Subgroups — C₂×#5470 #85671

Order 384.

1168 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×#926	C₂	1	2	yes	maximal
3	C₂×#924	C₂	1	4	yes	maximal
4	#5470	C₂	1	16	yes	maximal, direct product factor
5	C₂×#870	C₂	1	2	yes	maximal, complemented
6	C₂×#869	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Dic₆	C₂	1	4	yes	maximal, complemented
8	C₂×#533	—	3	1	no	maximal, Sylow
9	C₂³×C₃×C₄	C₂²	1	2	yes	—
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂²×Dic₆	C₂²	1	4	yes	—
12	C₂×C₃×C₄²	C₂²	1	2	yes	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²	C₂²	1	8	yes	—
16	C₂×#217	C₂²	1	4	yes	—
17	#926	C₂²	1	16	yes	—
18	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
20	#924	C₂²	1	32	yes	—
21	C₂×Dic₃⋊C₄	C₂²	1	8	yes	—
22	C₂×C₄×Dic₃	C₂²	1	4	yes	—
23	#870	C₂²	1	16	yes	complemented
24	#869	C₂²	1	16	yes	complemented
25	C₄×Dic₆	C₂²	1	32	yes	complemented
26	#533	—	3	16	no	—
27	C₂×C₄⋊Q₈	—	3	2	no	—
28	C₂×#125	—	3	2	no	—
29	C₂×C₂²⋊Q₈	—	3	4	no	—
30	C₂×C₄×Q₈	—	3	4	no	—
31	C₂×#117	—	3	2	no	—
32	C₂²×C₄²	D₃	1	1	yes	characteristic
33	C₂⁴×C₃	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄	—	2	8	no	—
35	C₂²×C₃×C₄	Q₈	1	8	yes	complemented
36	C₂²×C₃×C₄	C₂³	1	2	yes	—
37	C₂²×C₃×C₄	C₂³	1	8	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂³	1	4	yes	—
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂²×Dic₃	C₂³	1	8	yes	—
43	C₂×Dic₆	—	2	16	no	—
44	C₂×Dic₆	C₂³	1	16	yes	—
45	C₃×C₄²	C₂³	1	8	yes	—
46	C₃×C₄²	C₂³	1	4	yes	—
47	C₃×C₄²	C₂³	1	4	yes	—
48	#217	C₂³	1	16	yes	—
49	C₄⋊Dic₃	C₂³	1	16	yes	—
50	C₄⋊Dic₃	C₂³	1	16	yes	—
51	Dic₃⋊C₄	C₂³	1	32	yes	—
52	C₄×Dic₃	C₂³	1	16	yes	—
53	C₂²×Q₈	—	3	4	no	—
54	C₂³×C₄	C₂×D₃	1	2	yes	—
55	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
56	C₄⋊Q₈	—	3	16	no	—
57	#125	—	3	16	no	—
58	C₂²⋊Q₈	—	3	32	no	—
59	C₄×Q₈	—	3	32	no	—
60	#117	—	3	16	no	—
61	C₂×C₄⋊C₄	—	3	4	no	—
62	C₂×C₄⋊C₄	—	3	8	no	—
63	C₂×C₄⋊C₄	—	3	4	no	—
64	C₂×C₂²⋊C₄	—	3	4	no	—
65	C₂×C₄²	—	3	4	no	—
66	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₃	1	8	yes	—
69	C₂×C₄²	C₂×D₃	1	2	yes	—
70	C₂³×C₃	—	2	4	no	—
71	C₂³×C₃	C₂×Q₈	1	4	yes	—
72	C₂³×C₃	C₂⁴	1	2	yes	—
73	C₂³×C₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	—	2	16	no	—
76	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
77	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
78	C₂×C₃×C₄	C₂×Q₈	1	16	yes	complemented
79	C₂×C₃×C₄	#41	1	8	yes	—
80	C₂×C₃×C₄	C₂⁴	1	4	yes	—
81	C₂×C₃×C₄	C₂⁴	1	2	yes	—
82	C₂×C₃×C₄	C₂⁴	1	2	yes	—
83	C₂×C₃×C₄	C₂⁴	1	4	yes	—
84	C₂×Dic₃	—	2	16	no	—
85	C₂×Dic₃	C₂⁴	1	16	yes	—
86	Dic₆	—	2	32	no	—
87	C₂⁴	C₂²×D₃	1	1	yes	characteristic
