×C₄×C₁₅₁	1	16	yes	complemented
128	C₂²×C₄	D₄×C₁₅₁	1	8	yes	—
129	C₂²×C₄	C₂³×C₁₅₁	1	8	yes	—
130	C₂²×C₄	C₂³×C₁₅₁	1	2	yes	—
131	C₂²×C₄	C₂³×C₁₅₁	1	2	yes	—
132	C₄⋊C₄	—	2	16	no	—
133	C₂²⋊C₄	—	2	16	no	—
134	C₂³	—	2	8	no	—
135	C₂³	—	2	8	no	—
136	C₂³	—	2	16	no	—
137	C₂³	—	2	16	no	—
138	C₂³	—	2	4	no	—
139	C₂³	—	2	4	no	—
140	C₂³	C₄⋊C₄×C₁₅₁	1	16	yes	complemented
141	C₂³	C₂²×C₄×C₁₅₁	1	4	yes	—
142	C₂³	C₂²×C₄×C₁₅₁	1	2	yes	—
143	C₂³	C₂×D₄×C₁₅₁	1	4	yes	—
144	C₂³	C₂×D₄×C₁₅₁	1	4	yes	—
145	C₂³	C₂×D₄×C₁₅₁	1	2	yes	—
146	C₂³	C₂×Q₈×C₁₅₁	1	4	yes	—
147	C₂³	C₂×Q₈×C₁₅₁	1	2	yes	—
148	C₂³	#41×C₁₅₁	1	4	yes	—
149	C₂³	C₂⁴×C₁₅₁	1	1	yes	characteristic
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	32	no	—
153	C₂×C₄	—	2	8	no	—
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	4	16	no	—
157	C₂×C₄	C₂²⋊C₄×C₁₅₁	1	16	yes	complemented
158	C₂×C₄	C₂²×C₄×C₁₅₁	1	4	yes	—
159	C₂×C₄	C₂²×C₄×C₁₅₁	1	4	yes	—
160	C₂×C₄	C₂×D₄×C₁₅₁	1	8	yes	—
161	C₂²	—	2	32	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	—	2	8	no	—
166	C₂²	C₂×C₂²⋊C₄×C₁₅₁	1	4	yes	—
167	C₂²	C₂×C₄⋊C₄×C₁₅₁	1	4	yes	—
168	C₂²	C₂×C₄⋊C₄×C₁₅₁	1	8	yes	complemented
169	C₂²	#117×C₁₅₁	1	4	yes	—
170	C₂²	#121×C₁₅₁	1	8	yes	—
171	C₂²	C₂²⋊Q₈×C₁₅₁	1	8	yes	—
172	C₂²	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
173	C₂²	C₂²×D₄×C₁₅₁	1	2	yes	—
174	C₂²	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
175	C₂²	C₂²×Q₈×C₁₅₁	1	1	yes	characteristic
176	C₂²	C₂×#41×C₁₅₁	1	2	yes	—
177	C₄	—	2	4	no	—
178	C₄	—	4	8	no	—
179	C₄	C₂×C₂²⋊C₄×C₁₅₁	1	8	yes	complemented
180	C₂	—	2	8	no	—
181	C₂	#380×C₁₅₁	1	8	yes	direct product factor
182	C₂	C₂²×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
183	C₂	C₂²×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
184	C₂	C₂×#117×C₁₅₁	1	1	yes	characteristic
185	C₂	C₂×#121×C₁₅₁	1	2	yes	—
186	C₂	C₂×C₂²⋊Q₈×C₁₅₁	1	2	yes	—

	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
4. #380×C₁₅₁	C₂	direct (N × H)
5. C₂×#95×C₁₅₁	C₂	semidirect
8. C₂³×C₄×C₁₅₁	C₄	semidirect
14. C₂×C₄⋊C₄×C₁₅₁	C₂²	semidirect
17. #95×C₁₅₁	C₂²	semidirect
33. C₂²×C₄×C₁₅₁	C₂×C₄	semidirect
46. C₂³×C₁₅₁	C₄⋊C₄	semidirect
63. C₂×C₄×C₁₅₁	C₂²⋊C₄	semidirect
74. C₂²×C₁₅₁	C₂×C₄⋊C₄	semidirect
85. C₄×C₁₅₁	C₂×C₂²⋊C₄	semidirect
87. C₂×C₁₅₁	#380	direct (N × H)
93. C₁₅₁	C₂×#380	direct (N × H)
94. C₂×#380	C₁₅₁	direct (N × H)
96. C₂²×C₄⋊C₄	C₂×C₁₅₁	semidirect
98. #380	C₂×C₁₅₁	direct (N × H)
99. C₂×#95	C₂×C₁₅₁	semidirect
102. C₂³×C₄	C₄×C₁₅₁	semidirect
108. C₂×C₄⋊C₄	C₂²×C₁₅₁	semidirect
111. #95	C₂²×C₁₅₁	semidirect
127. C₂²×C₄	C₂×C₄×C₁₅₁	semidirect
140. C₂³	C₄⋊C₄×C₁₅₁	semidirect
157. C₂×C₄	C₂²⋊C₄×C₁₅₁	semidirect
168. C₂²	C₂×C₄⋊C₄×C₁₅₁	semidirect
179. C₄	C₂×C₂²⋊C₄×C₁₅₁	semidirect
181. C₂	#380×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
#380×C₁₅₁	2
C₂×#95×C₁₅₁	2
C₂×#380	151

Subgroups — C2×#380×C151 #34065610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#380×C₁₅₁ #34065610