Subgroups — C₂×#550×C₁₇₃ #36351252

Order 22144.

1736 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×#41×C₁₇₃	C₂	1	2	yes	maximal, complemented
2	#550×C₁₇₃	C₂	1	16	yes	maximal, direct product factor
3	C₂×#127×C₁₇₃	C₂	1	3	yes	maximal, complemented
4	C₂×#121×C₁₇₃	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₁₇₃	C₂	1	3	yes	maximal, complemented
7	C₂×#41×C₁₇₃	—	2	16	no	—
8	C₂×#41×C₁₇₃	C₂²	1	16	yes	complemented
9	C₂²×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₇₃	C₂²	1	3	yes	—
11	C₂²×D₄×C₁₇₃	C₂²	1	6	yes	—
12	C₂²×D₄×C₁₇₃	C₂²	1	6	yes	—
13	C₂³×C₄×C₁₇₃	C₂²	1	6	yes	—
14	#127×C₁₇₃	C₂²	1	24	yes	complemented
15	#121×C₁₇₃	C₂²	1	48	yes	complemented
16	C₄×Q₈×C₁₇₃	C₂²	1	8	yes	complemented
17	C₄×D₄×C₁₇₃	C₂²	1	24	yes	complemented
18	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄×C₁₇₃	C₂²	1	6	yes	—
21	C₂×C₄²×C₁₇₃	C₂²	1	3	yes	—
22	C₂⁴×C₁₇₃	C₂³	1	6	yes	—
23	#41×C₁₇₃	—	2	64	no	—
24	C₂×Q₈×C₁₇₃	D₄	1	8	yes	complemented
25	C₂×Q₈×C₁₇₃	C₂³	1	4	yes	—
26	C₂×D₄×C₁₇₃	—	2	24	no	—
27	C₂×D₄×C₁₇₃	—	2	12	no	—
28	C₂×D₄×C₁₇₃	—	4	12	no	—
29	C₂×D₄×C₁₇₃	C₂³	1	24	yes	—
30	C₂×D₄×C₁₇₃	C₂³	1	12	yes	—
31	C₂×D₄×C₁₇₃	C₂³	1	24	yes	—
32	C₂²×C₄×C₁₇₃	—	2	24	no	—
33	C₂²×C₄×C₁₇₃	C₂³	1	24	yes	—
34	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₇₃	C₂³	1	3	yes	—
36	C₂²×C₄×C₁₇₃	C₂³	1	3	yes	—
37	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₇₃	C₂³	1	4	yes	—
39	C₄⋊C₄×C₁₇₃	C₂³	1	12	yes	—
40	C₂²⋊C₄×C₁₇₃	C₂³	1	24	yes	—
41	C₄²×C₁₇₃	C₂³	1	12	yes	—
42	C₂³×C₁₇₃	—	2	12	no	—
43	C₂³×C₁₇₃	—	2	12	no	—
44	C₂³×C₁₇₃	—	2	12	no	—
45	C₂³×C₁₇₃	C₂⁴	1	12	yes	—
46	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
47	Q₈×C₁₇₃	C₂×D₄	1	16	yes	complemented
48	D₄×C₁₇₃	—	2	48	no	—
49	D₄×C₁₇₃	—	2	24	no	—
50	D₄×C₁₇₃	—	4	24	no	—
51	C₂×C₄×C₁₇₃	—	2	4	no	—
52	C₂×C₄×C₁₇₃	—	2	6	no	—
53	C₂×C₄×C₁₇₃	—	2	48	no	—
54	C₂×C₄×C₁₇₃	C₂×D₄	1	12	yes	—
55	C₂×C₄×C₁₇₃	#41	1	4	yes	complemented
56	C₂×C₄×C₁₇₃	C₂⁴	1	6	yes	—
57	C₂×C₄×C₁₇₃	C₂⁴	1	4	yes	—
58	C₂×C₄×C₁₇₃	C₂⁴	1	6	yes	—
59	C₂×C₄×C₁₇₃	C₂⁴	1	2	yes	—
60	C₂²×C₁₇₃	—	2	12	no	—
61	C₂²×C₁₇₃	—	2	12	no	—
62	C₂²×C₁₇₃	—	2	12	no	—
63	C₂²×C₁₇₃	—	4	24	no	—
64	C₂²×C₁₇₃	C₂²×D₄	1	2	yes	—
65	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
