Subgroups — #36448091

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020369	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10623494	C₂	1	1	yes	maximal, characteristic
3	#26018086	C₂	1	4	yes	maximal
4	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
6	C₂×C₄×#10623494	C₄	1	2	yes	—
7	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
8	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	2	yes	—
10	#19151353	—	2	2	no	—
11	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
12	C₄³×C₈₉	C₂²	1	1	yes	characteristic
13	#336×C₈₉	—	2	2	no	—
14	C₂²×#10623494	—	2	2	no	—
15	C₂²×#10623494	C₂×C₄	1	2	yes	—
16	C₂²×#10623494	C₂×C₄	1	2	yes	—
17	C₄⋊#10623494	—	2	4	no	—
18	C₄×#10623494	—	2	4	no	—
19	C₄×#10623494	—	2	2	no	—
20	C₄×#10623494	—	2	2	no	—
21	C₄×#10623494	—	2	2	no	—
22	C₄×#10623494	C₈	1	4	yes	complemented
23	C₂²×C₈×C₈₉	—	2	2	no	—
24	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	2	yes	—
27	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
28	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	—	2	4	no	—
30	C₂×#10623494	—	2	2	no	—
31	C₂×#10623494	—	2	2	no	—
32	C₂×#10623494	—	2	2	no	—
33	C₂×#10623494	—	2	2	no	—
34	C₂×#10623494	—	4	2	no	—
35	C₂×#10623494	—	4	4	no	—
36	C₂×#10623494	C₂×C₈	1	4	yes	—
37	C₂×#10623494	M₁₆	1	4	yes	—
38	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₈₉	—	4	2	no	—
44	C₂×C₈×C₈₉	—	4	4	no	—
45	C₄²×C₈₉	—	2	4	no	—
46	C₄²×C₈₉	—	2	2	no	—
47	C₄²×C₈₉	—	2	4	no	—
48	C₄²×C₈₉	C₂×C₈	1	2	yes	—
49	C₄²×C₈₉	M₁₆	1	2	yes	—
50	C₄²×C₈₉	#41	1	2	yes	—
51	C₄²×C₈₉	#41	1	2	yes	—
52	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	—	2	2	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	2	2	no	—
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₈₉	#108	1	4	yes	—
60	C₂×C₄×C₈₉	#117	1	2	yes	—
61	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#125	1	2	yes	—
64	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#135	1	4	yes	—
68	C₈×C₈₉	—	8	2	no	—
69	#10623494	—	2	4	no	—
70	#10623494	—	2	4	no	—
71	#10623494	—	8	2	no	—
72	C₂²×C₈₉	#382	1	1	yes	characteristic
73	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₂²×C₈₉	#423	1	1	yes	characteristic
75	C₂²×C₈₉	#427	1	2	yes	—
76	C₂²×C₈₉	C₂×#108	1	2	yes	—
77	C₄×C₈₉	—	2	4	no	—
78	C₄×C₈₉	—	2	4	no	—
79	C₄×C₈₉	—	2	2	no	—
80	C₄×C₈₉	#431	1	2	yes	—
81	C₄×C₈₉	#432	1	2	yes	—
82	C₄×C₈₉	#488	1	2	yes	—
83	C₄×C₈₉	#488	1	2	yes	—
84	#8785	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	#1564	1	2	yes	—
86	C₂×C₈₉	#1568	1	2	yes	—
87	C₂×C₈₉	#1831	1	1	yes	characteristic
88	C₂×C₈₉	#1835	1	1	yes	characteristic
89	C₂×C₈₉	#1837	1	1	yes	characteristic
90	#1831	—	89	1	no	—
91	#1831	D₈₉	1	1	yes	characteristic
92	C₄²×C₈	—	89	1	no	—
93	#1277	—	89	4	no	—
94	C₈₉	#8785	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄⋊C₈	—	89	2	no	—
96	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
97	C₂×C₄×C₈	—	89	1	no	—
98	C₂×C₄×C₈	—	89	2	no	—
99	C₂×C₄×C₈	—	89	1	no	—
100	C₂×C₄×C₈	—	89	2	no	—
101	C₄³	C₂×D₈₉	1	1	yes	characteristic
102	#336	—	2	2	no	—
103	#336	—	178	2	no	—
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	89	2	no	—
106	C₂²×C₈	—	89	2	no	—
107	C₂²×C₈	—	178	2	no	—
108	C₂×C₄²	Dic₁₇₈	1	2	yes	—
109	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
111	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
112	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	178	4	no	—
115	C₄×C₈	—	89	4	no	—
116	C₄×C₈	—	178	4	no	—
117	C₄×C₈	—	178	2	no	—
118	C₄×C₈	—	178	2	no	—
119	C₄×C₈	—	178	2	no	—
120	C₂²×C₄	#13083553	1	2	yes	—
121	C₂²×C₄	#13083553	1	2	yes	—
122	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	4	4	no	—
127	C₂×C₈	—	89	4	no	—
128	C₂×C₈	—	89	4	no	—
129	C₂×C₈	—	356	2	no	—
130	C₂×C₈	—	356	4	no	—
131	C₂×C₈	—	178	2	no	—
132	C₂×C₈	—	178	2	no	—
133	C₂×C₈	—	178	2	no	—
134	C₂×C₈	—	178	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	C₈×D₈₉	1	2	yes	—
139	C₄²	#13083557	1	2	yes	—
140	C₄²	#13083573	1	2	yes	—
141	C₄²	#13083575	1	2	yes	—
142	C₂³	C₂×#13083553	1	1	yes	characteristic
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	2	no	—
148	C₂×C₄	#15667941	1	4	yes	—
149	C₂×C₄	#15667947	1	4	yes	—
150	C₂×C₄	#15667970	1	2	yes	—
151	C₂×C₄	#15667999	1	1	yes	characteristic
152	C₂×C₄	#15668001	1	1	yes	characteristic
153	C₂×C₄	#15668003	1	2	yes	—
154	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
155	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
156	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
157	C₂×C₄	#15668032	1	4	yes	—
158	C₈	—	8	2	no	—
159	C₈	—	178	4	no	—
160	C₈	—	178	4	no	—
161	C₈	—	712	2	no	—
162	C₂²	#19151463	1	1	yes	characteristic
163	C₂²	C₂×#15667941	1	1	yes	characteristic
164	C₂²	#19151621	1	1	yes	characteristic
165	C₂²	#19151634	1	2	yes	—
166	C₂²	C₂×#15667947	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	4	no	—
170	C₄	#19151650	1	2	yes	—
171	C₄	#19151654	1	2	yes	—
172	C₄	#19151952	1	2	yes	—
173	C₄	#19151955	1	2	yes	—
174	C₂	#26018998	1	2	yes	—
175	C₂	#26019013	1	2	yes	—
176	C₂	#26020368	1	1	yes	characteristic
177	C₂	#26020382	1	1	yes	characteristic
178	C₂	#26020389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#8785
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
22. C₄×#10623494	C₈	semidirect
94. C₈₉	#8785	semidirect

Maximal subgroups

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

subgroup	index
#26020369	2
C₄²×#10623494	2
#26018086	2
C₈₉×#1831	2
#8785	89