Subgroups — #36356041

Order 22144.

500 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011398	C₂	1	2	yes	maximal
2	#26011390	C₂	1	2	yes	maximal
3	#26011377	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₁₇₃	C₂	1	2	yes	maximal
5	C₄×C₄⋊Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
6	#26011337	C₂	1	2	yes	maximal
7	C₄²×Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₇₃	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
16	#19149586	C₂²	1	4	yes	—
17	#19149586	C₂²	1	4	yes	—
18	#19149586	C₂²	1	4	yes	—
19	#19149586	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₇₃	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
26	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
27	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
28	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
29	C₄⋊Dic₁₇₃	—	2	8	no	—
30	C₄⋊Dic₁₇₃	—	2	4	no	—
31	C₄×Dic₁₇₃	—	2	4	no	—
32	C₄×Dic₁₇₃	—	2	4	no	—
33	C₄×Dic₁₇₃	—	2	2	no	—
34	C₄×Dic₁₇₃	—	2	4	no	—
35	C₄×Dic₁₇₃	Q₈	1	4	yes	complemented
36	C₂×Q₈×C₁₇₃	—	2	4	no	—
37	C₂×Q₈×C₁₇₃	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₇₃	C₂×C₄	1	8	yes	—
44	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
45	C₄²×C₁₇₃	C₂×C₄	1	8	yes	—
46	C₄²×C₁₇₃	C₂×C₄	1	4	yes	—
47	C₂×Dic₁₇₃	—	2	2	no	—
48	C₂×Dic₁₇₃	—	2	2	no	—
49	C₂×Dic₁₇₃	—	2	2	no	—
50	C₂×Dic₁₇₃	—	2	2	no	—
51	C₂×Dic₁₇₃	—	2	4	no	—
52	C₂×Dic₁₇₃	—	2	4	no	—
53	C₂×Dic₁₇₃	—	2	4	no	—
54	C₂×Dic₁₇₃	C₂×Q₈	1	4	yes	—
55	C₂×Dic₁₇₃	#41	1	4	yes	—
56	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
57	Q₈×C₁₇₃	—	2	8	no	—
58	C₂×C₄×C₁₇₃	—	2	4	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	4	yes	—
63	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₁₇₃	#41	1	4	yes	—
68	C₂×C₄×C₁₇₃	#41	1	2	yes	—
69	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
70	C₂²×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
72	C₂²×C₁₇₃	C₂×#41	1	1	yes	characteristic
73	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
74	C₄×C₁₇₃	—	2	4	no	—
75	C₄×C₁₇₃	—	2	2	no	—
76	C₄×C₁₇₃	—	2	4	no	—
77	C₄×C₁₇₃	#117	1	4	yes	—
78	C₄×C₁₇₃	C₄×Q₈	1	4	yes	—
79	Dic₁₇₃	—	2	4	no	—
80	Dic₁₇₃	—	2	4	no	—
81	Dic₁₇₃	—	4	4	no	—
82	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
83	C₂×C₁₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
84	C₂×C₁₇₃	C₄×#41	1	1	yes	characteristic
85	C₂×C₁₇₃	#529	1	2	yes	—
86	C₂×C₁₇₃	#533	1	2	yes	—
87	C₁₇₃	#2295	1	1	yes	characteristic, complemented, Sylow
88	#2295	—	173	1	no	maximal, Sylow
89	C₂×C₄×Q₈	D₁₇₃	1	1	yes	characteristic
90	#391	—	173	2	no	—
91	#389	—	173	2	no	—
92	#387	—	173	4	no	—
93	C₄×C₄⋊C₄	—	173	1	no	—
94	C₄×C₄⋊C₄	—	173	2	no	—
95	#376	—	173	2	no	—
96	C₄³	—	173	1	no	—
