Subgroups — C₂×#22624843 #34009641

Order 18688.

366 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924031	C₂	1	2	yes	maximal
2	#22624843	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	C₄	1	4	yes	—
5	C₄×#11924031	C₄	1	4	yes	—
6	C₄×#11924031	C₂²	1	4	yes	—
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	C₈×#10596831	C₄	1	2	yes	—
10	C₈×#10596831	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924031	C₈	1	16	yes	complemented
13	C₂×#11924031	C₂×C₄	1	8	yes	—
14	C₂²×#10596831	C₈	1	2	yes	—
15	C₂²×#10596831	C₂×C₄	1	2	yes	—
16	C₄×#10596831	C₈	1	2	yes	—
17	C₄×#10596831	C₂×C₄	1	2	yes	—
18	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₄×#10596831	C₂×C₄	1	2	yes	—
20	C₄×#10596831	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	C₈	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₃	C₈	1	2	yes	—
24	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
25	#11924031	C₂×C₈	1	16	yes	complemented
26	C₂×#10596831	—	2	4	no	—
27	C₂×#10596831	C₄²	1	2	yes	—
28	C₂×#10596831	C₂×C₈	1	4	yes	—
29	C₂×#10596831	C₂×C₈	1	8	yes	—
30	C₂×#10596831	C₂²×C₄	1	2	yes	—
31	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	—	2	4	no	—
35	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
36	C₄²×C₇₃	C₄²	1	1	yes	characteristic
37	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
42	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
46	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
47	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₇₃	—	2	4	no	—
51	#10596831	—	2	4	no	—
52	#10596831	C₄×C₈	1	4	yes	—
53	#10596831	C₂²×C₈	1	4	yes	—
54	C₂²×C₇₃	C₈²	1	2	yes	—
55	C₂²×C₇₃	#346	1	2	yes	—
56	C₂²×C₇₃	#346	1	2	yes	—
57	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
58	C₄×C₇₃	C₈²	1	2	yes	—
59	C₄×C₇₃	#346	1	2	yes	—
60	C₄×C₇₃	#346	1	2	yes	—
61	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
62	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
63	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
64	C₂×#1311	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	#1311	1	4	yes	complemented
66	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
67	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
68	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
69	C₂×C₄×C₁₆	—	73	2	no	—
70	C₂×C₈²	—	73	1	no	—
71	#1311	—	73	4	no	—
72	C₇₃	C₂×#1311	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₁₆	—	73	4	no	—
74	C₂×C₄×C₈	—	73	1	no	—
75	C₂×C₄×C₈	—	73	1	no	—
76	C₂×C₄×C₈	#64408	1	1	yes	characteristic
77	C₄×C₁₆	—	73	4	no	—
78	C₄×C₁₆	—	73	4	no	—
79	C₈²	—	73	2	no	—
80	C₈²	—	73	2	no	—
81	C₂²×C₈	—	73	2	no	—
82	C₂²×C₈	—	73	2	no	—
83	C₂²×C₈	#10596833	1	2	yes	—
84	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
85	C₂×C₁₆	—	73	8	no	—
86	C₂×C₁₆	—	73	16	no	—
87	C₄×C₈	—	73	2	no	—
88	C₄×C₈	—	73	2	no	—
89	C₄×C₈	—	73	1	no	—
90	C₄×C₈	—	73	2	no	—
91	C₄×C₈	—	73	1	no	—
92	C₄×C₈	#10596833	1	2	yes	—
93	C₄×C₈	C₂×#64408	1	2	yes	—
94	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
95	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
96	C₂²×C₄	#11924036	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	73	8	no	—
99	C₂×C₈	—	73	2	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	73	4	no	—
102	C₂×C₈	—	146	4	no	—
103	C₂×C₈	C₂×#10596833	1	4	yes	—
104	C₄²	C₄×#64408	1	1	yes	characteristic
105	C₄²	C₂×#10596833	1	1	yes	characteristic
106	C₄²	#11924036	1	1	yes	characteristic
107	C₄²	C₂²×#64408	1	1	yes	characteristic
108	C₁₆	—	73	16	no	—
109	C₂³	C₄×#10596833	1	1	yes	characteristic
110	C₂×C₄	C₈×#64408	1	4	yes	—
111	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
112	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
113	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
114	C₂×C₄	#14556665	1	4	yes	—
115	C₂×C₄	#14556666	1	4	yes	—
116	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
117	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
118	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	73	4	no	—
121	C₈	—	73	4	no	—
122	C₈	—	146	4	no	—
123	C₂²	C₈×#10596833	1	2	yes	—
124	C₂²	#17989475	1	2	yes	—
125	C₂²	#17989476	1	2	yes	—
126	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
127	C₄	C₈×#10596833	1	2	yes	—
128	C₄	#17989475	1	2	yes	—
129	C₄	#17989476	1	2	yes	—
130	C₄	C₂×C₈×#64408	1	2	yes	—
131	C₄	C₂×#14556665	1	2	yes	—
132	C₄	C₂×#14556666	1	2	yes	—
133	C₂	#22624843	1	4	yes	direct product factor
134	C₂	C₂×C₈×#10596833	1	1	yes	characteristic
135	C₂	C₂×#17989475	1	1	yes	characteristic
136	C₂	C₂×#17989476	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#64408
commutator	C₂×C₇₃	C₂×C₈²
Frattini	C₄²	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#1311
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22624843	C₂	direct (N × H)
12. C₂×#11924031	C₈	semidirect
25. #11924031	C₂×C₈	semidirect
65. C₂×C₇₃	#1311	semidirect
72. C₇₃	C₂×#1311	semidirect
133. C₂	#22624843	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#11924031	2
#22624843	2
C₂×C₈×#10596831	2
C₂×#1311	73

Subgroups — C2×#22624843 #34009641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624843 #34009641