Subgroups — C₄×#18012787 #34082322

Order 19328.

1492 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18012787	C₂	1	1	yes	maximal, characteristic
2	#23752694	C₂	1	4	yes	maximal, complemented
3	#23752693	C₂	1	6	yes	maximal, complemented
4	#23752690	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752688	C₂	1	4	yes	maximal, complemented
6	C₄×#14559005	C₂	1	4	yes	maximal, complemented
7	C₄×#14559004	C₂	1	6	yes	maximal, complemented
8	C₄×Q₈×D₁₅₁	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#18012787	C₄	1	16	yes	direct product factor
11	C₂×#14559005	C₂²	1	4	yes	—
12	C₂×#14559004	C₂²	1	6	yes	—
13	C₂×Q₈×D₁₅₁	C₂²	1	4	yes	—
14	Q₈×Dic₁₅₁	C₂²	1	4	yes	—
15	Q₈×Dic₁₅₁	C₂²	1	4	yes	—
16	C₄×Dic₃₀₂	C₂²	1	6	yes	—
17	C₄×Dic₃₀₂	C₂²	1	6	yes	—
18	#18012704	C₂²	1	12	yes	—
19	C₄×D₆₀₄	C₂²	1	6	yes	—
20	C₄×D₆₀₄	C₂²	1	6	yes	—
21	#18012701	C₂²	1	12	yes	—
22	C₄×C₂²⋊D₁₅₁	C₂²	1	4	yes	—
23	C₄×C₂²⋊D₁₅₁	C₂²	1	12	yes	—
24	#18012698	C₂²	1	6	yes	—
25	#18012697	C₂²	1	12	yes	—
26	#18012696	C₂²	1	12	yes	—
27	C₄⋊C₄×D₁₅₁	C₂²	1	12	yes	—
28	C₄²×D₁₅₁	C₂²	1	12	yes	—
29	C₂²×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
30	C₄×Q₈×C₁₅₁	C₂²	1	4	yes	—
31	C₄×Q₈×C₁₅₁	C₂²	1	4	yes	—
32	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	3	yes	—
33	C₂×C₄²×C₁₅₁	C₂²	1	3	yes	—
34	#14559005	C₂×C₄	1	32	yes	complemented
35	#14559004	C₂×C₄	1	48	yes	complemented
36	C₂×Dic₃₀₂	C₂³	1	6	yes	—
37	Q₈×D₁₅₁	C₂×C₄	1	32	yes	complemented
38	C₂×D₆₀₄	C₂³	1	6	yes	—
39	C₂×C₂²⋊D₁₅₁	C₂³	1	4	yes	—
40	C₂×C₄×D₁₅₁	C₂³	1	4	yes	—
41	C₂×C₄×D₁₅₁	C₂³	1	12	yes	—
42	C₂×C₄×D₁₅₁	C₂³	1	12	yes	—
43	C₄⋊Dic₁₅₁	C₂³	1	6	yes	—
44	C₄⋊Dic₁₅₁	C₂³	1	6	yes	—
45	Dic₁₅₁⋊C₄	C₂³	1	12	yes	—
46	Dic₁₅₁⋊C₄	C₂³	1	4	yes	—
47	C₂²⋊Dic₁₅₁	C₂³	1	4	yes	—
48	(C₂×D₁₅₁)⋊C₄	C₂³	1	4	yes	—
49	(C₂×D₁₅₁)⋊C₄	C₂³	1	12	yes	—
50	C₄×Dic₁₅₁	C₂³	1	4	yes	—
51	C₄×Dic₁₅₁	C₂³	1	12	yes	—
52	C₄×Dic₁₅₁	C₂³	1	12	yes	—
53	C₂×Q₈×C₁₅₁	C₂×C₄	1	8	yes	complemented
54	C₂×Q₈×C₁₅₁	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₅₁	C₂³	1	3	yes	—
57	C₂²×C₄×C₁₅₁	C₂³	1	3	yes	—
58	C₄⋊C₄×C₁₅₁	C₂³	1	12	yes	—
59	C₄²×C₁₅₁	C₂³	1	6	yes	—
60	C₄²×C₁₅₁	C₂³	1	6	yes	—
61	C₂²×D₁₅₁	C₂⁴	1	4	yes	—
62	Dic₃₀₂	C₂²×C₄	1	24	yes	—
63	D₆₀₄	C₂²×C₄	1	24	yes	—
64	C₂²⋊D₁₅₁	C₂²×C₄	1	16	yes	—
65	C₂×Dic₁₅₁	C₂⁴	1	4	yes	—
66	C₂×Dic₁₅₁	C₂⁴	1	4	yes	—
67	C₂×Dic₁₅₁	C₂⁴	1	4	yes	—
68	C₄×D₁₅₁	—	2	8	no	—
69	C₄×D₁₅₁	—	4	12	no	—
70	C₄×D₁₅₁	C₂²×C₄	1	48	yes	—
