Subgroups — C₂×#14521523 #16858897

Order 3968.

1920 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	C₂³×D₁₂₄	C₂	1	1	yes	maximal, characteristic
2	#14521523	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×D₁₂₄	C₂	1	4	yes	maximal, complemented
4	C₂×#11758470	C₂	1	4	yes	maximal, complemented
5	C₂×#11758503	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊C₄×D₃₁	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×D₃₁	C₂²	1	2	yes	—
9	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×D₁₂₄	C₄	1	16	yes	complemented
11	C₂²×D₁₂₄	C₂²	1	2	yes	—
12	C₂²×D₁₂₄	C₂²	1	2	yes	—
13	C₂²×D₁₂₄	C₂²	1	8	yes	—
14	C₂²×C₄×D₃₁	C₂²	1	8	yes	complemented
15	C₃₁×#117	C₂²	1	8	yes	complemented
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
17	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₃₁	C₂²	1	2	yes	complemented
19	C₂×C₂²⋊Dic₃₁	C₂²	1	2	yes	—
20	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	8	yes	complemented
21	#11758503	C₂²	1	8	yes	complemented
22	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	complemented
23	C₂×C₄×Dic₃₁	C₂²	1	2	yes	complemented
24	#11758470	C₂²	1	32	yes	complemented
25	C₂²⋊C₄×D₃₁	C₂²	1	32	yes	complemented
26	C₄×D₁₂₄	C₂²	1	32	yes	complemented
27	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
28	C₂³×D₃₁	—	4	8	no	—
29	C₂³×D₃₁	C₂×C₄	1	16	yes	complemented
30	C₂³×D₃₁	C₂³	1	4	yes	—
31	C₂³×D₃₁	C₂³	1	8	yes	—
32	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
33	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
37	C₂²×Dic₃₁	C₂³	1	4	yes	—
38	C₂×D₁₂₄	C₂×C₄	1	16	yes	complemented
39	C₂×D₁₂₄	C₂×C₄	1	16	yes	complemented
40	C₂×D₁₂₄	C₂×C₄	1	64	yes	complemented
41	C₂×D₁₂₄	C₂³	1	8	yes	—
42	C₂×D₁₂₄	C₂³	1	8	yes	—
43	C₂×C₄×D₃₁	—	4	16	no	—
44	C₂×C₄×D₃₁	C₂³	1	32	yes	complemented
45	C₄⋊C₄×C₃₁	C₂³	1	8	yes	complemented
46	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
47	C₄²×C₃₁	C₂³	1	8	yes	complemented
48	C₂²⋊Dic₃₁	C₂³	1	8	yes	—
49	(C₂×D₃₁)⋊C₄	C₂³	1	32	yes	complemented
50	C₄⋊Dic₃₁	C₂³	1	8	yes	complemented
51	C₄×Dic₃₁	C₂³	1	8	yes	complemented
52	C₂³×C₃₁	—	2	4	no	—
53	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
54	C₂³×C₃₁	C₂⁴	1	2	yes	—
55	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
56	C₂²×D₃₁	—	2	16	no	—
57	C₂²×D₃₁	—	4	32	no	—
58	C₂²×D₃₁	—	4	16	no	—
59	C₂²×D₃₁	C₂²×C₄	1	16	yes	—
60	C₂²×D₃₁	C₂²×C₄	1	32	yes	complemented
61	C₂²×D₃₁	C₂⁴	1	8	yes	—
62	C₂×C₄×C₃₁	—	2	8	no	—
63	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
66	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
67	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
68	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
69	C₂×Dic₃₁	—	2	8	no	—
70	C₂×Dic₃₁	C₂⁴	1	8	yes	—
