Subgroups — #36448074

Order 22784.

310 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019000	C₂	1	2	yes	maximal
2	#26018097	C₂	1	2	yes	maximal
3	#26018081	C₂	1	2	yes	maximal
4	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	4	yes	—
6	#19151351	—	2	2	no	—
7	#19151349	—	2	2	no	—
8	C₂×#97×C₈₉	C₄	1	1	yes	characteristic
9	C₂×#97×C₈₉	C₄	1	1	yes	characteristic
10	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#376×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊#10623494	—	2	4	no	—
16	C₂²⋊#10623494	—	2	4	no	—
17	C₄×#10623494	—	2	4	no	—
18	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
25	#97×C₈₉	—	2	2	no	—
26	#97×C₈₉	—	2	2	no	—
27	#97×C₈₉	—	2	2	no	—
28	C₄×C₈×C₈₉	—	2	2	no	—
29	C₄×C₈×C₈₉	C₈	1	4	yes	complemented
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
34	#95×C₈₉	—	2	2	no	—
35	C₂×#10623494	—	4	4	no	—
36	C₂²×C₄×C₈₉	—	2	1	no	—
37	C₂²×C₄×C₈₉	—	2	1	no	—
38	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	#41	1	2	yes	—
40	C₂×C₈×C₈₉	—	2	1	no	—
41	C₂×C₈×C₈₉	—	2	1	no	—
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	—	2	1	no	—
45	C₂×C₈×C₈₉	—	2	1	no	—
46	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
53	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
54	C₄²×C₈₉	—	2	1	no	—
55	C₄²×C₈₉	—	2	1	no	—
56	C₄²×C₈₉	C₂×C₈	1	2	yes	—
57	C₄²×C₈₉	M₁₆	1	2	yes	—
58	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₈₉	#41	1	2	yes	—
61	C₂³×C₈₉	#117	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	4	2	no	—
65	C₂×C₄×C₈₉	—	4	2	no	—
66	C₂×C₄×C₈₉	—	4	2	no	—
67	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#124	1	2	yes	—
70	C₂×C₄×C₈₉	#125	1	2	yes	—
71	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
75	C₈×C₈₉	—	2	1	no	—
76	C₈×C₈₉	—	2	1	no	—
77	C₈×C₈₉	—	2	2	no	—
78	C₈×C₈₉	—	2	2	no	—
79	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
80	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
81	#10623494	—	8	4	no	—
82	C₂²×C₈₉	—	2	2	no	—
83	C₂²×C₈₉	#382	1	1	yes	characteristic
84	C₂²×C₈₉	#431	1	1	yes	characteristic
85	C₂²×C₈₉	#432	1	1	yes	characteristic
86	C₄×C₈₉	—	2	2	no	—
87	C₄×C₈₉	—	2	2	no	—
88	C₄×C₈₉	—	4	2	no	—
89	C₄×C₈₉	—	4	2	no	—
90	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₈₉	#423	1	1	yes	characteristic
92	C₄×C₈₉	#431	1	1	yes	characteristic
93	C₄×C₈₉	#432	1	1	yes	characteristic
94	#8780	—	89	1	no	maximal, Sylow
95	C₂×C₈₉	—	2	1	no	—
96	C₂×C₈₉	—	2	1	no	—
97	C₂×C₈₉	#1831	1	1	yes	characteristic
98	C₂×C₈₉	#1833	1	2	yes	—
99	#1726	D₈₉	1	1	yes	characteristic
100	#1564	—	89	2	no	—
101	#1281	—	89	2	no	—
102	#1274	—	89	2	no	—
103	C₈₉	#8780	1	1	yes	characteristic, complemented, Sylow
104	#431	—	89	4	no	—
105	C₂×#97	Dic₈₉	1	1	yes	characteristic
106	C₂×#97	Dic₈₉	1	1	yes	characteristic
107	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
108	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
109	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
111	#376	C₂×D₈₉	1	1	yes	characteristic
112	#335	—	178	2	no	—
113	#334	—	178	2	no	—
114	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
115	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
119	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
121	C₄⋊C₈	—	178	4	no	—
122	C₂²⋊C₈	—	178	4	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	178	4	no	—
128	C₄×C₈	#10623494	1	4	yes	complemented
129	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
130	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
131	C₄×C₈	D₃₅₆	1	1	yes	characteristic
132	C₄×C₈	D₃₅₆	1	1	yes	characteristic
133	#95	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
137	C₂²×C₄	#13083574	1	2	yes	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	356	4	no	—
145	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
146	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
147	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
148	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
149	C₂×C₈	C₂×#10623494	1	2	yes	—
150	C₂×C₈	C₂×#10623494	1	2	yes	—
151	C₂×C₈	#13083558	1	2	yes	—
152	C₂×C₈	#13083558	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₂×#10623494	1	2	yes	—
156	C₄²	#13083558	1	2	yes	—
157	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
158	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
159	C₄²	#13083574	1	2	yes	—
160	C₂³	#15667971	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
167	C₂×C₄	#15667971	1	1	yes	characteristic
168	C₂×C₄	#15668000	1	2	yes	—
169	C₂×C₄	#15668004	1	2	yes	—
170	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
171	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
172	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
173	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	712	4	no	—
179	C₈	C₄⋊#10623494	1	2	yes	—
180	C₈	C₄⋊#10623494	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	#19151464	1	1	yes	characteristic
183	C₂²	#19151651	1	1	yes	characteristic
184	C₂²	#19151655	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	C₂×C₄⋊#10623494	1	1	yes	characteristic
190	C₄	#19151622	1	1	yes	characteristic
191	C₄	#19151651	1	1	yes	characteristic
192	C₄	#19151655	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#26020369	1	1	yes	characteristic
196	C₂	#26020376	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667971
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1726	C₂
socle	C₂²×C₈₉	#382
prime radical	C₂³×C₈₉	#117
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#8780
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
29. C₄×C₈×C₈₉	C₈	semidirect
103. C₈₉	#8780	semidirect
128. C₄×C₈	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26019000	2
#26018097	2
#26018081	2
C₈₉×#1726	2
#8780	89