Subgroups — #36351790

Order 22144.

568 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149586	C₂	1	2	yes	maximal
2	C₂×#19149586	C₂	1	2	yes	maximal
3	C₂×#19149586	C₂	1	2	yes	maximal
4	C₂²×C₄²×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₇₃	C₂²	1	4	yes	—
6	#19149586	—	2	4	no	—
7	#19149586	—	2	4	no	—
8	#19149586	C₄	1	8	yes	complemented
9	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₇₃	—	2	4	no	—
13	C₂×C₄²×C₁₇₃	C₄	1	4	yes	complemented
14	C₂²×Dic₁₇₃	—	2	4	no	—
15	C₂²×Dic₁₇₃	—	2	4	no	—
16	C₂²×Dic₁₇₃	—	2	4	no	—
17	C₂²×Dic₁₇₃	—	2	4	no	—
18	C₂²×Dic₁₇₃	—	2	4	no	—
19	C₂²×Dic₁₇₃	—	2	4	no	—
20	C₂²×Dic₁₇₃	C₂×C₄	1	8	yes	—
21	C₂⁴×C₁₇₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₇₃	—	2	2	no	—
23	C₂²×C₄×C₁₇₃	—	2	2	no	—
24	C₂²×C₄×C₁₇₃	—	2	2	no	—
25	C₂²×C₄×C₁₇₃	—	2	2	no	—
26	C₂²×C₄×C₁₇₃	—	2	2	no	—
27	C₂²×C₄×C₁₇₃	—	2	2	no	—
28	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₇₃	D₄	1	4	yes	—
33	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
34	C₂²×C₄×C₁₇₃	Q₈	1	2	yes	—
35	C₄²×C₁₇₃	—	2	8	no	—
36	C₂×Dic₁₇₃	—	2	8	no	—
37	C₂×Dic₁₇₃	—	2	8	no	—
38	C₂×Dic₁₇₃	—	2	8	no	—
39	C₂×Dic₁₇₃	—	4	4	no	—
40	C₂×Dic₁₇₃	—	4	4	no	—
41	C₂×Dic₁₇₃	—	4	4	no	—
42	C₂×Dic₁₇₃	—	4	4	no	—
43	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
44	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂³×C₁₇₃	C₂×D₄	1	2	yes	—
46	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂³×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂³×C₁₇₃	#41	1	2	yes	—
49	C₂³×C₁₇₃	#41	1	2	yes	—
50	C₂³×C₁₇₃	#41	1	2	yes	—
51	C₂³×C₁₇₃	#41	1	2	yes	—
52	C₂×C₄×C₁₇₃	—	2	2	no	—
53	C₂×C₄×C₁₇₃	—	2	2	no	—
54	C₂×C₄×C₁₇₃	—	2	2	no	—
55	C₂×C₄×C₁₇₃	—	2	2	no	—
56	C₂×C₄×C₁₇₃	—	2	2	no	—
57	C₂×C₄×C₁₇₃	—	2	2	no	—
58	C₂×C₄×C₁₇₃	—	2	8	no	—
59	C₂×C₄×C₁₇₃	—	2	4	no	—
60	C₂×C₄×C₁₇₃	—	2	4	no	—
61	C₂×C₄×C₁₇₃	—	2	8	no	—
62	C₂×C₄×C₁₇₃	C₄²	1	4	yes	—
63	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	4	yes	—
64	C₂×C₄×C₁₇₃	C₄⋊C₄	1	4	yes	—
65	C₂²×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₁₇₃	#117	1	1	yes	characteristic
69	C₂²×C₁₇₃	#117	1	1	yes	characteristic
70	C₂²×C₁₇₃	#117	1	4	yes	—
71	C₂²×C₁₇₃	#117	1	1	yes	characteristic
72	C₂²×C₁₇₃	#117	1	1	yes	characteristic
73	C₂²×C₁₇₃	C₄×D₄	1	4	yes	—
74	C₂²×C₁₇₃	C₄×D₄	1	2	yes	—
75	C₂²×C₁₇₃	C₄×Q₈	1	2	yes	—
76	C₂²×C₁₇₃	#121	1	2	yes	—
77	C₂²×C₁₇₃	C₂²⋊Q₈	1	2	yes	—
78	C₂²×C₁₇₃	#123	1	4	yes	—
79	C₂²×C₁₇₃	#124	1	2	yes	—
80	C₂²×C₁₇₃	#125	1	2	yes	—
81	C₂²×C₁₇₃	#126	1	4	yes	—
82	C₄×C₁₇₃	—	2	4	no	—
83	C₄×C₁₇₃	—	2	4	no	—
84	C₄×C₁₇₃	—	2	4	no	—
85	Dic₁₇₃	—	4	8	no	—
86	C₂×C₁₇₃	#376	1	1	yes	characteristic
87	C₂×C₁₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₁₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₁₇₃	#380	1	1	yes	characteristic
90	C₂×C₁₇₃	#381	1	1	yes	characteristic
91	C₂×C₁₇₃	#382	1	1	yes	characteristic
92	C₂×C₁₇₃	#383	1	1	yes	characteristic
93	C₂×C₁₇₃	#387	1	4	yes	—
94	C₂×C₁₇₃	#388	1	4	yes	—
95	C₁₇₃	#1434	1	1	yes	characteristic, complemented, Sylow