71	D₁₂₄	C₂²×C₄	1	32	yes	complemented
72	D₁₂₄	C₂²×C₄	1	32	yes	complemented
73	C₄×D₃₁	—	4	32	no	—
74	C₂×#518	—	31	1	no	maximal, Sylow
75	C₂²×C₃₁	—	2	8	no	—
76	C₂²×C₃₁	—	2	4	no	—
77	C₂²×C₃₁	C₂³×C₄	1	4	yes	—
78	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
79	C₂²×C₃₁	H(5,𝔽₂)	1	4	yes	—
80	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
81	C₂×D₃₁	—	2	32	no	—
82	C₂×D₃₁	—	2	16	no	—
83	C₂×D₃₁	—	4	32	no	—
84	C₂×D₃₁	C₂³×C₄	1	16	yes	—
85	C₄×C₃₁	—	2	8	no	—
86	C₄×C₃₁	C₂³×C₄	1	4	yes	—
87	C₄×C₃₁	C₂³×C₄	1	4	yes	—
88	Dic₃₁	—	2	8	no	—
89	C₂³×D₄	—	31	1	no	—
90	#518	—	31	16	no	—
91	C₂×C₄×D₄	—	31	4	no	—
92	C₂×C₄×D₄	—	31	4	no	—
93	C₂×#117	—	31	1	no	—
94	C₂×#117	D₃₁	1	1	yes	characteristic, complemented
95	C₂²×C₂²⋊C₄	—	31	4	no	—
96	C₂×C₃₁	—	2	4	no	—
97	C₂×C₃₁	#518	1	4	yes	complemented
98	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
99	C₂×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
100	D₃₁	—	2	16	no	—
101	C₂⁵	—	31	2	no	—
102	C₂²×D₄	—	31	16	no	—
103	C₂²×D₄	—	31	2	no	—
104	C₂²×D₄	—	31	2	no	—
105	C₂²×D₄	—	31	8	no	—
106	C₂³×C₄	—	31	8	no	—
107	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
108	C₄×D₄	—	31	32	no	—
109	C₄×D₄	—	31	32	no	—
110	#117	—	31	8	no	—
111	#117	C₂×D₃₁	1	8	yes	complemented
112	C₂×C₄⋊C₄	—	31	2	no	—
113	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	complemented
114	C₂×C₂²⋊C₄	—	31	2	no	—
115	C₂×C₂²⋊C₄	—	31	8	no	—
116	C₂×C₂²⋊C₄	—	31	32	no	—
117	C₂×C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
118	C₂×C₄²	—	31	2	no	—
119	C₂×C₄²	C₂×D₃₁	1	2	yes	complemented
120	C₃₁	C₂×#518	1	1	yes	characteristic, complemented, Sylow
121	C₂⁴	—	31	4	no	—
122	C₂⁴	—	31	16	no	—
123	C₂⁴	—	31	8	no	—
124	C₂⁴	—	124	8	no	—
125	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
126	C₂×D₄	—	31	16	no	—
127	C₂×D₄	—	31	16	no	—
128	C₂×D₄	—	31	64	no	—
129	C₂×D₄	—	31	8	no	—
130	C₂×D₄	—	31	8	no	—
131	C₂²×C₄	—	31	4	no	—
132	C₂²×C₄	—	31	32	no	—
133	C₂²×C₄	—	124	16	no	—
134	C₂²×C₄	C₄×D₃₁	1	8	yes	complemented
135	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
136	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
139	C₄⋊C₄	—	31	8	no	—
140	C₄⋊C₄	C₂²×D₃₁	1	8	yes	complemented
141	C₂²⋊C₄	—	31	8	no	—
142	C₂²⋊C₄	—	31	32	no	—
143	C₂²⋊C₄	C₂²×D₃₁	1	8	yes	—
144	C₄²	—	31	8	no	—
145	C₄²	C₂²×D₃₁	1	8	yes	complemented
146	C₂³	—	2	4	no	—
147	C₂³	—	31	16	no	—
148	C₂³	—	31	8	no	—
149	C₂³	—	31	32	no	—
150	C₂³	—	62	16	no	—
151	C₂³	—	124	16	no	—
152	C₂³	—	124	32	no	—
153	C₂³	C₂×C₄×D₃₁	1	4	yes	—
154	C₂³	C₂³×D₃₁	1	2	yes	—
155	C₂³	C₂³×D₃₁	1	1	yes	characteristic
156	D₄	—	31	32	no	—
157	D₄	—	31	32	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	31	8	no	—
160	C₂×C₄	—	62	8	no	—
161	C₂×C₄	—	124	32	no	—
