Subgroups — #36351447

Order 22144.

188 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#13063436	C₂	1	1	yes	maximal, characteristic
2	#26011361	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13063436	—	2	1	no	—
5	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
6	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
7	C₂×C₄⋊Dic₁₇₃	C₄	1	2	yes	—
8	C₄⋊#13063436	—	2	1	no	—
9	C₄⋊#13063436	C₄	1	2	yes	complemented
10	C₂²×C₈×C₁₇₃	—	2	1	no	—
11	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₇₃	—	2	1	no	—
13	C₄⋊C₈×C₁₇₃	C₄	1	2	yes	complemented
14	C₂²×Dic₁₇₃	—	2	2	no	—
15	C₂×#13063436	—	2	2	no	—
16	C₂×#13063436	—	4	1	no	—
17	C₂×#13063436	—	4	1	no	—
18	C₄⋊Dic₁₇₃	—	4	2	no	—
19	C₄⋊Dic₁₇₃	—	4	1	no	—
20	C₄⋊Dic₁₇₃	—	4	1	no	—
21	C₄⋊Dic₁₇₃	—	4	1	no	—
22	C₄⋊Dic₁₇₃	—	4	1	no	—
23	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₇₃	—	2	2	no	—
27	C₂×C₈×C₁₇₃	—	4	1	no	—
28	C₂×C₈×C₁₇₃	—	4	1	no	—
29	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
32	C₄²×C₁₇₃	Q₈	1	1	yes	characteristic
33	C₂×Dic₁₇₃	—	4	2	no	—
34	C₂×Dic₁₇₃	—	4	2	no	—
35	C₂×Dic₁₇₃	—	4	2	no	—
36	#13063436	—	4	2	no	—
37	C₂³×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇₃	—	2	1	no	—
39	C₂×C₄×C₁₇₃	—	2	1	no	—
40	C₂×C₄×C₁₇₃	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₄×C₁₇₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₇₃	SD₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₇₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₇₃	Q₁₆	1	1	yes	characteristic
53	C₂×C₄×C₁₇₃	Q₁₆	1	1	yes	characteristic
54	C₈×C₁₇₃	—	4	2	no	—
55	C₂²×C₁₇₃	#95	1	1	yes	characteristic
56	C₂²×C₁₇₃	#100	1	1	yes	characteristic
57	C₂²×C₁₇₃	#101	1	1	yes	characteristic
58	C₂²×C₁₇₃	#102	1	1	yes	characteristic
59	C₂²×C₁₇₃	#102	1	1	yes	characteristic
60	C₂²×C₁₇₃	#103	1	1	yes	characteristic
61	C₂²×C₁₇₃	#103	1	1	yes	characteristic
62	C₄×C₁₇₃	—	2	1	no	—
63	C₄×C₁₇₃	—	2	1	no	—
64	C₄×C₁₇₃	#102	1	1	yes	characteristic
65	C₄×C₁₇₃	#102	1	1	yes	characteristic
66	C₄×C₁₇₃	#103	1	1	yes	characteristic
67	C₄×C₁₇₃	#103	1	1	yes	characteristic
68	C₄×C₁₇₃	#106	1	1	yes	characteristic
69	C₄×C₁₇₃	#106	1	1	yes	characteristic
70	C₄×C₁₇₃	#107	1	1	yes	characteristic
71	C₄×C₁₇₃	#107	1	1	yes	characteristic
72	Dic₁₇₃	—	8	2	no	—
73	C₂×C₁₇₃	#331	1	1	yes	characteristic
74	C₂×C₁₇₃	#332	1	1	yes	characteristic
75	C₂×C₁₇₃	#332	1	1	yes	characteristic
76	C₂×C₁₇₃	#333	1	1	yes	characteristic
77	C₂×C₁₇₃	#340	1	1	yes	characteristic
78	C₂×C₁₇₃	#340	1	1	yes	characteristic
79	C₂×C₁₇₃	#343	1	1	yes	characteristic
80	C₁₇₃	#1296	1	1	yes	characteristic, complemented, Sylow
81	#1296	—	173	1	no	maximal, Sylow
82	C₂×C₄⋊C₈	—	173	1	no	—
83	C₂×C₄⋊C₈	D₁₇₃	1	1	yes	characteristic
84	#384	—	173	1	no	—
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	346	1	no	—
87	C₂×C₄⋊C₄	—	346	1	no	—
88	C₂×C₄⋊C₄	—	346	1	no	—
89	C₂×C₄⋊C₄	—	173	2	no	—
