Subgroups — C₂×#23752936 #34082660

Order 19328.

1112 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752936	C₂	1	16	yes	maximal, direct product factor
2	C₂×#18012734	C₂	1	4	yes	maximal, complemented
3	C₂×#18012728	C₂	1	2	yes	maximal
4	C₂×#18012719	C₂	1	2	yes	maximal, complemented
5	C₂×Q₈×Dic₁₅₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Dic₃₀₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₆₀₄	C₂	1	1	yes	maximal, characteristic
8	C₂×#18012697	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₃₀₂	C₂²	1	1	yes	characteristic
11	C₂²×D₆₀₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₅₁	C₂²	1	2	yes	—
13	#18012734	C₂²	1	32	yes	complemented
14	#18012728	C₂²	1	16	yes	—
15	#18012719	C₂²	1	16	yes	complemented
16	Q₈×Dic₁₅₁	C₂²	1	16	yes	complemented
17	C₄×Dic₃₀₂	C₂²	1	8	yes	complemented
18	C₄×D₆₀₄	C₂²	1	8	yes	—
19	#18012697	C₂²	1	16	yes	complemented
20	C₂×C₄⋊Dic₁₅₁	C₂²	1	4	yes	—
21	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₁⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₁₅₁)⋊C₄	C₂²	1	4	yes	—
25	C₂×(C₂×D₁₅₁)⋊C₄	C₂²	1	4	yes	—
26	C₂×(C₂×D₁₅₁)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₅₁	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₅₁	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₅₁	C₂²	1	2	yes	—
30	C₄⋊Q₈×C₁₅₁	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
34	C₂³×D₁₅₁	C₂³	1	2	yes	—
35	C₂×Dic₃₀₂	—	2	4	no	—
36	C₂×Dic₃₀₂	C₂³	1	4	yes	—
37	C₂×D₆₀₄	—	2	4	no	—
38	C₂×D₆₀₄	C₂³	1	4	yes	—
39	C₂²×Dic₁₅₁	C₂³	1	2	yes	—
40	C₂²×Dic₁₅₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₅₁	C₂³	1	2	yes	—
42	C₂×C₄×D₁₅₁	—	2	8	no	—
43	C₂×C₄×D₁₅₁	C₂³	1	8	yes	—
44	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
45	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
46	C₄⋊Dic₁₅₁	C₂³	1	16	yes	—
47	Dic₁₅₁⋊C₄	C₂³	1	8	yes	—
48	(C₂×D₁₅₁)⋊C₄	C₂³	1	16	yes	—
49	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
50	(C₂×D₁₅₁)⋊C₄	C₂³	1	16	yes	—
51	C₄×Dic₁₅₁	C₂³	1	16	yes	—
52	C₄×Dic₁₅₁	C₂³	1	8	yes	—
53	C₂×Q₈×C₁₅₁	—	2	8	no	—
54	C₂×Q₈×C₁₅₁	C₂³	1	8	yes	—
55	C₂²×C₄×C₁₅₁	C₂³	1	4	yes	—
56	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₁₅₁	C₂³	1	8	yes	—
60	C₄⋊C₄×C₁₅₁	C₂³	1	8	yes	—
61	C₄²×C₁₅₁	C₂³	1	4	yes	—
62	C₂²×D₁₅₁	—	2	4	no	—
63	C₂²×D₁₅₁	—	2	4	no	—
64	C₂²×D₁₅₁	—	2	4	no	—
65	C₂²×D₁₅₁	C₂⁴	1	4	yes	—
66	Dic₃₀₂	—	2	8	no	—
67	D₆₀₄	—	2	8	no	—
68	C₂×Dic₁₅₁	—	2	4	no	—
69	C₂×Dic₁₅₁	—	2	4	no	—
70	C₂×Dic₁₅₁	—	2	4	no	—
71	C₂×Dic₁₅₁	C₂⁴	1	4	yes	—
72	C₂×Dic₁₅₁	C₂⁴	1	4	yes	—
73	C₂×Dic₁₅₁	C₂⁴	1	4	yes	—
74	C₄×D₁₅₁	—	2	16	no	—
75	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
