Subgroups — #2320×C₁₇₃ #36350105

Order 22144.

444 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
2	#391×C₁₇₃	C₂	1	2	yes	maximal
3	#389×C₁₇₃	C₂	1	2	yes	maximal
4	#387×C₁₇₃	C₂	1	4	yes	maximal
5	#383×C₁₇₃	C₂	1	2	yes	maximal
6	#382×C₁₇₃	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	#376×C₁₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₇₃	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
19	#95×C₁₇₃	C₂²	1	1	yes	characteristic
20	#95×C₁₇₃	C₂²	1	1	yes	characteristic
21	#95×C₁₇₃	C₂²	1	2	yes	—
22	#95×C₁₇₃	C₂²	1	2	yes	—
23	#95×C₁₇₃	C₂²	1	4	yes	—
24	#95×C₁₇₃	C₂²	1	4	yes	—
25	#95×C₁₇₃	C₂²	1	4	yes	—
26	C₂×Q₈×C₁₇₃	—	2	4	no	—
27	C₂×Q₈×C₁₇₃	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₇₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₇₃	—	2	4	no	—
37	C₄⋊C₄×C₁₇₃	—	4	4	no	—
38	C₄⋊C₄×C₁₇₃	C₂×C₄	1	8	yes	—
39	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
40	C₄²×C₁₇₃	—	2	4	no	—
41	C₄²×C₁₇₃	—	4	2	no	—
42	C₄²×C₁₇₃	C₂×C₄	1	8	yes	—
43	C₄²×C₁₇₃	C₂×C₄	1	4	yes	—
44	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
45	Q₈×C₁₇₃	—	2	8	no	—
46	C₂×C₄×C₁₇₃	—	2	4	no	—
47	C₂×C₄×C₁₇₃	—	2	1	no	—
48	C₂×C₄×C₁₇₃	—	2	2	no	—
49	C₂×C₄×C₁₇₃	—	2	2	no	—
50	C₂×C₄×C₁₇₃	—	2	1	no	—
51	C₂×C₄×C₁₇₃	—	2	2	no	—
52	C₂×C₄×C₁₇₃	—	2	2	no	—
53	C₂×C₄×C₁₇₃	—	2	2	no	—
54	C₂×C₄×C₁₇₃	—	2	4	no	—
55	C₂×C₄×C₁₇₃	—	2	4	no	—
56	C₂×C₄×C₁₇₃	—	2	2	no	—
57	C₂×C₄×C₁₇₃	—	2	2	no	—
58	C₂×C₄×C₁₇₃	—	2	2	no	—
59	C₂×C₄×C₁₇₃	—	2	2	no	—
60	C₂×C₄×C₁₇₃	—	2	4	no	—
61	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₇₃	#41	1	4	yes	—
67	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
68	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
69	C₂²×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
70	C₂²×C₁₇₃	2₋¹⁺⁴	1	2	yes	—
71	C₂²×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
72	C₄×C₁₇₃	—	2	4	no	—
73	C₄×C₁₇₃	—	2	2	no	—
74	C₄×C₁₇₃	—	2	2	no	—
75	C₄×C₁₇₃	—	2	4	no	—
76	C₄×C₁₇₃	—	4	2	no	—
77	C₄×C₁₇₃	—	4	2	no	—
78	C₄×C₁₇₃	—	4	4	no	—
79	C₄×C₁₇₃	#117	1	4	yes	—
80	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
81	C₂×C₁₇₃	#519	1	1	yes	characteristic
82	C₂×C₁₇₃	#520	1	1	yes	characteristic
83	C₂×C₁₇₃	#541	1	2	yes	—
84	C₂×C₁₇₃	#542	1	2	yes	—
85	C₁₇₃	#2320	1	1	yes	characteristic, direct product factor, Sylow
86	#2320	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₄×Q₈	C₂×C₁₇₃	1	1	yes	characteristic
88	#391	C₂×C₁₇₃	1	2	yes	—
89	#389	C₂×C₁₇₃	1	2	yes	—
90	#387	C₂×C₁₇₃	1	4	yes	—
91	#383	C₂×C₁₇₃	1	2	yes	—
92	#382	C₂×C₁₇₃	1	2	yes	—
93	C₄×C₄⋊C₄	C₂×C₁₇₃	1	1	yes	characteristic
