Subgroups — #26028918

Order 11392.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151659	C₂	1	4	yes	maximal, complemented
2	#19151655	C₂	1	2	yes	maximal, complemented
3	#19151651	C₂	1	2	yes	maximal, complemented
4	#19151622	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151565	C₂	1	2	yes	maximal
7	#19151554	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#13083558	C₂²	1	2	yes	—
10	C₂²×#10623494	C₂²	1	2	yes	—
11	C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₄⋊#10623494	C₂²	1	2	yes	—
14	C₄⋊#10623494	C₂²	1	2	yes	—
15	C₄⋊#10623494	C₂²	1	2	yes	—
16	C₂²⋊#10623494	C₂²	1	4	yes	—
17	C₂²⋊#10623494	C₂²	1	4	yes	—
18	#15667923	C₂²	1	4	yes	—
19	C₄×#10623494	C₂²	1	4	yes	—
20	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
23	C₄×D₄×C₈₉	C₄	1	2	yes	—
24	C₄×D₄×C₈₉	C₄	1	1	yes	characteristic
25	C₄×D₄×C₈₉	C₄	1	2	yes	—
26	C₄×D₄×C₈₉	C₄	1	1	yes	characteristic
27	#117×C₈₉	C₂²	1	2	yes	—
28	#117×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
31	#13083558	—	2	4	no	—
32	C₂×#10623494	—	2	4	no	—
33	C₂×#10623494	C₂³	1	2	yes	—
34	C₂×#10623494	C₂³	1	2	yes	—
35	C₂×#10623494	C₂³	1	4	yes	—
36	#41×C₈₉	—	2	2	no	—
37	#41×C₈₉	—	2	2	no	—
38	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
40	C₂×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
54	C₄²×C₈₉	C₂×C₄	1	2	yes	—
55	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×C₄	1	2	yes	—
57	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄²×C₈₉	C₂³	1	2	yes	—
60	C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	C₂³×C₈₉	C₂²×C₄	1	2	yes	—
62	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	Q₈×C₈₉	—	2	2	no	—
64	D₄×C₈₉	—	2	4	no	—
65	D₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	2	4	no	—
68	C₂×C₄×C₈₉	—	2	2	no	—
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	—	4	1	no	—
71	C₂×C₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#41	1	2	yes	—
81	C₂×C₄×C₈₉	#41	1	2	yes	—
82	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
83	#10623494	—	2	2	no	—
84	#10623494	—	2	2	no	—
85	#10623494	—	2	4	no	—
86	C₂²×C₈₉	—	2	1	no	—
87	C₂²×C₈₉	—	2	1	no	—
88	C₂²×C₈₉	—	2	2	no	—
89	C₂²×C₈₉	—	2	2	no	—
90	C₂²×C₈₉	—	2	2	no	—
91	C₂²×C₈₉	#117	1	2	yes	—
92	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	—	2	1	no	—
95	C₄×C₈₉	—	2	2	no	—
96	C₄×C₈₉	—	2	1	no	—
97	C₄×C₈₉	—	2	1	no	—
98	C₄×C₈₉	—	2	2	no	—
99	C₄×C₈₉	#117	1	2	yes	—
100	C₄×C₈₉	#131	1	2	yes	—
101	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
102	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
103	C₂×C₈₉	—	2	2	no	—
104	C₂×C₈₉	—	4	2	no	—
105	C₂×C₈₉	C₂×#117	1	1	yes	characteristic
106	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
107	C₂×C₈₉	#568	1	1	yes	characteristic
108	#2925	—	89	1	no	maximal, Sylow
109	C₈₉	#2925	1	1	yes	characteristic, complemented, Sylow
110	C₄×#41	D₈₉	1	1	yes	characteristic
111	#433	—	89	4	no	—
112	#432	—	89	2	no	—
113	#431	—	89	2	no	—
114	#423	—	89	1	no	—
115	C₂×C₄⋊C₈	—	89	1	no	—
116	#408	—	89	2	no	—
117	#405	—	89	2	no	—
118	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
119	C₂×M₁₆	—	89	2	no	—
120	C₂²×C₈	—	89	2	no	—
121	C₄×Q₈	Dic₈₉	1	1	yes	characteristic
122	C₄×Q₈	Dic₈₉	1	1	yes	characteristic
123	C₄×D₄	Dic₈₉	1	2	yes	—
124	C₄×D₄	Dic₈₉	1	1	yes	characteristic
125	C₄×D₄	Dic₈₉	1	2	yes	—
126	C₄×D₄	Dic₈₉	1	1	yes	characteristic
127	#117	C₂×D₈₉	1	1	yes	characteristic
128	#117	C₂×D₈₉	1	2	yes	—
129	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
130	C₂×C₄²	C₂×D₈₉	1	2	yes	—
131	C₄⋊C₈	—	89	1	no	—
132	C₄⋊C₈	—	89	1	no	—
133	C₄⋊C₈	—	89	2	no	—
134	C₄⋊C₈	—	89	2	no	—
135	C₄⋊C₈	—	89	2	no	—
136	C₂²⋊C₈	—	89	4	no	—
137	C₂²⋊C₈	—	89	4	no	—
138	#97	—	89	4	no	—
139	C₄×C₈	—	89	4	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
143	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
144	C₂×D₄	C₂×Dic₈₉	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
146	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
148	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
150	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
151	M₁₆	—	178	4	no	—
152	C₂×C₈	—	89	2	no	—
153	C₂×C₈	—	89	4	no	—
154	C₂×C₈	—	89	2	no	—
155	C₂×C₈	—	178	4	no	—
156	C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
159	C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
160	C₂²⋊C₄	C₂×Dic₈₉	1	2	yes	—
161	C₂²⋊C₄	C₂×Dic₈₉	1	2	yes	—
162	C₂²⋊C₄	C₂×Dic₈₉	1	2	yes	—
163	C₄²	C₂×Dic₈₉	1	2	yes	—
164	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
165	C₄²	C₂×Dic₈₉	1	2	yes	—
166	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
167	C₄²	C₂²×D₈₉	1	2	yes	—
168	C₄²	C₂²×D₈₉	1	1	yes	characteristic
169	C₄²	C₂²×D₈₉	1	1	yes	characteristic
170	C₂³	C₂²×Dic₈₉	1	2	yes	—
171	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
183	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
184	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
186	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
189	C₂×C₄	#13083574	1	2	yes	—
190	C₂×C₄	#13083574	1	2	yes	—
191	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
192	C₈	—	178	2	no	—
193	C₈	—	178	2	no	—
194	C₈	—	178	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	#15667971	1	2	yes	—
201	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	#15667971	1	2	yes	—
209	C₄	#15668019	1	2	yes	—
210	C₄	C₂×#13083574	1	1	yes	characteristic
211	C₄	C₂×#13083574	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×#15667971	1	1	yes	characteristic
215	C₂	C₂×#15668019	1	1	yes	characteristic
216	C₂	#19152365	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₄×#41×C₈₉	C₂
prime radical	C₂×#41×C₈₉	C₂²
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₄×#41	D₈₉
hypocenter	C₈₉	#2925
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #19151659	C₂	semidirect
2. #19151655	C₂	semidirect
3. #19151651	C₂	semidirect
4. #19151622	C₂	semidirect
5. C₂×C₄⋊#10623494	C₂	semidirect
7. #19151554	C₂	semidirect
109. C₈₉	#2925	semidirect

Maximal subgroups

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

subgroup	index
#19151659	2
#19151655	2
#19151651	2
#19151622	2
C₂×C₄⋊#10623494	2
#19151565	2
#19151554	2
C₄×#41×C₈₉	2
#2925	89