76	Q₈×C₁₅₁	—	2	16	no	—
77	C₂×C₄×C₁₅₁	—	2	8	no	—
78	C₂×C₄×C₁₅₁	—	2	2	no	—
79	C₂×C₄×C₁₅₁	#41	1	4	yes	—
80	C₂×C₄×C₁₅₁	#41	1	4	yes	—
81	C₂×C₄×C₁₅₁	C₂⁴	1	8	yes	—
82	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
83	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
84	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
85	C₂²×C₁₅₁	C₂×#41	1	2	yes	—
86	C₂²×C₁₅₁	C₂×#41	1	2	yes	—
87	C₂²×C₁₅₁	2₋¹⁺⁴	1	2	yes	—
88	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
89	C₂×D₁₅₁	—	2	4	no	—
90	C₂×D₁₅₁	—	2	4	no	—
91	C₂×D₁₅₁	—	2	4	no	—
92	C₂×D₁₅₁	—	4	8	no	—
93	C₄×C₁₅₁	—	2	8	no	—
94	C₄×C₁₅₁	—	2	2	no	—
95	C₄×C₁₅₁	C₂×#41	1	4	yes	—
96	C₄×C₁₅₁	C₂×#41	1	4	yes	—
97	Dic₁₅₁	—	2	4	no	—
98	Dic₁₅₁	—	2	4	no	—
99	Dic₁₅₁	—	2	4	no	—
100	C₂×C₁₅₁	#556	1	4	yes	complemented
101	C₂×C₁₅₁	C₂²×#41	1	1	yes	characteristic
102	C₂×C₁₅₁	C₂²×#41	1	1	yes	characteristic
103	C₂×C₁₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
104	D₁₅₁	—	4	4	no	—
105	C₁₅₁	C₂×#556	1	1	yes	characteristic, complemented, Sylow
106	C₂×#556	—	151	1	no	maximal, Sylow
107	#556	—	151	16	no	—
108	C₂×C₄⋊Q₈	D₁₅₁	1	1	yes	characteristic, complemented
109	C₂×#126	—	151	4	no	—
110	C₂×#124	—	151	2	no	—
111	C₂×C₂²⋊Q₈	—	151	2	no	—
112	C₂×C₄×Q₈	—	151	2	no	—
113	C₂×C₄×Q₈	—	151	1	no	—
114	C₂×C₄×D₄	—	151	1	no	—
115	C₂×#117	—	151	2	no	—
116	C₂²×Q₈	—	151	1	no	—
117	C₂²×Q₈	C₂×D₁₅₁	1	2	yes	—
118	C₂²×D₄	—	151	1	no	—
119	C₂³×C₄	—	151	2	no	—
120	C₄⋊Q₈	C₂×D₁₅₁	1	8	yes	complemented
121	#126	—	151	32	no	—
122	#124	—	151	16	no	—
123	C₂²⋊Q₈	—	151	16	no	—
124	C₄×Q₈	—	151	16	no	—
125	C₄×Q₈	—	151	8	no	—
126	C₄×D₄	—	151	8	no	—
127	#117	—	151	16	no	—
128	C₂×C₄⋊C₄	—	151	1	no	—
129	C₂×C₄⋊C₄	—	151	1	no	—
130	C₂×C₄⋊C₄	—	151	4	no	—
131	C₂×C₄⋊C₄	—	151	2	no	—
132	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
133	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
134	C₂×C₂²⋊C₄	—	151	4	no	—
135	C₂×C₂²⋊C₄	—	151	2	no	—
136	C₂×C₂²⋊C₄	—	151	4	no	—
137	C₂×C₄²	—	151	4	no	—
138	C₂×C₄²	—	151	2	no	—
139	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
140	C₂⁴	—	151	2	no	—
141	C₂×Q₈	—	2	8	no	—
142	C₂×Q₈	—	302	4	no	—
143	C₂×Q₈	—	151	4	no	—
144	C₂×Q₈	C₂²×D₁₅₁	1	8	yes	—
145	C₂×D₄	—	302	4	no	—
146	C₂×D₄	—	151	4	no	—
147	C₂²×C₄	—	302	8	no	—
148	C₂²×C₄	—	151	2	no	—
149	C₂²×C₄	—	151	2	no	—
150	C₂²×C₄	—	151	2	no	—
151	C₂²×C₄	—	151	8	no	—
152	C₂²×C₄	C₂²×D₁₅₁	1	4	yes	—
153	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
156	C₄⋊C₄	—	151	8	no	—
157	C₄⋊C₄	—	151	4	no	—
158	C₄⋊C₄	—	151	4	no	—
159	C₄⋊C₄	—	151	16	no	—
160	C₄⋊C₄	C₂²×D₁₅₁	1	8	yes	—
161	C₄⋊C₄	C₂²×D₁₅₁	1	8	yes	—
162	C₂²⋊C₄	—	151	16	no	—
