Subgroups — #3380×C₁₇₃ #36351165

Order 22144.

352 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554×C₁₇₃	C₂	1	2	yes	maximal, complemented
2	#500×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#499×C₁₇₃	C₂	1	2	yes	maximal, complemented
4	#478×C₁₇₃	C₂	1	4	yes	maximal, complemented
5	#476×C₁₇₃	C₂	1	2	yes	maximal, complemented
6	C₈×Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₈×C₁₇₃	C₂	1	2	yes	maximal
8	C₄×D₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×D₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
11	#125×C₁₇₃	C₂²	1	4	yes	—
12	C₂²⋊Q₈×C₁₇₃	—	2	2	no	—
13	C₂²⋊Q₈×C₁₇₃	—	2	4	no	—
14	C₄×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₇₃	C₂²	1	2	yes	—
16	C₄×D₄×C₁₇₃	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₇₃	—	2	2	no	—
18	#107×C₁₇₃	C₂²	1	1	yes	characteristic
19	#107×C₁₇₃	C₂²	1	2	yes	—
20	#107×C₁₇₃	C₂²	1	2	yes	—
21	#106×C₁₇₃	C₂²	1	4	yes	—
22	C₄⋊C₈×C₁₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₇₃	C₂²	1	2	yes	—
24	#102×C₁₇₃	C₂²	1	4	yes	—
25	#102×C₁₇₃	C₂²	1	2	yes	—
26	C₄×C₈×C₁₇₃	C₂²	1	2	yes	—
27	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₇₃	—	2	2	no	—
29	C₂×Q₈×C₁₇₃	D₄	1	1	yes	characteristic
30	C₂×D₄×C₁₇₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₇₃	—	2	2	no	—
32	C₂²×C₄×C₁₇₃	—	2	4	no	—
33	D₈×C₁₇₃	Q₈	1	4	yes	complemented
34	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
35	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₇₃	—	2	2	no	—
38	C₄⋊C₄×C₁₇₃	—	2	4	no	—
39	C₄⋊C₄×C₁₇₃	—	2	4	no	—
40	C₄⋊C₄×C₁₇₃	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₇₃	D₄	1	2	yes	—
42	C₄⋊C₄×C₁₇₃	C₂³	1	4	yes	—
43	C₄⋊C₄×C₁₇₃	C₂³	1	2	yes	—
44	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
45	C₂²⋊C₄×C₁₇₃	—	2	4	no	—
46	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
47	C₄²×C₁₇₃	C₂³	1	2	yes	—
48	C₂³×C₁₇₃	—	2	2	no	—
49	Q₈×C₁₇₃	—	2	2	no	—
50	Q₈×C₁₇₃	—	8	2	no	—
51	D₄×C₁₇₃	—	2	2	no	—
52	D₄×C₁₇₃	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₁₇₃	—	2	2	no	—
54	C₂×C₄×C₁₇₃	—	2	4	no	—
55	C₂×C₄×C₁₇₃	—	4	4	no	—
56	C₂×C₄×C₁₇₃	—	8	4	no	—
57	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
62	C₈×C₁₇₃	—	2	2	no	—
63	C₈×C₁₇₃	—	2	1	no	—
64	C₈×C₁₇₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₁₇₃	—	2	4	no	—
66	C₂²×C₁₇₃	—	4	2	no	—
67	C₂²×C₁₇₃	—	4	2	no	—
68	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₁₇₃	—	2	2	no	—
70	C₄×C₁₇₃	—	2	2	no	—
71	C₄×C₁₇₃	—	2	1	no	—
72	C₄×C₁₇₃	—	4	2	no	—
73	C₄×C₁₇₃	—	4	4	no	—
74	C₄×C₁₇₃	#135	1	2	yes	—
75	C₄×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
77	C₂×C₁₇₃	—	4	4	no	—
78	C₂×C₁₇₃	D₄×Q₈	1	1	yes	characteristic
79	C₂×C₁₇₃	C₂×#135	1	1	yes	characteristic
80	C₂×C₁₇₃	#576	1	1	yes	characteristic
81	C₁₇₃	#3380	1	1	yes	characteristic, direct product factor, Sylow
82	#3380	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	#554	C₂×C₁₇₃	1	2	yes	complemented
84	#500	C₂×C₁₇₃	1	1	yes	characteristic, complemented
85	#499	C₂×C₁₇₃	1	2	yes	complemented
86	#478	C₂×C₁₇₃	1	4	yes	complemented
87	#476	C₂×C₁₇₃	1	2	yes	complemented