88	C₂×Q₈	—	3	16	no	—
89	C₂×Q₈	—	6	16	no	—
90	C₂²×C₄	—	2	8	no	—
91	C₂²×C₄	—	3	8	no	—
92	C₂²×C₄	Dic₆	1	8	yes	complemented
93	C₂²×C₄	C₂²×D₃	1	2	yes	—
94	C₂²×C₄	C₂²×D₃	1	8	yes	—
95	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₃	1	4	yes	—
97	C₂²×C₄	C₂²×D₃	1	2	yes	—
98	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₄⋊C₄	—	3	16	no	—
100	C₄⋊C₄	—	3	32	no	—
101	C₄⋊C₄	—	3	16	no	—
102	C₂²⋊C₄	—	3	16	no	—
103	C₄²	—	3	16	no	—
104	C₄²	C₂²×D₃	1	8	yes	—
105	C₄²	C₂²×D₃	1	4	yes	—
106	C₄²	C₂²×D₃	1	4	yes	—
107	C₂²×C₃	—	2	8	no	—
108	C₂²×C₃	—	2	4	no	—
109	C₂²×C₃	C₂²×Q₈	1	4	yes	—
110	C₂²×C₃	C₂²×Q₈	1	2	yes	—
111	C₂²×C₃	C₂×#41	1	4	yes	—
112	C₂²×C₃	C₂⁵	1	1	yes	characteristic
113	C₃×C₄	—	2	4	no	—
114	C₃×C₄	C₂²×Q₈	1	4	yes	—
115	C₃×C₄	C₂²×Q₈	1	4	yes	—
116	C₃×C₄	C₂×#41	1	8	yes	—
117	Dic₃	—	2	16	no	—
118	C₂³	—	2	4	no	—
119	C₂³	C₂×Dic₆	1	4	yes	—
120	C₂³	C₂³×D₃	1	2	yes	—
121	C₂³	C₂³×D₃	1	1	yes	characteristic
122	Q₈	—	6	32	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	16	no	—
125	C₂×C₄	—	3	16	no	—
126	C₂×C₄	—	6	16	no	—
127	C₂×C₄	C₂×Dic₆	1	4	yes	—
128	C₂×C₄	C₂×Dic₆	1	4	yes	—
129	C₂×C₄	C₂×Dic₆	1	16	yes	complemented
130	C₂×C₄	#235	1	8	yes	—
131	C₂×C₄	C₂³×D₃	1	4	yes	—
132	C₂×C₄	C₂³×D₃	1	2	yes	—
133	C₂×C₄	C₂³×D₃	1	2	yes	—
134	C₂×C₄	C₂³×D₃	1	4	yes	—
135	C₂×C₃	—	2	4	no	—
136	C₂×C₃	#533	1	4	yes	complemented
137	C₂×C₃	C₂³×Q₈	1	1	yes	characteristic
138	C₂×C₃	C₂²×#41	1	2	yes	—
139	C₂²	—	2	4	no	—
140	C₂²	—	2	8	no	—
141	C₂²	C₂²×Dic₆	1	4	yes	—
142	C₂²	C₂²×Dic₆	1	2	yes	—
143	C₂²	C₂×#235	1	4	yes	—
144	C₂²	C₂⁴×D₃	1	1	yes	characteristic
145	C₄	—	2	4	no	—
146	C₄	—	6	16	no	—
147	C₄	C₂²×Dic₆	1	4	yes	—
148	C₄	C₂²×Dic₆	1	4	yes	—
149	C₄	C₂×#235	1	8	yes	—
150	C₃	C₂×#533	1	1	yes	characteristic, complemented, Sylow
151	C₂	—	2	4	no	—
152	C₂	#5470	1	4	yes	direct product factor
153	C₂	C₂³×Dic₆	1	1	yes	characteristic
154	C₂	C₂²×#235	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	C₂×#533
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. #5470	C₂	direct (N × H)
5. C₂×#870	C₂	semidirect
6. C₂×#869	C₂	semidirect
7. C₂×C₄×Dic₆	C₂	semidirect
23. #870	C₂²	semidirect
24. #869	C₂²	semidirect
25. C₄×Dic₆	C₂²	semidirect
35. C₂²×C₃×C₄	Q₈	semidirect
78. C₂×C₃×C₄	C₂×Q₈	semidirect
92. C₂²×C₄	Dic₆	semidirect
129. C₂×C₄	C₂×Dic₆	semidirect
136. C₂×C₃	#533	semidirect
150. C₃	C₂×#533	semidirect
152. C₂	#5470	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×C₃×C₄²	2
C₂×#926	2
C₂×#924	2
#5470	2
C₂×#870	2
C₂×#869	2
C₂×C₄×Dic₆	2
C₂×#533	3

Subgroups — C2×#5470 #85671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5470 #85671