163	C₂²⋊C₄	—	151	8	no	—
164	C₂²⋊C₄	—	151	16	no	—
165	C₄²	—	151	16	no	—
166	C₄²	—	151	8	no	—
167	C₄²	C₂²×D₁₅₁	1	4	yes	—
168	C₂³	—	302	4	no	—
169	C₂³	—	302	4	no	—
170	C₂³	—	302	4	no	—
171	C₂³	—	151	4	no	—
172	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
173	Q₈	—	2	16	no	—
174	Q₈	—	302	8	no	—
175	D₄	—	302	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	302	4	no	—
179	C₂×C₄	—	302	4	no	—
180	C₂×C₄	—	302	4	no	—
181	C₂×C₄	—	302	16	no	—
182	C₂×C₄	—	151	4	no	—
183	C₂×C₄	—	151	4	no	—
184	C₂×C₄	—	151	4	no	—
185	C₂×C₄	#14559003	1	4	yes	—
186	C₂×C₄	#14559005	1	4	yes	—
187	C₂×C₄	C₂³×D₁₅₁	1	8	yes	—
188	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
189	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
190	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
191	C₂²	—	302	4	no	—
192	C₂²	—	302	4	no	—
193	C₂²	—	302	4	no	—
194	C₂²	—	604	8	no	—
195	C₂²	C₂×#14559003	1	2	yes	—
196	C₂²	C₂×#14559005	1	2	yes	—
197	C₂²	#18012787	1	2	yes	—
198	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
199	C₄	—	2	8	no	—
200	C₄	—	2	2	no	—
201	C₄	—	302	4	no	—
202	C₄	—	302	4	no	—
203	C₄	—	302	4	no	—
204	C₄	C₂×#14559003	1	4	yes	—
205	C₄	C₂×#14559005	1	4	yes	—
206	C₂	—	604	4	no	—
207	C₂	#23752936	1	4	yes	direct product factor
208	C₂	C₂²×#14559003	1	1	yes	characteristic
209	C₂	C₂²×#14559005	1	1	yes	characteristic
210	C₂	C₂×#18012787	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₅₁
commutator, cosocle	C₂²×C₁₅₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅₁
Fitting	C₂×C₄⋊Q₈×C₁₅₁	C₂
socle	C₂³×C₁₅₁	C₂⁴
prime radical	C₂²×D₆₀₄	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₅₁
hypocenter	C₁₅₁	C₂×#556
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #23752936	C₂	direct (N × H)
2. C₂×#18012734	C₂	semidirect
4. C₂×#18012719	C₂	semidirect
5. C₂×Q₈×Dic₁₅₁	C₂	semidirect
6. C₂×C₄×Dic₃₀₂	C₂	semidirect
8. C₂×#18012697	C₂	semidirect
9. C₂×C₄⋊Q₈×C₁₅₁	C₂	semidirect
13. #18012734	C₂²	semidirect
15. #18012719	C₂²	semidirect
16. Q₈×Dic₁₅₁	C₂²	semidirect
17. C₄×Dic₃₀₂	C₂²	semidirect
19. #18012697	C₂²	semidirect
30. C₄⋊Q₈×C₁₅₁	C₂²	semidirect
100. C₂×C₁₅₁	#556	semidirect
105. C₁₅₁	C₂×#556	semidirect
108. C₂×C₄⋊Q₈	D₁₅₁	semidirect
120. C₄⋊Q₈	C₂×D₁₅₁	semidirect
207. C₂	#23752936	direct (N × H)

Maximal subgroups

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

subgroup	index
#23752936	2
C₂×#18012734	2
C₂×#18012728	2
C₂×#18012719	2
C₂×Q₈×Dic₁₅₁	2
C₂×C₄×Dic₃₀₂	2
C₂×C₄×D₆₀₄	2
C₂×#18012697	2
C₂×C₄⋊Q₈×C₁₅₁	2
C₂×#556	151

Subgroups — C2×#23752936 #34082660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23752936 #34082660