Subgroups — C₂×#23752776 #34082416

Order 19328.

1896 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₆₀₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752776	C₂	1	16	yes	maximal, direct product factor
3	C₂×#18012735	C₂	1	2	yes	maximal, complemented
4	C₂×#18012728	C₂	1	2	yes	maximal, complemented
5	C₂×#18012713	C₂	1	4	yes	maximal, complemented
6	C₂×#18012707	C₂	1	4	yes	maximal, complemented
7	C₂×#18012698	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₅₁	C₂²	1	2	yes	—
10	C₂²×D₆₀₄	—	2	8	no	—
11	C₂²×D₆₀₄	C₂²	1	2	yes	—
12	C₂²×D₆₀₄	C₂²	1	2	yes	—
13	C₂²×D₆₀₄	C₂²	1	8	yes	complemented
14	C₂²×C₂²⋊D₁₅₁	C₂²	1	8	yes	complemented
15	#18012735	C₂²	1	16	yes	complemented
16	#18012728	C₂²	1	16	yes	complemented
17	#18012713	C₂²	1	32	yes	complemented
18	#18012707	C₂²	1	32	yes	complemented
19	#18012698	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	complemented
21	C₂×C₂²⋊Dic₁₅₁	C₂²	1	2	yes	—
22	C₂×(C₂×D₁₅₁)⋊C₄	C₂²	1	8	yes	complemented
23	C₂×C₄×Dic₁₅₁	C₂²	1	2	yes	complemented
24	C₂×#41×C₁₅₁	—	2	8	no	—
25	C₂×#41×C₁₅₁	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₄×C₁₅₁	C₂²	1	2	yes	—
28	C₂²×D₄×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
29	C₂³×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₅₁	C₂²	1	2	yes	complemented
31	C₂³×D₁₅₁	—	2	8	no	—
32	C₂³×D₁₅₁	—	4	8	no	—
33	C₂³×D₁₅₁	C₂³	1	4	yes	—
34	C₂³×D₁₅₁	C₂³	1	8	yes	—
35	C₂×D₆₀₄	—	2	8	no	—
36	C₂×D₆₀₄	—	2	8	no	—
37	C₂×D₆₀₄	—	2	32	no	—
38	C₂×D₆₀₄	C₂³	1	8	yes	—
39	C₂×D₆₀₄	C₂³	1	8	yes	—
40	C₂×C₂²⋊D₁₅₁	—	4	16	no	—
41	C₂×C₂²⋊D₁₅₁	C₂³	1	32	yes	complemented
42	C₂²×Dic₁₅₁	C₂³	1	4	yes	—
43	C₄⋊Dic₁₅₁	C₂³	1	8	yes	complemented
44	C₂²⋊Dic₁₅₁	C₂³	1	8	yes	—
45	(C₂×D₁₅₁)⋊C₄	C₂³	1	32	yes	complemented
46	C₄×Dic₁₅₁	C₂³	1	8	yes	complemented
47	C₂⁴×C₁₅₁	C₂³	1	2	yes	—
48	C₂⁴×C₁₅₁	C₂³	1	1	yes	characteristic
49	#41×C₁₅₁	—	2	32	no	—
50	C₂×Q₈×C₁₅₁	—	2	4	no	—
51	C₂×Q₈×C₁₅₁	C₂³	1	4	yes	complemented
52	C₂×D₄×C₁₅₁	—	2	8	no	—
53	C₂×D₄×C₁₅₁	—	2	4	no	—
54	C₂×D₄×C₁₅₁	C₂³	1	8	yes	—
55	C₂×D₄×C₁₅₁	C₂³	1	4	yes	complemented
56	C₂²×C₄×C₁₅₁	—	2	8	no	—
57	C₂²×C₄×C₁₅₁	D₄	1	8	yes	complemented
58	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₅₁	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₅₁	C₂³	1	8	yes	complemented