96	#1434	—	173	1	no	maximal, Sylow
97	C₂²×C₄²	D₁₇₃	1	1	yes	characteristic
98	C₂×#95	—	173	2	no	—
99	C₂×#95	—	173	2	no	—
100	C₂×#95	—	173	2	no	—
101	C₂³×C₄	—	173	4	no	—
102	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
104	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
105	C₂×C₄²	—	2	4	no	—
106	C₂×C₄²	Dic₁₇₃	1	4	yes	complemented
107	#95	—	346	4	no	—
108	#95	—	346	4	no	—
109	#95	—	173	8	no	—
110	C₂⁴	C₂²×D₁₇₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	346	4	no	—
118	C₂²×C₄	—	346	4	no	—
119	C₂²×C₄	—	346	4	no	—
120	C₂²×C₄	—	346	4	no	—
121	C₂²×C₄	—	346	4	no	—
122	C₂²×C₄	—	346	4	no	—
123	C₂²×C₄	—	173	8	no	—
124	C₂²×C₄	C₄×D₁₇₃	1	4	yes	—
125	C₂²×C₄	C₂×Dic₁₇₃	1	2	yes	—
126	C₂²×C₄	C₂×Dic₁₇₃	1	2	yes	—
127	C₂²×C₄	C₂×Dic₁₇₃	1	2	yes	—
128	C₂²×C₄	C₂²⋊D₁₇₃	1	4	yes	—
129	C₂²×C₄	D₆₉₂	1	2	yes	—
130	C₂²×C₄	Dic₃₄₆	1	2	yes	—
131	C₄²	—	2	8	no	—
132	C₂³	C₂×C₄×D₁₇₃	1	2	yes	—
133	C₂³	C₂²×Dic₁₇₃	1	1	yes	characteristic
134	C₂³	C₂×C₂²⋊D₁₇₃	1	2	yes	—
135	C₂³	C₂×D₆₉₂	1	1	yes	characteristic
136	C₂³	C₂×Dic₃₄₆	1	1	yes	characteristic
137	C₂³	#14583826	1	2	yes	—
138	C₂³	#14583826	1	2	yes	—
139	C₂³	#14583826	1	2	yes	—
140	C₂³	#14583826	1	2	yes	—
141	C₂×C₄	—	2	2	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	2	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	346	8	no	—
152	C₂×C₄	—	346	8	no	—
153	C₂×C₄	—	346	8	no	—
154	C₂×C₄	—	692	4	no	—
155	C₂×C₄	—	692	4	no	—
156	C₂×C₄	—	692	4	no	—
157	C₂×C₄	—	692	4	no	—
158	C₂×C₄	C₄×Dic₁₇₃	1	4	yes	—
159	C₂×C₄	C₂²⋊Dic₁₇₃	1	4	yes	—
160	C₂×C₄	C₄⋊Dic₁₇₃	1	4	yes	—
161	C₂²	C₂×C₄×Dic₁₇₃	1	1	yes	characteristic
162	C₂²	C₂×C₂²⋊Dic₁₇₃	1	1	yes	characteristic
163	C₂²	C₂×C₄⋊Dic₁₇₃	1	1	yes	characteristic
164	C₂²	#19149636	1	4	yes	—
165	C₂²	#19149638	1	1	yes	characteristic
166	C₂²	#19149638	1	1	yes	characteristic
167	C₂²	#19149638	1	1	yes	characteristic
168	C₂²	#19149638	1	1	yes	characteristic
169	C₂²	C₄×C₂²⋊D₁₇₃	1	4	yes	—
170	C₂²	C₄×D₆₉₂	1	2	yes	—
171	C₂²	C₄×Dic₃₄₆	1	2	yes	—
172	C₂²	#19149653	1	2	yes	—
173	C₂²	#19149658	1	2	yes	—
174	C₂²	#19149662	1	4	yes	—
175	C₂²	#19149667	1	2	yes	—
176	C₂²	#19149671	1	2	yes	—
177	C₂²	#19149673	1	4	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	692	8	no	—
182	C₂	#26011337	1	1	yes	characteristic
183	C₂	C₄×C₂²⋊Dic₁₇₃	1	1	yes	characteristic
184	C₂	C₄×C₄⋊Dic₁₇₃	1	1	yes	characteristic
185	C₂	#26011351	1	1	yes	characteristic
186	C₂	#26011354	1	1	yes	characteristic
187	C₂	#26011357	1	1	yes	characteristic
188	C₂	#26011359	1	1	yes	characteristic
189	C₂	#26011375	1	4	yes	—
190	C₂	#26011382	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₇₃
commutator	C₂²×C₁₇₃	C₂×C₄²
Fitting	C₂²×C₄²×C₁₇₃	C₂
socle, prime radical, cosocle	C₂⁴×C₁₇₃	C₂³
hypercenter	C₂²×C₄²	D₁₇₃
hypocenter	C₁₇₃	#1434
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
8. #19149586	C₄	semidirect
13. C₂×C₄²×C₁₇₃	C₄	semidirect
95. C₁₇₃	#1434	semidirect
106. C₂×C₄²	Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149586	2
C₂²×C₄²×C₁₇₃	2
#1434	173