71	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
72	Q₈×C₁₅₁	C₂²×C₄	1	16	yes	—
73	C₂×C₄×C₁₅₁	—	2	2	no	—
74	C₂×C₄×C₁₅₁	—	4	3	no	—
75	C₂×C₄×C₁₅₁	C₂²×C₄	1	12	yes	—
76	C₂×C₄×C₁₅₁	C₂⁴	1	6	yes	—
77	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
78	C₂×C₄×C₁₅₁	C₂⁴	1	6	yes	—
79	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
80	C₂²×C₁₅₁	—	2	1	no	—
81	C₂²×C₁₅₁	—	2	1	no	—
82	C₂²×C₁₅₁	C₂³×C₄	1	2	yes	—
83	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
84	C₂×D₁₅₁	—	2	4	no	—
85	C₂×D₁₅₁	—	2	4	no	—
86	C₂×D₁₅₁	C₂³×C₄	1	8	yes	—
87	C₄×C₁₅₁	—	2	1	no	—
88	C₄×C₁₅₁	—	2	6	no	—
89	C₄×C₁₅₁	C₂³×C₄	1	12	yes	—
90	C₄×C₁₅₁	2₋¹⁺⁴	1	2	yes	complemented
91	Dic₁₅₁	—	2	4	no	—
92	Dic₁₅₁	—	2	4	no	—
93	Dic₁₅₁	C₂³×C₄	1	8	yes	—
94	C₂×C₁₅₁	—	2	2	no	—
95	C₂×C₁₅₁	C₂⁴×C₄	1	1	yes	characteristic
96	C₂×C₁₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
97	C₂×C₁₅₁	#585	1	1	yes	characteristic
98	D₁₅₁	—	2	8	no	—
99	C₁₅₁	C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
100	C₄×2₋¹⁺⁴	—	151	1	no	maximal, Sylow
101	C₂×2₋¹⁺⁴	—	151	1	no	—
102	#520	—	151	4	no	—
103	#520	—	151	6	no	—
104	#519	—	151	1	no	—
105	#519	—	151	4	no	—
106	C₄×#41	—	151	4	no	—
107	C₄×#41	—	151	6	no	—
108	C₂×C₄×Q₈	—	151	4	no	—
109	C₂×C₄×Q₈	D₁₅₁	1	1	yes	characteristic, complemented
110	2₋¹⁺⁴	—	151	16	no	—
111	C₂×#41	—	151	4	no	—
112	C₂×#41	—	151	6	no	—
113	C₂²×Q₈	—	151	4	no	—
114	C₂²×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
115	C₄×Q₈	—	151	4	no	—
116	C₄×Q₈	—	151	4	no	—
117	C₄×Q₈	—	151	12	no	—
118	C₄×Q₈	—	151	6	no	—
119	C₄×Q₈	—	151	6	no	—
120	C₄×Q₈	C₂×D₁₅₁	1	4	yes	—
121	C₄×Q₈	C₂×D₁₅₁	1	4	yes	—
122	C₄×D₄	—	151	4	no	—
123	C₄×D₄	—	151	12	no	—
124	C₄×D₄	—	151	12	no	—
125	C₄×D₄	—	151	6	no	—
126	C₄×D₄	—	151	6	no	—
127	#117	—	151	6	no	—
128	#117	—	151	12	no	—
129	#117	—	151	12	no	—
130	C₂×C₄⋊C₄	—	151	12	no	—
131	C₂×C₄⋊C₄	C₂×D₁₅₁	1	3	yes	—
132	C₂×C₄²	—	151	12	no	—
133	C₂×C₄²	C₂×D₁₅₁	1	3	yes	—
134	#41	—	151	32	no	—
135	#41	—	151	48	no	—
136	C₂×Q₈	—	151	32	no	—
137	C₂×Q₈	—	151	6	no	—
138	C₂×Q₈	C₄×D₁₅₁	1	8	yes	complemented
139	C₂×Q₈	C₂²×D₁₅₁	1	4	yes	—
140	C₂×D₄	—	151	4	no	—
141	C₂×D₄	—	151	6	no	—
142	C₂²×C₄	—	151	4	no	—
143	C₂²×C₄	—	151	12	no	—
144	C₂²×C₄	—	151	12	no	—
145	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₁₅₁	1	3	yes	—
147	C₂²×C₄	C₂²×D₁₅₁	1	3	yes	—
148	C₄⋊C₄	—	151	12	no	—
149	C₄⋊C₄	—	151	4	no	—
150	C₄⋊C₄	—	151	6	no	—
151	C₄⋊C₄	—	151	6	no	—