63	C₂²×D₁₅₁	—	2	16	no	—
64	C₂²×D₁₅₁	—	2	8	no	—
65	C₂²×D₁₅₁	—	2	16	no	—
66	C₂²×D₁₅₁	—	4	16	no	—
67	C₂²×D₁₅₁	—	4	32	no	—
68	C₂²×D₁₅₁	C₂⁴	1	8	yes	—
69	D₆₀₄	—	2	16	no	—
70	D₆₀₄	—	2	16	no	—
71	C₂²⋊D₁₅₁	—	4	32	no	—
72	C₂×Dic₁₅₁	—	2	8	no	—
73	C₂×Dic₁₅₁	C₂⁴	1	8	yes	—
74	C₂³×C₁₅₁	—	2	4	no	—
75	C₂³×C₁₅₁	—	2	4	no	—
76	C₂³×C₁₅₁	—	2	8	no	—
77	C₂³×C₁₅₁	C₂×D₄	1	4	yes	—
78	C₂³×C₁₅₁	C₂⁴	1	4	yes	—
79	C₂³×C₁₅₁	C₂⁴	1	2	yes	—
80	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
81	Q₈×C₁₅₁	—	2	8	no	—
82	D₄×C₁₅₁	—	2	16	no	—
83	D₄×C₁₅₁	—	2	8	no	—
84	C₂×C₄×C₁₅₁	—	2	4	no	—
85	C₂×C₄×C₁₅₁	—	2	16	no	—
86	C₂×C₄×C₁₅₁	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₁₅₁	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₁₅₁	C₂×D₄	1	16	yes	complemented
89	C₂×C₄×C₁₅₁	C₂⁴	1	4	yes	—
90	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
92	C₂²×C₁₅₁	—	2	8	no	—
93	C₂²×C₁₅₁	—	2	4	no	—
94	C₂²×C₁₅₁	—	2	4	no	—
95	C₂²×C₁₅₁	—	2	8	no	—
96	C₂²×C₁₅₁	—	4	8	no	—
97	C₂²×C₁₅₁	C₂²×D₄	1	4	yes	—
98	C₂²×C₁₅₁	C₂²×D₄	1	2	yes	—
99	C₂²×C₁₅₁	H(5,𝔽₂)	1	4	yes	—
100	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
101	C₂×D₁₅₁	—	2	8	no	—
102	C₂×D₁₅₁	—	2	16	no	—
103	C₂×D₁₅₁	—	4	32	no	—
104	C₂×D₁₅₁	—	4	16	no	—
105	C₄×C₁₅₁	—	2	4	no	—
106	C₄×C₁₅₁	C₂²×D₄	1	4	yes	—
107	C₄×C₁₅₁	C₂²×D₄	1	4	yes	—
108	Dic₁₅₁	—	2	8	no	—
109	C₂×C₁₅₁	—	2	4	no	—
110	C₂×C₁₅₁	—	4	4	no	—
111	C₂×C₁₅₁	#535	1	4	yes	complemented
112	C₂×C₁₅₁	C₂³×D₄	1	1	yes	characteristic
113	C₂×C₁₅₁	C₂×H(5,𝔽₂)	1	2	yes	—
114	D₁₅₁	—	4	8	no	—
115	C₁₅₁	C₂×#535	1	1	yes	characteristic, complemented, Sylow
116	C₂×#535	—	151	1	no	maximal, Sylow
117	C₂²×#41	D₁₅₁	1	1	yes	characteristic, complemented
118	C₂³×D₄	—	151	1	no	—
119	#535	—	151	16	no	—
120	C₂×#127	—	151	2	no	—
121	C₂×#124	—	151	2	no	—
122	C₂×#121	—	151	4	no	—
123	C₂×#120	—	151	4	no	—
124	C₂×#117	—	151	1	no	—
125	C₂⁵	—	151	2	no	—
126	C₂×#41	—	2	8	no	—
127	C₂×#41	C₂×D₁₅₁	1	8	yes	complemented
128	C₂²×Q₈	C₂×D₁₅₁	1	1	yes	characteristic, complemented
129	C₂²×D₄	—	302	8	no	—
130	C₂²×D₄	—	151	8	no	—
131	C₂²×D₄	—	151	2	no	—
132	C₂²×D₄	—	151	2	no	—
133	C₂²×D₄	—	151	8	no	—
134	C₂²×D₄	C₂×D₁₅₁	1	2	yes	—
135	C₂²×D₄	C₂×D₁₅₁	1	1	yes	characteristic, complemented
136	C₂³×C₄	C₂×D₁₅₁	1	1	yes	characteristic
137	C₂³×C₄	C₂×D₁₅₁	1	2	yes	complemented