66	C₂²×C₁₇₃	H(5,𝔽₂)	1	2	yes	—
67	C₂²×C₁₇₃	C₂⁵	1	1	yes	characteristic
68	C₄×C₁₇₃	—	2	4	no	—
69	C₄×C₁₇₃	—	2	6	no	—
70	C₄×C₁₇₃	C₂²×D₄	1	12	yes	—
71	C₄×C₁₇₃	C₂×#41	1	4	yes	complemented
72	C₂×C₁₇₃	—	4	12	no	—
73	C₂×C₁₇₃	#550	1	4	yes	direct product factor
74	C₂×C₁₇₃	C₂³×D₄	1	1	yes	characteristic
75	C₂×C₁₇₃	C₂²×#41	1	1	yes	characteristic
76	C₂×C₁₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
77	C₁₇₃	C₂×#550	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×#550	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×#41	C₂×C₁₇₃	1	2	yes	complemented
80	#550	C₂×C₁₇₃	1	16	yes	direct product factor
81	C₂×#127	C₂×C₁₇₃	1	3	yes	complemented
82	C₂×#121	C₂×C₁₇₃	1	6	yes	complemented
83	C₂×C₄×Q₈	C₂×C₁₇₃	1	1	yes	characteristic, complemented
84	C₂×C₄×D₄	C₂×C₁₇₃	1	3	yes	complemented
85	C₂×#41	—	2	16	no	—
86	C₂×#41	C₂²×C₁₇₃	1	16	yes	complemented
87	C₂²×Q₈	C₂²×C₁₇₃	1	1	yes	characteristic
88	C₂²×D₄	C₂²×C₁₇₃	1	3	yes	—
89	C₂²×D₄	C₂²×C₁₇₃	1	6	yes	—
90	C₂²×D₄	C₂²×C₁₇₃	1	6	yes	—
91	C₂³×C₄	C₂²×C₁₇₃	1	6	yes	—
92	#127	C₂²×C₁₇₃	1	24	yes	complemented
93	#121	C₂²×C₁₇₃	1	48	yes	complemented
94	C₄×Q₈	C₂²×C₁₇₃	1	8	yes	complemented
95	C₄×D₄	C₂²×C₁₇₃	1	24	yes	complemented
96	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	3	yes	—
98	C₂×C₂²⋊C₄	C₂²×C₁₇₃	1	6	yes	—
99	C₂×C₄²	C₂²×C₁₇₃	1	3	yes	—
100	C₂⁴	C₂³×C₁₇₃	1	6	yes	—
101	#41	—	2	64	no	—
102	C₂×Q₈	D₄×C₁₇₃	1	8	yes	complemented
103	C₂×Q₈	C₂³×C₁₇₃	1	4	yes	—
104	C₂×D₄	—	2	24	no	—
105	C₂×D₄	—	2	12	no	—
106	C₂×D₄	—	4	12	no	—
107	C₂×D₄	C₂³×C₁₇₃	1	12	yes	—
108	C₂×D₄	C₂³×C₁₇₃	1	24	yes	—
109	C₂×D₄	C₂³×C₁₇₃	1	24	yes	—
110	C₂²×C₄	—	2	24	no	—
111	C₂²×C₄	C₂³×C₁₇₃	1	3	yes	—
112	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
113	C₂²×C₄	C₂³×C₁₇₃	1	24	yes	—
114	C₂²×C₄	C₂³×C₁₇₃	1	3	yes	—
115	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
116	C₄⋊C₄	C₂³×C₁₇₃	1	4	yes	—
117	C₄⋊C₄	C₂³×C₁₇₃	1	12	yes	—
118	C₂²⋊C₄	C₂³×C₁₇₃	1	24	yes	—
119	C₄²	C₂³×C₁₇₃	1	12	yes	—
120	C₂³	—	2	12	no	—
121	C₂³	—	2	12	no	—
122	C₂³	—	2	12	no	—
123	C₂³	C₂⁴×C₁₇₃	1	12	yes	—
124	C₂³	C₂⁴×C₁₇₃	1	1	yes	characteristic
125	Q₈	C₂×D₄×C₁₇₃	1	16	yes	complemented
126	D₄	—	2	48	no	—
127	D₄	—	2	24	no	—