88	C₈×Q₈	C₂×C₁₇₃	1	1	yes	characteristic, complemented
89	C₄×D₈	C₂×C₁₇₃	1	2	yes	—
90	C₄×D₈	C₂×C₁₇₃	1	1	yes	characteristic
91	C₂×D₈	C₂²×C₁₇₃	1	1	yes	characteristic
92	C₄⋊Q₈	C₂²×C₁₇₃	1	2	yes	—
93	#125	C₂²×C₁₇₃	1	4	yes	—
94	C₂²⋊Q₈	—	2	2	no	—
95	C₂²⋊Q₈	—	2	4	no	—
96	C₄×Q₈	C₂²×C₁₇₃	1	1	yes	characteristic
97	C₄×D₄	C₂²×C₁₇₃	1	2	yes	—
98	C₄×D₄	C₂²×C₁₇₃	1	4	yes	—
99	C₂×C₄⋊C₄	—	2	2	no	—
100	#107	C₂²×C₁₇₃	1	1	yes	characteristic
101	#107	C₂²×C₁₇₃	1	2	yes	—
102	#107	C₂²×C₁₇₃	1	2	yes	—
103	#106	C₂²×C₁₇₃	1	4	yes	—
104	C₄⋊C₈	C₂²×C₁₇₃	1	1	yes	characteristic
105	C₄⋊C₈	C₂²×C₁₇₃	1	2	yes	—
106	#102	C₂²×C₁₇₃	1	4	yes	—
107	#102	C₂²×C₁₇₃	1	2	yes	—
108	C₄×C₈	C₂²×C₁₇₃	1	2	yes	—
109	C₄×C₈	C₂²×C₁₇₃	1	1	yes	characteristic
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	D₄×C₁₇₃	1	1	yes	characteristic
112	C₂×D₄	C₂³×C₁₇₃	1	2	yes	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	4	no	—
115	D₈	Q₈×C₁₇₃	1	4	yes	complemented
116	C₂×C₈	C₂³×C₁₇₃	1	2	yes	—
117	C₂×C₈	C₂³×C₁₇₃	1	1	yes	characteristic
118	C₂×C₈	C₂³×C₁₇₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	D₄×C₁₇₃	1	1	yes	characteristic
123	C₄⋊C₄	D₄×C₁₇₃	1	2	yes	—
124	C₄⋊C₄	C₂³×C₁₇₃	1	2	yes	—
125	C₄⋊C₄	C₂³×C₁₇₃	1	4	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	2	4	no	—
128	C₄²	C₂³×C₁₇₃	1	2	yes	—
129	C₄²	C₂³×C₁₇₃	1	1	yes	characteristic
130	C₂³	—	2	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	8	2	no	—
133	D₄	—	2	2	no	—
134	D₄	C₂×Q₈×C₁₇₃	1	4	yes	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	4	no	—
139	C₂×C₄	C₂×D₄×C₁₇₃	1	2	yes	—
140	C₂×C₄	C₂×D₄×C₁₇₃	1	1	yes	characteristic
141	C₂×C₄	C₂×D₄×C₁₇₃	1	2	yes	—
142	C₂×C₄	C₂×D₄×C₁₇₃	1	1	yes	characteristic
143	C₂×C₄	C₂⁴×C₁₇₃	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	1	no	—
146	C₈	C₂×Q₈×C₁₇₃	1	2	yes	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	C₂²×D₄×C₁₇₃	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	2	no	—
155	C₄	—	4	4	no	—
156	C₄	#135×C₁₇₃	1	2	yes	—
157	C₄	C₂²×Q₈×C₁₇₃	1	1	yes	characteristic
158	C₄	2₋¹⁺⁴×C₁₇₃	1	1	yes	characteristic
159	C₂	—	4	4	no	—
160	C₂	D₄×Q₈×C₁₇₃	1	1	yes	characteristic
161	C₂	C₂×#135×C₁₇₃	1	1	yes	characteristic
162	C₂	#576×C₁₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇₃
prime radical	C₂×D₈×C₁₇₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇₃ ◁ C₄×Q₈×C₁₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #554×C₁₇₃	C₂	semidirect
2. #500×C₁₇₃	C₂	semidirect
3. #499×C₁₇₃	C₂	semidirect
4. #478×C₁₇₃	C₂	semidirect
5. #476×C₁₇₃	C₂	semidirect
6. C₈×Q₈×C₁₇₃	C₂	semidirect
33. D₈×C₁₇₃	Q₈	semidirect
81. C₁₇₃	#3380	direct (N × H)
82. #3380	C₁₇₃	direct (N × H)
83. #554	C₂×C₁₇₃	semidirect
84. #500	C₂×C₁₇₃	semidirect
85. #499	C₂×C₁₇₃	semidirect
86. #478	C₂×C₁₇₃	semidirect
87. #476	C₂×C₁₇₃	semidirect
88. C₈×Q₈	C₂×C₁₇₃	semidirect
115. D₈	Q₈×C₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#554×C₁₇₃	2
#500×C₁₇₃	2
#499×C₁₇₃	2
#478×C₁₇₃	2
#476×C₁₇₃	2
C₈×Q₈×C₁₇₃	2
C₄×D₈×C₁₇₃	2
#3380	173

Subgroups — #3380×C173 #36351165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3380×C₁₇₃ #36351165