90	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	346	1	no	—
93	C₄⋊C₈	—	173	2	no	—
94	C₄⋊C₈	Dic₁₇₃	1	2	yes	complemented
95	C₂²×C₄	—	346	2	no	—
96	C₂²×C₄	C₄×D₁₇₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
98	C₂²×C₄	D₆₉₂	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	346	2	no	—
103	C₂×C₈	—	692	1	no	—
104	C₂×C₈	—	692	1	no	—
105	C₄⋊C₄	—	692	2	no	—
106	C₄⋊C₄	—	692	1	no	—
107	C₄⋊C₄	—	692	1	no	—
108	C₄⋊C₄	—	692	1	no	—
109	C₄⋊C₄	—	692	1	no	—
110	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
111	C₄²	C₂×Dic₁₇₃	1	1	yes	characteristic
112	C₄²	C₂²⋊D₁₇₃	1	1	yes	characteristic
113	C₄²	Dic₃₄₆	1	1	yes	characteristic
114	C₂³	#14583803	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	692	2	no	—
118	C₂×C₄	—	692	2	no	—
119	C₂×C₄	—	692	2	no	—
120	C₂×C₄	C₄×Dic₁₇₃	1	1	yes	characteristic
121	C₂×C₄	#14583803	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
123	C₂×C₄	#14583805	1	1	yes	characteristic
124	C₂×C₄	#14583805	1	1	yes	characteristic
125	C₂×C₄	C₄⋊Dic₁₇₃	1	1	yes	characteristic
126	C₂×C₄	#14583811	1	1	yes	characteristic
127	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
128	C₂×C₄	#14583813	1	1	yes	characteristic
129	C₂×C₄	#14583814	1	1	yes	characteristic
130	C₂×C₄	#14583815	1	1	yes	characteristic
131	C₂×C₄	#14583815	1	1	yes	characteristic
132	C₂×C₄	#14583816	1	1	yes	characteristic
133	C₂×C₄	Dic₆₉₂	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	692	2	no	—
136	C₂²	#19149586	1	1	yes	characteristic
137	C₂²	#19149595	1	1	yes	characteristic
138	C₂²	#19149597	1	1	yes	characteristic
139	C₂²	#19149599	1	1	yes	characteristic
140	C₂²	#19149601	1	1	yes	characteristic
141	C₂²	#19149602	1	1	yes	characteristic
142	C₂²	#19149604	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	1384	2	no	—
146	C₄	#19149600	1	1	yes	characteristic
147	C₄	#19149601	1	1	yes	characteristic
148	C₄	#19149603	1	1	yes	characteristic
149	C₄	#19149604	1	1	yes	characteristic
150	C₄	#19149610	1	1	yes	characteristic
151	C₄	#19149611	1	1	yes	characteristic
152	C₄	#19149612	1	1	yes	characteristic
153	C₄	#19149613	1	1	yes	characteristic
154	C₂	#26011234	1	1	yes	characteristic
155	C₂	#26011236	1	1	yes	characteristic
156	C₂	#26011238	1	1	yes	characteristic
157	C₂	#26011239	1	1	yes	characteristic
158	C₂	#26011253	1	1	yes	characteristic
159	C₂	#26011254	1	1	yes	characteristic
160	C₂	#26011259	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14583803
commutator	C₂×C₄×C₁₇₃	C₄²
Frattini	C₂×C₄²	C₂×D₁₇₃
Fitting	C₂×C₄⋊C₈×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇₃	C₂²
hypercenter	C₂×C₄⋊C₈	D₁₇₃
hypocenter	C₁₇₃	#1296
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. C₄⋊#13063436	C₄	semidirect
13. C₄⋊C₈×C₁₇₃	C₄	semidirect
80. C₁₇₃	#1296	semidirect
94. C₄⋊C₈	Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#13063436	2
#26011361	2
C₂×C₄⋊C₈×C₁₇₃	2
#1296	173