162	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
163	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
164	C₂×C₄	C₂×C₄×D₃₁	1	16	yes	complemented
165	C₂×C₄	C₂³×D₃₁	1	8	yes	—
166	C₂×C₄	C₂³×D₃₁	1	2	yes	—
167	C₂×C₄	C₂³×D₃₁	1	2	yes	—
168	C₂²	—	2	4	no	—
169	C₂²	—	2	8	no	—
170	C₂²	—	31	16	no	—
171	C₂²	—	62	16	no	—
172	C₂²	—	62	32	no	—
173	C₂²	—	124	32	no	—
174	C₂²	C₂²×C₄×D₃₁	1	4	yes	—
175	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
176	C₂²	#11758566	1	4	yes	—
177	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
178	C₄	—	2	8	no	—
179	C₄	—	62	8	no	—
180	C₄	C₂²×C₄×D₃₁	1	4	yes	—
181	C₄	C₂²×C₄×D₃₁	1	4	yes	—
182	C₂	—	2	4	no	—
183	C₂	—	62	16	no	—
184	C₂	#14521523	1	4	yes	direct product factor
185	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
186	C₂	C₂×#11758566	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₃₁×#117	C₂
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂³×D₁₂₄	C₂
cosocle	C₂²×C₃₁	C₂⁵
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	C₂×#518
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #14521523	C₂	direct (N × H)
3. C₂×C₄×D₁₂₄	C₂	semidirect
4. C₂×#11758470	C₂	semidirect
5. C₂×#11758503	C₂	semidirect
6. C₂×C₂²⋊C₄×D₃₁	C₂	semidirect
7. C₂×C₃₁×#117	C₂	semidirect
10. C₂²×D₁₂₄	C₄	semidirect
14. C₂²×C₄×D₃₁	C₂²	semidirect
15. C₃₁×#117	C₂²	semidirect
16. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
18. C₂×C₄²×C₃₁	C₂²	semidirect
20. C₂×(C₂×D₃₁)⋊C₄	C₂²	semidirect
21. #11758503	C₂²	semidirect
22. C₂×C₄⋊Dic₃₁	C₂²	semidirect
23. C₂×C₄×Dic₃₁	C₂²	semidirect
24. #11758470	C₂²	semidirect
25. C₂²⋊C₄×D₃₁	C₂²	semidirect
26. C₄×D₁₂₄	C₂²	semidirect
29. C₂³×D₃₁	C₂×C₄	semidirect
32. C₂²×C₄×C₃₁	C₂×C₄	semidirect
38. C₂×D₁₂₄	C₂×C₄	semidirect
39. C₂×D₁₂₄	C₂×C₄	semidirect
40. C₂×D₁₂₄	C₂×C₄	semidirect
44. C₂×C₄×D₃₁	C₂³	semidirect
45. C₄⋊C₄×C₃₁	C₂³	semidirect
47. C₄²×C₃₁	C₂³	semidirect
49. (C₂×D₃₁)⋊C₄	C₂³	semidirect
50. C₄⋊Dic₃₁	C₂³	semidirect
51. C₄×Dic₃₁	C₂³	semidirect
60. C₂²×D₃₁	C₂²×C₄	semidirect
65. C₂×C₄×C₃₁	C₂²×C₄	semidirect
71. D₁₂₄	C₂²×C₄	semidirect
72. D₁₂₄	C₂²×C₄	semidirect
94. C₂×#117	D₃₁	semidirect
97. C₂×C₃₁	#518	semidirect
111. #117	C₂×D₃₁	semidirect
113. C₂×C₄⋊C₄	C₂×D₃₁	semidirect
119. C₂×C₄²	C₂×D₃₁	semidirect
120. C₃₁	C₂×#518	semidirect
134. C₂²×C₄	C₄×D₃₁	semidirect
140. C₄⋊C₄	C₂²×D₃₁	semidirect
145. C₄²	C₂²×D₃₁	semidirect
164. C₂×C₄	C₂×C₄×D₃₁	semidirect
184. C₂	#14521523	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×D₁₂₄	2
#14521523	2
C₂×C₄×D₁₂₄	2
C₂×#11758470	2
C₂×#11758503	2
C₂×C₂²⋊C₄×D₃₁	2
C₂×C₃₁×#117	2
C₂×#518	31

Subgroups — C2×#14521523 #16858897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521523 #16858897