152	C₄⋊C₄	C₂²×D₁₅₁	1	12	yes	—
153	C₂²⋊C₄	—	151	4	no	—
154	C₂²⋊C₄	—	151	12	no	—
155	C₂²⋊C₄	—	151	4	no	—
156	C₄²	—	151	12	no	—
157	C₄²	—	151	12	no	—
158	C₄²	—	151	4	no	—
159	C₄²	C₂²×D₁₅₁	1	6	yes	—
160	C₄²	C₂²×D₁₅₁	1	6	yes	—
161	C₂³	—	151	4	no	—
162	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
163	Q₈	—	151	24	no	—
164	Q₈	C₂×C₄×D₁₅₁	1	16	yes	—
165	D₄	—	151	16	no	—
166	D₄	—	151	24	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	3	no	—
169	C₂×C₄	—	302	8	no	—
170	C₂×C₄	—	604	12	no	—
171	C₂×C₄	—	151	4	no	—
172	C₂×C₄	—	151	4	no	—
173	C₂×C₄	—	151	4	no	—
174	C₂×C₄	—	151	48	no	—
175	C₂×C₄	C₂×C₄×D₁₅₁	1	12	yes	—
176	C₂×C₄	C₂³×D₁₅₁	1	6	yes	—
177	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
178	C₂×C₄	C₂³×D₁₅₁	1	6	yes	—
179	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	302	4	no	—
183	C₂²	—	302	4	no	—
184	C₂²	—	151	8	no	—
185	C₂²	C₂²×C₄×D₁₅₁	1	2	yes	—
186	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	6	no	—
189	C₄	—	302	4	no	—
190	C₄	—	302	4	no	—
191	C₄	—	151	8	no	—
192	C₄	C₂²×C₄×D₁₅₁	1	12	yes	—
193	C₄	#18012787	1	2	yes	direct product factor
194	C₂	—	2	2	no	—
195	C₂	—	302	8	no	—
196	C₂	C₂³×C₄×D₁₅₁	1	1	yes	characteristic
197	C₂	C₂×#18012787	1	1	yes	characteristic
198	C₂	#23753048	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₅₁
commutator	C₂×C₁₅₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₅₁
Fitting	C₂×C₄×Q₈×C₁₅₁	C₂
socle, cosocle	C₂²×C₁₅₁	C₂⁵
prime radical	C₂×#18012787	C₂
hypercenter	C₂×C₄×Q₈	D₁₅₁
hypocenter	C₁₅₁	C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #23752694	C₂	semidirect
3. #23752693	C₂	semidirect
4. #23752690	C₂	semidirect
5. #23752688	C₂	semidirect
6. C₄×#14559005	C₂	semidirect
7. C₄×#14559004	C₂	semidirect
8. C₄×Q₈×D₁₅₁	C₂	semidirect
9. C₂×C₄×Q₈×C₁₅₁	C₂	semidirect
10. #18012787	C₄	direct (N × H)
34. #14559005	C₂×C₄	semidirect
35. #14559004	C₂×C₄	semidirect
37. Q₈×D₁₅₁	C₂×C₄	semidirect
53. C₂×Q₈×C₁₅₁	C₂×C₄	semidirect
90. C₄×C₁₅₁	2₋¹⁺⁴	semidirect
99. C₁₅₁	C₄×2₋¹⁺⁴	semidirect
109. C₂×C₄×Q₈	D₁₅₁	semidirect
138. C₂×Q₈	C₄×D₁₅₁	semidirect
193. C₄	#18012787	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×#18012787	2
#23752694	2
#23752693	2
#23752690	2
#23752688	2
C₄×#14559005	2
C₄×#14559004	2
C₄×Q₈×D₁₅₁	2
C₂×C₄×Q₈×C₁₅₁	2
C₄×2₋¹⁺⁴	151

Subgroups — C4×#18012787 #34082322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#18012787 #34082322