138	#127	—	151	16	no	—
139	#124	—	151	16	no	—
140	#121	—	151	32	no	—
141	#120	—	151	32	no	—
142	#117	—	151	8	no	—
143	C₂×C₄⋊C₄	—	151	2	no	—
144	C₂×C₂²⋊C₄	—	151	2	no	—
145	C₂×C₂²⋊C₄	—	151	8	no	—
146	C₂×C₄²	—	151	2	no	—
147	C₂⁴	—	302	8	no	—
148	C₂⁴	—	604	8	no	—
149	C₂⁴	—	151	4	no	—
150	C₂⁴	—	151	8	no	—
151	C₂⁴	C₂²×D₁₅₁	1	2	yes	—
152	C₂⁴	C₂²×D₁₅₁	1	1	yes	characteristic
153	#41	—	2	32	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	C₂²×D₁₅₁	1	4	yes	complemented
156	C₂×D₄	—	2	8	no	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	302	8	no	—
159	C₂×D₄	—	302	8	no	—
160	C₂×D₄	—	302	32	no	—
161	C₂×D₄	—	604	16	no	—
162	C₂×D₄	—	151	32	no	—
163	C₂×D₄	—	151	8	no	—
164	C₂×D₄	—	151	8	no	—
165	C₂×D₄	C₂²×D₁₅₁	1	8	yes	—
166	C₂×D₄	C₂²×D₁₅₁	1	4	yes	complemented
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	151	4	no	—
169	C₂²×C₄	C₂²⋊D₁₅₁	1	8	yes	complemented
170	C₂²×C₄	C₂²×D₁₅₁	1	2	yes	—
171	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₁₅₁	1	4	yes	—
174	C₂²×C₄	C₂²×D₁₅₁	1	8	yes	complemented
175	C₄⋊C₄	—	151	8	no	—
176	C₂²⋊C₄	—	151	8	no	—
177	C₂²⋊C₄	—	151	32	no	—
178	C₄²	—	151	8	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	8	no	—
182	C₂³	—	302	16	no	—
183	C₂³	—	302	8	no	—
184	C₂³	—	302	16	no	—
185	C₂³	—	604	16	no	—
186	C₂³	—	604	32	no	—
187	C₂³	—	151	8	no	—
188	C₂³	C₂×C₂²⋊D₁₅₁	1	4	yes	—
189	C₂³	C₂³×D₁₅₁	1	4	yes	—
190	C₂³	C₂³×D₁₅₁	1	2	yes	—
191	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
192	Q₈	—	2	8	no	—
193	D₄	—	2	16	no	—
194	D₄	—	2	8	no	—
195	D₄	—	302	16	no	—
196	D₄	—	302	16	no	—
197	D₄	—	604	32	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	16	no	—
200	C₂×C₄	—	302	8	no	—
201	C₂×C₄	—	151	8	no	—
202	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	4	yes	—
203	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	4	yes	—
204	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	16	yes	complemented
205	C₂×C₄	C₂³×D₁₅₁	1	4	yes	—
206	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
207	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	8	no	—
212	C₂²	—	4	8	no	—
213	C₂²	—	302	8	no	—
214	C₂²	—	302	16	no	—
215	C₂²	—	604	32	no	—
216	C₂²	—	604	16	no	—
217	C₂²	C₂²×C₂²⋊D₁₅₁	1	4	yes	—
218	C₂²	C₂²×C₂²⋊D₁₅₁	1	2	yes	—
219	C₂²	#18012785	1	4	yes	—