128	D₄	—	4	24	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	6	no	—
131	C₂×C₄	—	2	48	no	—
132	C₂×C₄	C₂×D₄×C₁₇₃	1	12	yes	—
133	C₂×C₄	#41×C₁₇₃	1	4	yes	complemented
134	C₂×C₄	C₂⁴×C₁₇₃	1	6	yes	—
135	C₂×C₄	C₂⁴×C₁₇₃	1	4	yes	—
136	C₂×C₄	C₂⁴×C₁₇₃	1	6	yes	—
137	C₂×C₄	C₂⁴×C₁₇₃	1	2	yes	—
138	C₂²	—	2	12	no	—
139	C₂²	—	2	12	no	—
140	C₂²	—	2	12	no	—
141	C₂²	—	4	24	no	—
142	C₂²	C₂²×D₄×C₁₇₃	1	2	yes	—
143	C₂²	C₂×#41×C₁₇₃	1	2	yes	—
144	C₂²	H(5,𝔽₂)×C₁₇₃	1	2	yes	—
145	C₂²	C₂⁵×C₁₇₃	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	2	6	no	—
148	C₄	C₂²×D₄×C₁₇₃	1	12	yes	—
149	C₄	C₂×#41×C₁₇₃	1	4	yes	complemented
150	C₂	—	4	12	no	—
151	C₂	#550×C₁₇₃	1	4	yes	direct product factor
152	C₂	C₂³×D₄×C₁₇₃	1	1	yes	characteristic
153	C₂	C₂²×#41×C₁₇₃	1	1	yes	characteristic
154	C₂	C₂×H(5,𝔽₂)×C₁₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₇₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₁₇₃	C₂	semidirect
2. #550×C₁₇₃	C₂	direct (N × H)
3. C₂×#127×C₁₇₃	C₂	semidirect
4. C₂×#121×C₁₇₃	C₂	semidirect
5. C₂×C₄×Q₈×C₁₇₃	C₂	semidirect
6. C₂×C₄×D₄×C₁₇₃	C₂	semidirect
8. C₂×#41×C₁₇₃	C₂²	semidirect
14. #127×C₁₇₃	C₂²	semidirect
15. #121×C₁₇₃	C₂²	semidirect
16. C₄×Q₈×C₁₇₃	C₂²	semidirect
17. C₄×D₄×C₁₇₃	C₂²	semidirect
24. C₂×Q₈×C₁₇₃	D₄	semidirect
47. Q₈×C₁₇₃	C₂×D₄	semidirect
55. C₂×C₄×C₁₇₃	#41	semidirect
71. C₄×C₁₇₃	C₂×#41	semidirect
73. C₂×C₁₇₃	#550	direct (N × H)
77. C₁₇₃	C₂×#550	direct (N × H)
78. C₂×#550	C₁₇₃	direct (N × H)
79. C₂²×#41	C₂×C₁₇₃	semidirect
80. #550	C₂×C₁₇₃	direct (N × H)
81. C₂×#127	C₂×C₁₇₃	semidirect
82. C₂×#121	C₂×C₁₇₃	semidirect
83. C₂×C₄×Q₈	C₂×C₁₇₃	semidirect
84. C₂×C₄×D₄	C₂×C₁₇₃	semidirect
86. C₂×#41	C₂²×C₁₇₃	semidirect
92. #127	C₂²×C₁₇₃	semidirect
93. #121	C₂²×C₁₇₃	semidirect
94. C₄×Q₈	C₂²×C₁₇₃	semidirect
95. C₄×D₄	C₂²×C₁₇₃	semidirect
102. C₂×Q₈	D₄×C₁₇₃	semidirect
125. Q₈	C₂×D₄×C₁₇₃	semidirect
133. C₂×C₄	#41×C₁₇₃	semidirect
149. C₄	C₂×#41×C₁₇₃	semidirect
151. C₂	#550×C₁₇₃	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×#41×C₁₇₃	2
#550×C₁₇₃	2
C₂×#127×C₁₇₃	2
C₂×#121×C₁₇₃	2
C₂×C₄×Q₈×C₁₇₃	2
C₂×C₄×D₄×C₁₇₃	2
C₂×#550	173

Subgroups — C2×#550×C173 #36351252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#550×C₁₇₃ #36351252