Subgroups — C₂×#4708 #78485

Order 384.

592 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#872	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#4708	C₂	1	8	yes	maximal, direct product factor
9	C₂×#405	—	3	1	no	maximal, Sylow
10	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
15	C₂×Dic₃⋊C₄	C₄	1	2	yes	—
16	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×ΓL(1,𝔽₂₅)	C₄	1	8	yes	complemented
20	C₂×ΓL(1,𝔽₂₅)	C₂²	1	4	yes	—
21	C₂×D₃×C₈	C₄	1	8	yes	complemented
22	C₂×D₃×C₈	C₂²	1	4	yes	—
23	#872	C₄	1	4	yes	—
24	#872	C₂²	1	4	yes	—
25	C₃×#97	C₂²	1	4	yes	complemented
26	#815	C₂²	1	4	yes	complemented
27	C₈×Dic₃	C₂²	1	4	yes	complemented
28	C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
29	C₂²×M₁₆	—	3	1	no	—
30	C₂³×C₈	—	3	1	no	—
31	C₂×#117	—	3	1	no	—
32	#405	—	3	8	no	—
33	C₂×#97	—	3	1	no	—
34	C₂×#97	D₃	1	1	yes	characteristic, complemented
35	C₂×C₄×C₈	—	3	1	no	—
36	C₂×C₄×C₈	—	3	1	no	—
37	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃	C₂×C₄	1	2	yes	—
41	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×D₃	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₃	C₂³	1	2	yes	—
46	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
47	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
48	C₂×C₃×C₈	C₂³	1	2	yes	—
49	C₂×C₃×C₈	C₂³	1	2	yes	—
50	C₃×C₄²	C₂×C₄	1	2	yes	—
51	C₃×C₄²	C₂³	1	2	yes	—
52	(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
53	Dic₃⋊C₄	C₂×C₄	1	8	yes	—
54	C₄×Dic₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₃	C₂³	1	2	yes	—
56	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
57	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
58	C₂×C₃⋊C₈	C₂³	1	2	yes	—
59	C₂×C₃⋊C₈	C₂³	1	2	yes	—
60	ΓL(1,𝔽₂₅)	C₂×C₄	1	16	yes	complemented
61	D₃×C₈	C₂×C₄	1	16	yes	complemented
62	C₂³×C₄	—	3	1	no	—
63	C₂×M₁₆	—	3	8	no	—
64	C₂×M₁₆	—	3	4	no	—
65	C₂²×C₈	—	3	1	no	—
66	C₂²×C₈	—	3	1	no	—
67	C₂²×C₈	—	3	8	no	—
68	C₂²×C₈	—	3	4	no	—
69	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
71	#117	—	3	4	no	—
72	#117	—	3	4	no	—
73	C₂×C₄⋊C₄	—	3	2	no	—
74	C₂×C₂²⋊C₄	—	3	2	no	—
75	C₂×C₄²	—	3	1	no	—
76	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
77	#97	—	3	4	no	—
78	#97	C₂×D₃	1	4	yes	complemented
79	C₄×C₈	—	3	4	no	—
80	C₄×C₈	—	3	4	no	—
81	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×D₃	—	2	4	no	—
83	C₂²×D₃	C₄²	1	4	yes	—
84	C₂²×D₃	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
87	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
90	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₃	—	2	4	no	—
92	C₂×Dic₃	C₄²	1	4	yes	—
93	C₂×Dic₃	C₂²×C₄	1	4	yes	—
94	C₂×Dic₃	C₂²×C₄	1	2	yes	—
95	C₄×D₃	C₄²	1	8	yes	—
96	C₄×D₃	C₂²×C₄	1	4	yes	—
97	C₄×D₃	C₂²×C₄	1	4	yes	—
98	C₃×C₈	C₂²×C₄	1	4	yes	—
99	C₃×C₈	C₂²×C₄	1	4	yes	—
100	C₃⋊C₈	C₂²×C₄	1	4	yes	—
101	C₃⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂⁴	—	3	1	no	—
103	C₂²×C₄	—	3	1	no	—
104	C₂²×C₄	—	3	2	no	—
105	C₂²×C₄	—	3	4	no	—
106	C₂²×C₄	—	3	4	no	—
107	C₂²×C₄	—	3	2	no	—
108	C₂²×C₄	—	3	2	no	—