94	#376	C₂×C₁₇₃	1	1	yes	characteristic
95	C₂²×Q₈	C₂²×C₁₇₃	1	1	yes	characteristic
96	C₄×Q₈	C₄×C₁₇₃	1	8	yes	complemented
97	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	4	yes	—
99	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₁₇₃	1	2	yes	—
102	C₂×C₄²	C₂²×C₁₇₃	1	2	yes	—
103	C₂×C₄²	C₂²×C₁₇₃	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₁₇₃	1	2	yes	—
105	#95	C₂²×C₁₇₃	1	1	yes	characteristic
106	#95	C₂²×C₁₇₃	1	1	yes	characteristic
107	#95	C₂²×C₁₇₃	1	2	yes	—
108	#95	C₂²×C₁₇₃	1	2	yes	—
109	#95	C₂²×C₁₇₃	1	4	yes	—
110	#95	C₂²×C₁₇₃	1	4	yes	—
111	#95	C₂²×C₁₇₃	1	4	yes	—
112	C₂×Q₈	—	2	4	no	—
113	C₂×Q₈	C₂×C₄×C₁₇₃	1	4	yes	—
114	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
115	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
116	C₂²×C₄	C₂³×C₁₇₃	1	4	yes	—
117	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
119	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
120	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
121	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	4	4	no	—
124	C₄⋊C₄	C₂×C₄×C₁₇₃	1	8	yes	—
125	C₄⋊C₄	C₂×C₄×C₁₇₃	1	4	yes	—
126	C₄²	—	2	4	no	—
127	C₄²	—	4	2	no	—
128	C₄²	C₂×C₄×C₁₇₃	1	8	yes	—
129	C₄²	C₂×C₄×C₁₇₃	1	4	yes	—
130	C₂³	C₂⁴×C₁₇₃	1	1	yes	characteristic
131	Q₈	—	2	8	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
148	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
149	C₂×C₄	C₂²×C₄×C₁₇₃	1	4	yes	—
150	C₂×C₄	C₂²×C₄×C₁₇₃	1	4	yes	—
151	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
152	C₂×C₄	#41×C₁₇₃	1	4	yes	—
153	C₂²	C₂³×C₄×C₁₇₃	1	1	yes	characteristic
154	C₂²	C₂×#41×C₁₇₃	1	2	yes	—
155	C₂²	H(5,𝔽₂)×C₁₇₃	1	1	yes	characteristic
156	C₂²	2₋¹⁺⁴×C₁₇₃	1	2	yes	—
157	C₂²	2₋¹⁺⁴×C₁₇₃	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	4	no	—
162	C₄	—	4	2	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	4	no	—
165	C₄	#117×C₁₇₃	1	4	yes	—
166	C₂	C₂×#117×C₁₇₃	1	1	yes	characteristic
167	C₂	#519×C₁₇₃	1	1	yes	characteristic
168	C₂	#520×C₁₇₃	1	1	yes	characteristic
169	C₂	#541×C₁₇₃	1	2	yes	—
170	C₂	#542×C₁₇₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₇₃
Frattini, cosocle	C₂³	C₂⁴×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
10. C₄×Q₈×C₁₇₃	C₄	semidirect
85. C₁₇₃	#2320	direct (N × H)
86. #2320	C₁₇₃	direct (N × H)
96. C₄×Q₈	C₄×C₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₇₃	2
#391×C₁₇₃	2
#389×C₁₇₃	2
#387×C₁₇₃	2
#383×C₁₇₃	2
#382×C₁₇₃	2
C₄×C₄⋊C₄×C₁₇₃	2
#376×C₁₇₃	2
#2320	173

Subgroups — #2320×C173 #36350105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2320×C₁₇₃ #36350105