220	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	302	8	no	—
223	C₄	C₂²×C₂²⋊D₁₅₁	1	4	yes	—
224	C₄	C₂²×C₂²⋊D₁₅₁	1	4	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	4	4	no	—
227	C₂	—	604	8	no	—
228	C₂	#23752776	1	4	yes	direct product factor
229	C₂	C₂³×C₂²⋊D₁₅₁	1	1	yes	characteristic
230	C₂	C₂×#18012785	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₅₁
commutator, cosocle	C₂²×C₁₅₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅₁
Fitting	C₂²×#41×C₁₅₁	C₂
socle	C₂³×C₁₅₁	C₂⁴
hypercenter	C₂²×#41	D₁₅₁
hypocenter	C₁₅₁	C₂×#535
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×D₆₀₄	C₂	semidirect
2. #23752776	C₂	direct (N × H)
3. C₂×#18012735	C₂	semidirect
4. C₂×#18012728	C₂	semidirect
5. C₂×#18012713	C₂	semidirect
6. C₂×#18012707	C₂	semidirect
7. C₂×#18012698	C₂	semidirect
8. C₂²×#41×C₁₅₁	C₂	semidirect
13. C₂²×D₆₀₄	C₂²	semidirect
14. C₂²×C₂²⋊D₁₅₁	C₂²	semidirect
15. #18012735	C₂²	semidirect
16. #18012728	C₂²	semidirect
17. #18012713	C₂²	semidirect
18. #18012707	C₂²	semidirect
19. #18012698	C₂²	semidirect
20. C₂×C₄⋊Dic₁₅₁	C₂²	semidirect
22. C₂×(C₂×D₁₅₁)⋊C₄	C₂²	semidirect
23. C₂×C₄×Dic₁₅₁	C₂²	semidirect
25. C₂×#41×C₁₅₁	C₂²	semidirect
26. C₂²×Q₈×C₁₅₁	C₂²	semidirect
28. C₂²×D₄×C₁₅₁	C₂²	semidirect
30. C₂³×C₄×C₁₅₁	C₂²	semidirect
41. C₂×C₂²⋊D₁₅₁	C₂³	semidirect
43. C₄⋊Dic₁₅₁	C₂³	semidirect
45. (C₂×D₁₅₁)⋊C₄	C₂³	semidirect
46. C₄×Dic₁₅₁	C₂³	semidirect
51. C₂×Q₈×C₁₅₁	C₂³	semidirect
55. C₂×D₄×C₁₅₁	C₂³	semidirect
57. C₂²×C₄×C₁₅₁	D₄	semidirect
62. C₂²×C₄×C₁₅₁	C₂³	semidirect
88. C₂×C₄×C₁₅₁	C₂×D₄	semidirect
111. C₂×C₁₅₁	#535	semidirect
115. C₁₅₁	C₂×#535	semidirect
117. C₂²×#41	D₁₅₁	semidirect
127. C₂×#41	C₂×D₁₅₁	semidirect
128. C₂²×Q₈	C₂×D₁₅₁	semidirect
135. C₂²×D₄	C₂×D₁₅₁	semidirect
137. C₂³×C₄	C₂×D₁₅₁	semidirect
155. C₂×Q₈	C₂²×D₁₅₁	semidirect
166. C₂×D₄	C₂²×D₁₅₁	semidirect
169. C₂²×C₄	C₂²⋊D₁₅₁	semidirect
174. C₂²×C₄	C₂²×D₁₅₁	semidirect
204. C₂×C₄	C₂×C₂²⋊D₁₅₁	semidirect
228. C₂	#23752776	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₆₀₄	2
#23752776	2
C₂×#18012735	2
C₂×#18012728	2
C₂×#18012713	2
C₂×#18012707	2
C₂×#18012698	2
C₂²×#41×C₁₅₁	2
C₂×#535	151

Subgroups — C2×#23752776 #34082416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23752776 #34082416