97	C₂²×Q₈	C₂×D₁₇₃	1	1	yes	characteristic
98	C₄×Q₈	Dic₁₇₃	1	8	yes	complemented
99	C₂×C₄⋊C₄	—	173	4	no	—
100	C₂×C₄⋊C₄	—	173	2	no	—
101	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
103	C₂×C₄²	—	173	2	no	—
104	C₂×C₄²	—	173	2	no	—
105	C₂×C₄²	—	173	1	no	—
106	C₂×C₄²	—	173	2	no	—
107	C₂×C₄²	—	173	1	no	—
108	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₁₇₃	1	2	yes	—
110	#95	—	173	4	no	—
111	#95	—	173	4	no	—
112	#95	—	173	2	no	—
113	#95	—	173	4	no	—
114	C₂×Q₈	—	2	4	no	—
115	C₂×Q₈	C₂×Dic₁₇₃	1	4	yes	—
116	C₂²×C₄	—	173	4	no	—
117	C₂²×C₄	—	173	2	no	—
118	C₂²×C₄	—	173	2	no	—
119	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
122	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
124	C₄⋊C₄	—	346	8	no	—
125	C₄⋊C₄	—	346	4	no	—
126	C₄⋊C₄	C₂×Dic₁₇₃	1	8	yes	—
127	C₄⋊C₄	C₂×Dic₁₇₃	1	4	yes	—
128	C₄²	—	346	4	no	—
129	C₄²	—	346	4	no	—
130	C₄²	—	346	2	no	—
131	C₄²	—	346	4	no	—
132	C₄²	—	173	4	no	—
133	C₄²	C₂×Dic₁₇₃	1	8	yes	—
134	C₄²	C₂×Dic₁₇₃	1	4	yes	—
135	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
136	Q₈	—	2	8	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	346	2	no	—
141	C₂×C₄	—	346	2	no	—
142	C₂×C₄	—	346	2	no	—
143	C₂×C₄	—	346	2	no	—
144	C₂×C₄	—	346	4	no	—
145	C₂×C₄	—	346	4	no	—
146	C₂×C₄	—	346	4	no	—
147	C₂×C₄	—	173	4	no	—
148	C₂×C₄	—	173	4	no	—
149	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
150	C₂×C₄	C₂²×Dic₁₇₃	1	4	yes	—
151	C₂×C₄	C₂²×Dic₁₇₃	1	4	yes	—
152	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
153	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
154	C₂×C₄	Q₈×D₁₇₃	1	2	yes	—
155	C₂×C₄	#14583826	1	4	yes	—
156	C₂×C₄	#14583827	1	2	yes	—
157	C₂²	C₂³×Dic₁₇₃	1	1	yes	characteristic
158	C₂²	C₂×Q₈×D₁₇₃	1	1	yes	characteristic
159	C₂²	#41×D₁₇₃	1	2	yes	—
160	C₂²	C₂×#14583826	1	2	yes	—
161	C₂²	C₂×#14583827	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	346	4	no	—
166	C₄	—	346	4	no	—
167	C₄	—	692	4	no	—
168	C₄	#19149638	1	4	yes	—
169	C₄	Q₈×Dic₁₇₃	1	4	yes	—
170	C₂	C₂×#19149638	1	1	yes	characteristic
171	C₂	C₂×Q₈×Dic₁₇₃	1	1	yes	characteristic
172	C₂	#41×Dic₁₇₃	1	1	yes	characteristic
173	C₂	#26012028	1	2	yes	—
174	C₂	#26012045	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₇₃
commutator	C₂²×C₁₇₃	C₂³×C₄
Frattini	C₂³	C₂³×D₁₇₃
Fitting	C₂×C₄×Q₈×C₁₇₃	C₂
socle, prime radical, cosocle	C₂³×C₁₇₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₇₃
hypocenter	C₁₇₃	#2295
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×Q₈×C₁₇₃	C₄	semidirect
35. C₄×Dic₁₇₃	Q₈	semidirect
87. C₁₇₃	#2295	semidirect
98. C₄×Q₈	Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011398	2
#26011390	2
#26011377	2
C₄×C₄⋊Dic₁₇₃	2
#26011337	2
C₄²×Dic₁₇₃	2
C₂×C₄×Q₈×C₁₇₃	2
#2295	173