109	C₂²×C₄	C₄×D₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
111	M₁₆	—	3	16	no	—
112	C₂×C₈	—	3	4	no	—
113	C₂×C₈	—	3	4	no	—
114	C₂×C₈	—	3	2	no	—
115	C₂×C₈	—	3	2	no	—
116	C₂×C₈	—	3	16	no	—
117	C₂×C₈	C₄×D₃	1	4	yes	—
118	C₂×C₈	C₄×D₃	1	4	yes	—
119	C₂×C₈	C₂²×D₃	1	2	yes	—
120	C₂×C₈	C₂²×D₃	1	2	yes	—
121	C₄⋊C₄	—	3	8	no	—
122	C₂²⋊C₄	—	3	8	no	—
123	C₄²	—	3	2	no	—
124	C₄²	—	3	2	no	—
125	C₄²	C₄×D₃	1	2	yes	—
126	C₄²	C₂²×D₃	1	2	yes	—
127	C₂²×C₃	C₂×C₄²	1	2	yes	—
128	C₂²×C₃	#131	1	4	yes	—
129	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×D₃	—	2	4	no	—
131	C₂×D₃	—	2	8	no	—
132	C₂×D₃	C₂×C₄²	1	4	yes	—
133	C₃×C₄	—	2	4	no	—
134	C₃×C₄	C₂×C₄²	1	2	yes	—
135	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
136	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
137	Dic₃	—	2	4	no	—
138	Dic₃	C₂×C₄²	1	4	yes	—
139	C₂³	—	3	4	no	—
140	C₂³	—	3	2	no	—
141	C₂³	—	6	4	no	—
142	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	3	4	no	—
145	C₂×C₄	—	3	2	no	—
146	C₂×C₄	—	3	8	no	—
147	C₂×C₄	—	3	4	no	—
148	C₂×C₄	—	3	4	no	—
149	C₂×C₄	—	3	4	no	—
150	C₂×C₄	—	6	4	no	—
151	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
152	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
155	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
156	C₈	—	3	4	no	—
157	C₈	—	3	4	no	—
158	C₈	C₂×C₄×D₃	1	4	yes	—
159	C₈	C₂×C₄×D₃	1	4	yes	—
160	C₂×C₃	#405	1	4	yes	complemented
161	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
162	C₂×C₃	C₂×#131	1	2	yes	—
163	D₃	—	2	4	no	—
164	C₂²	—	3	4	no	—
165	C₂²	—	6	8	no	—
166	C₂²	—	6	4	no	—
167	C₂²	C₄²×D₃	1	2	yes	—
168	C₂²	#907	1	4	yes	—
169	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	3	4	no	—
172	C₄	—	6	4	no	—
173	C₄	C₄²×D₃	1	2	yes	—
174	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
175	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
176	C₃	C₂×#405	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	6	4	no	—
178	C₂	#4708	1	4	yes	direct product factor
179	C₂	C₂×C₄²×D₃	1	1	yes	characteristic
180	C₂	C₂×#907	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#97	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂³×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×#97	D₃
hypocenter	C₃	C₂×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
4. C₂×C₃×#97	C₂	semidirect
5. C₂×#815	C₂	semidirect
6. C₂×C₈×Dic₃	C₂	semidirect
7. C₂×C₄×C₃⋊C₈	C₂	semidirect
8. #4708	C₂	direct (N × H)
19. C₂×ΓL(1,𝔽₂₅)	C₄	semidirect
21. C₂×D₃×C₈	C₄	semidirect
25. C₃×#97	C₂²	semidirect
26. #815	C₂²	semidirect
27. C₈×Dic₃	C₂²	semidirect
28. C₄×C₃⋊C₈	C₂²	semidirect
34. C₂×#97	D₃	semidirect
60. ΓL(1,𝔽₂₅)	C₂×C₄	semidirect
61. D₃×C₈	C₂×C₄	semidirect
78. #97	C₂×D₃	semidirect
160. C₂×C₃	#405	semidirect
176. C₃	C₂×#405	semidirect
178. C₂	#4708	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×ΓL(1,𝔽₂₅)	2
C₂²×D₃×C₈	2
C₂×#872	2
C₂×C₃×#97	2
C₂×#815	2
C₂×C₈×Dic₃	2
C₂×C₄×C₃⋊C₈	2
#4708	2
C₂×#405	3

Subgroups — C2×#4708 #78485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4708 #78485