Subgroups — #36441371

Order 22784.

240 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151419	C₂	1	1	yes	maximal, characteristic
2	#26018535	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	—	2	1	no	—
5	#19151651	—	2	1	no	—
6	#19151419	—	2	1	no	—
7	#19151419	C₄	1	2	yes	complemented
8	#19151319	—	2	1	no	—
9	C₈×#10623494	C₄	1	2	yes	complemented
10	C₂×#108×C₈₉	C₄	1	1	yes	characteristic
11	#427×C₈₉	C₄	1	1	yes	characteristic
12	#405×C₈₉	—	2	1	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#339×C₈₉	—	2	1	no	—
15	#15667952	—	2	2	no	—
16	#15667952	—	4	1	no	—
17	C₂×#13083549	—	4	1	no	—
18	C₄⋊#10623494	—	2	2	no	—
19	C₂²⋊#10623494	—	2	2	no	—
20	C₄×#10623494	—	2	2	no	—
21	C₂×M₁₆×C₈₉	—	2	1	no	—
22	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	#117×C₈₉	—	2	1	no	—
25	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
26	#108×C₈₉	—	2	1	no	—
27	#108×C₈₉	C₈	1	2	yes	complemented
28	#107×C₈₉	—	2	1	no	—
29	#106×C₈₉	C₈	1	2	yes	complemented
30	#97×C₈₉	—	4	1	no	—
31	C₄×C₈×C₈₉	—	4	1	no	—
32	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
35	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
36	C₂×#10623494	—	2	4	no	—
37	#13083549	—	4	2	no	—
38	C₂²×C₄×C₈₉	—	2	1	no	—
39	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
40	M₁₆×C₈₉	—	2	2	no	—
41	M₁₆×C₈₉	—	4	1	no	—
42	C₂×C₈×C₈₉	—	4	1	no	—
43	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
52	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
55	C₄⋊C₄×C₈₉	—	2	2	no	—
56	C₂²⋊C₄×C₈₉	—	4	1	no	—
57	C₄²×C₈₉	—	2	1	no	—
58	C₄²×C₈₉	—	4	1	no	—
59	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₈₉	#104	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	2	1	no	—
63	C₂×C₄×C₈₉	—	2	1	no	—
64	C₂×C₄×C₈₉	—	4	2	no	—
65	C₂×C₄×C₈₉	—	4	2	no	—
66	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
70	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#104	1	1	yes	characteristic
72	C₈×C₈₉	—	4	2	no	—
73	C₈×C₈₉	C₄×C₈	1	1	yes	characteristic
74	C₈×C₈₉	#97	1	1	yes	characteristic
75	C₈×C₈₉	#102	1	1	yes	characteristic
76	C₈×C₈₉	#103	1	1	yes	characteristic
77	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₈₉	#106	1	1	yes	characteristic
80	C₈×C₈₉	#107	1	1	yes	characteristic
81	#10623494	—	4	4	no	—
82	C₂²×C₈₉	—	4	1	no	—
83	C₂²×C₈₉	D₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₈₉	Q₈⋊C₈	1	1	yes	characteristic
85	C₂²×C₈₉	#339	1	1	yes	characteristic
86	C₄×C₈₉	—	2	2	no	—
87	C₄×C₈₉	—	4	1	no	—
88	C₄×C₈₉	—	8	2	no	—
89	C₄×C₈₉	D₄⋊C₈	1	1	yes	characteristic
90	C₄×C₈₉	Q₈⋊C₈	1	1	yes	characteristic
91	C₄×C₈₉	#336	1	1	yes	characteristic
92	C₄×C₈₉	#340	1	1	yes	characteristic
93	#7180	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	—	2	1	no	—
95	C₂×C₈₉	—	2	1	no	—
96	C₂×C₈₉	—	2	1	no	—
97	C₂×C₈₉	#1279	1	1	yes	characteristic
98	C₂×#364	—	89	1	no	—
99	#1772	D₈₉	1	1	yes	characteristic
100	#1450	—	89	1	no	—
101	C₈₉	#7180	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	—	178	1	no	—
103	#431	—	178	1	no	—
104	C₂×#108	Dic₈₉	1	1	yes	characteristic
105	#427	Dic₈₉	1	1	yes	characteristic
106	#405	—	2	1	no	—
107	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
108	#364	—	89	2	no	—
109	#364	—	178	1	no	—
110	#339	—	2	1	no	—
111	#322	—	178	1	no	—
112	C₈²	—	89	2	no	—
113	C₂×M₁₆	—	2	1	no	—
114	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
116	#117	—	2	1	no	—
117	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
118	M₃₂	—	356	1	no	—
119	M₃₂	—	178	2	no	—
120	C₂×C₁₆	—	356	1	no	—
121	#108	—	2	1	no	—
122	#108	#10623494	1	2	yes	complemented
123	#107	—	2	1	no	—
124	#106	#10623494	1	2	yes	complemented
125	C₄⋊C₈	—	178	2	no	—
126	C₂²⋊C₈	—	178	2	no	—
127	#97	—	4	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	—	178	2	no	—
130	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
131	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
132	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
133	C₄×C₈	D₃₅₆	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	178	4	no	—
140	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
141	C₂×C₈	#13083551	1	1	yes	characteristic
142	C₂×C₈	#13083553	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
144	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
145	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
146	C₂×C₈	#13083558	1	1	yes	characteristic
147	C₂×C₈	#13083558	1	1	yes	characteristic
148	C₂×C₈	#13083559	1	1	yes	characteristic
149	C₂×C₈	#13083561	1	1	yes	characteristic
150	C₂×C₈	#13083562	1	1	yes	characteristic
151	C₂×C₈	#13083564	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	#13083551	1	1	yes	characteristic
157	C₄²	#13083553	1	1	yes	characteristic
158	C₁₆	—	356	2	no	—
159	C₂³	#15667939	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	#15667919	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
166	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
167	C₂×C₄	#15667933	1	1	yes	characteristic
168	C₂×C₄	#15667936	1	1	yes	characteristic
169	C₂×C₄	#15667939	1	1	yes	characteristic
170	C₈	—	4	2	no	—
171	C₈	—	356	4	no	—
172	C₈	C₄×#10623494	1	1	yes	characteristic
173	C₈	#15667923	1	1	yes	characteristic
174	C₈	#15667932	1	1	yes	characteristic
175	C₈	#15667935	1	1	yes	characteristic
176	C₈	C₄⋊#10623494	1	1	yes	characteristic
177	C₈	C₄⋊#10623494	1	1	yes	characteristic
178	C₈	#15667943	1	1	yes	characteristic
179	C₈	#15667945	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	D₄⋊#10623494	1	1	yes	characteristic
182	C₂²	Q₈⋊#10623494	1	1	yes	characteristic
183	C₂²	#19151357	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	D₄⋊#10623494	1	1	yes	characteristic
188	C₄	Q₈⋊#10623494	1	1	yes	characteristic
189	C₄	#19151353	1	1	yes	characteristic
190	C₄	#19151360	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#26018093	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667943
commutator	C₈×C₈₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₈₉×#1772	C₂
socle	C₂×C₈₉	#1279
prime radical	C₂³×C₈₉	#104
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	#1772	D₈₉
hypocenter	C₈₉	#7180
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
7. #19151419	C₄	semidirect
9. C₈×#10623494	C₄	semidirect
27. #108×C₈₉	C₈	semidirect
29. #106×C₈₉	C₈	semidirect
101. C₈₉	#7180	semidirect
122. #108	#10623494	semidirect
124. #106	#10623494	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151419	2
#26018535	2
C₈₉×#1772	2
#7180	89