Subgroups — C₄×#17988415 #33001387

Order 18688.

366 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988415	C₂	1	2	yes	maximal
2	C₄×#14556525	C₂	1	2	yes	maximal
3	#22608023	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556525	C₂²	1	4	yes	—
6	C₄×C₄⋊Dic₇₃	—	2	2	no	—
7	#17988415	C₄	1	8	yes	direct product factor
8	#17988350	—	2	2	no	—
9	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₇₃	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₇₃	—	2	4	no	—
14	C₂×C₄×Dic₇₃	—	4	2	no	—
15	#14556525	—	2	4	no	—
16	#14556525	C₂×C₄	1	8	yes	—
17	#14556498	—	4	2	no	—
18	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₃	—	2	2	no	—
22	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
24	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₇₃	Q₈	1	2	yes	—
27	C₂²×Dic₇₃	—	4	4	no	—
28	C₄⋊Dic₇₃	—	2	8	no	—
29	C₄⋊Dic₇₃	—	4	4	no	—
30	C₄×Dic₇₃	—	8	4	no	—
31	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
37	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₇₃	#41	1	2	yes	—
40	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
41	C₄²×C₇₃	Q₁₆	1	2	yes	—
42	C₁₆×C₇₃	—	2	2	no	—
43	C₁₆×C₇₃	C₄²	1	4	yes	complemented
44	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	#107	1	2	yes	—
47	C₂×C₄×C₇₃	#107	1	2	yes	—
48	C₂×C₄×C₇₃	SD₃₂	1	4	yes	—
49	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#135	1	2	yes	—
53	C₂×Dic₇₃	—	4	8	no	—
54	C₂×Dic₇₃	—	8	4	no	—
55	C₂×Dic₇₃	—	8	4	no	—
56	C₈×C₇₃	—	2	2	no	—
57	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
58	C₈×C₇₃	#117	1	1	yes	characteristic
59	C₈×C₇₃	C₄×Q₈	1	2	yes	—
60	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
61	C₂²×C₇₃	C₄×D₈	1	2	yes	—
62	C₂²×C₇₃	C₂×SD₃₂	1	2	yes	—
63	C₂²×C₇₃	#508	1	2	yes	—
64	C₄×C₇₃	—	2	2	no	—
65	C₄×C₇₃	#367	1	4	yes	complemented
66	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₇₃	#427	1	1	yes	characteristic
68	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
69	Dic₇₃	—	8	8	no	—
70	C₄×#367	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	C₄×#107	1	1	yes	characteristic
72	C₂×C₇₃	C₂×#367	1	1	yes	characteristic
73	C₂×C₇₃	#2158	1	1	yes	characteristic
74	C₂×C₇₃	C₄×SD₃₂	1	4	yes	—
75	C₂×#367	—	73	2	no	—
76	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
77	C₄×#107	—	73	2	no	—
78	#1380	—	73	2	no	—
79	C₇₃	C₄×#367	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
82	C₂×#107	—	73	4	no	—
83	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
84	C₄×C₄⋊C₄	—	146	2	no	—
85	#367	—	73	8	no	—
86	C₄×C₁₆	Dic₇₃	1	4	yes	complemented
87	#340	—	146	2	no	—
88	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
89	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	146	4	no	—
91	C₂×C₄²	—	292	2	no	—
92	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	C₄×D₇₃	1	4	yes	—
95	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
96	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
97	#107	—	73	8	no	—
98	#107	—	146	4	no	—
99	C₄×C₈	Dic₁₄₆	1	2	yes	—
100	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
101	#95	—	292	2	no	—
102	C₂²×C₄	—	292	4	no	—
103	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
104	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
108	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
109	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
110	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
111	C₂×C₈	#11924028	1	2	yes	—
112	C₄⋊C₄	—	292	4	no	—
113	C₄⋊C₄	—	146	8	no	—
114	C₄²	—	584	4	no	—
115	C₄²	C₄⋊Dic₇₃	1	2	yes	—
116	C₄²	Dic₂₉₂	1	2	yes	—
117	C₁₆	—	2	2	no	—
118	C₁₆	C₄×Dic₇₃	1	4	yes	complemented
119	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	292	8	no	—
122	C₂×C₄	—	584	4	no	—
123	C₂×C₄	—	584	4	no	—
124	C₂×C₄	#14556525	1	2	yes	—
125	C₂×C₄	#14556525	1	2	yes	—
126	C₂×C₄	#14556536	1	4	yes	—
127	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
128	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
129	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
130	C₂×C₄	#14556613	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	C₂×C₄×Dic₇₃	1	1	yes	characteristic
133	C₈	#14556550	1	1	yes	characteristic
134	C₈	C₄×Dic₁₄₆	1	2	yes	—
135	C₂²	C₂×#14556525	1	1	yes	characteristic
136	C₂²	C₄×D₅₈₄	1	2	yes	—
137	C₂²	C₂×#14556536	1	2	yes	—
138	C₂²	#17989019	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	584	8	no	—
141	C₄	#17988415	1	4	yes	direct product factor
142	C₄	C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
143	C₄	#17988625	1	1	yes	characteristic
144	C₄	C₄×Dic₂₉₂	1	2	yes	—
145	C₂	C₄×#14556525	1	1	yes	characteristic
146	C₂	C₂×#17988415	1	1	yes	characteristic
147	C₂	#22611715	1	1	yes	characteristic
148	C₂	C₄×#14556536	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₅₈₄
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	C₄×#367
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
7. #17988415	C₄	direct (N × H)
12. C₄×C₁₆×C₇₃	C₄	semidirect
43. C₁₆×C₇₃	C₄²	semidirect
65. C₄×C₇₃	#367	semidirect
79. C₇₃	C₄×#367	semidirect
86. C₄×C₁₆	Dic₇₃	semidirect
118. C₁₆	C₄×Dic₇₃	semidirect
141. C₄	#17988415	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×#17988415	2
C₄×#14556525	2
#22608023	2
C₂×C₄×C₁₆×C₇₃	2
C₄×#367	73

Subgroups — C4×#17988415 #33001387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17988415 #33001387