Subgroups — #36464391

Order 22784.

310 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022055	C₂	1	1	yes	maximal, characteristic
2	#26022051	C₂	1	1	yes	maximal, characteristic
3	#26018234	C₂	1	4	yes	maximal, complemented
4	C₂×#446×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#13083549	—	2	1	no	—
6	C₄⋊#13083549	C₂²	1	2	yes	—
7	C₂²⋊#13083549	—	2	1	no	—
8	C₂²⋊#13083549	—	2	1	no	—
9	#19151374	C₂²	1	2	yes	—
10	C₄×#13083549	—	2	1	no	—
11	C₂×C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
12	#446×C₈₉	—	2	1	no	—
13	#446×C₈₉	—	2	1	no	—
14	#446×C₈₉	C₄	1	2	yes	—
15	#446×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₈₉	C₄	1	1	yes	characteristic
18	C₂×#97×C₈₉	—	2	1	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083549	—	2	2	no	—
21	C₂×#13083549	—	2	2	no	—
22	C₂²×Q₈×C₈₉	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	—	2	1	no	—
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₈₉	—	2	1	no	—
27	C₄×Q₈×C₈₉	—	2	1	no	—
28	C₄×Q₈×C₈₉	C₈	1	2	yes	—
29	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	—	2	1	no	—
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₈₉	—	2	2	no	—
35	C₄⋊C₈×C₈₉	—	2	2	no	—
36	C₄⋊C₈×C₈₉	—	2	1	no	—
37	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
38	#97×C₈₉	—	2	2	no	—
39	#97×C₈₉	—	2	2	no	—
40	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	#13083549	—	4	2	no	—
45	#13083549	—	4	2	no	—
46	C₂×Q₈×C₈₉	—	2	1	no	—
47	C₂×Q₈×C₈₉	—	4	2	no	—
48	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₈₉	—	2	1	no	—
50	C₂²×C₄×C₈₉	—	2	1	no	—
51	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₂×C₈×C₈₉	—	4	1	no	—
57	C₂×C₈×C₈₉	—	4	2	no	—
58	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₄⋊C₄×C₈₉	—	2	1	no	—
64	C₄⋊C₄×C₈₉	—	2	2	no	—
65	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
66	C₄²×C₈₉	—	2	2	no	—
67	C₄²×C₈₉	—	2	2	no	—
68	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
71	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
73	Q₈×C₈₉	—	4	2	no	—
74	Q₈×C₈₉	—	4	2	no	—
75	C₂×C₄×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	—	2	2	no	—
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	4	2	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
89	C₈×C₈₉	—	2	1	no	—
90	C₈×C₈₉	—	2	1	no	—
91	C₈×C₈₉	—	2	1	no	—
92	C₈×C₈₉	—	4	2	no	—
93	C₈×C₈₉	#137	1	2	yes	—
94	C₂²×C₈₉	—	2	1	no	—
95	C₂²×C₈₉	—	2	1	no	—
96	C₂²×C₈₉	#350	1	2	yes	—
97	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₄×C₈₉	—	2	1	no	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	4	2	no	—
103	C₄×C₈₉	—	4	2	no	—
104	C₄×C₈₉	#349	1	2	yes	—
105	C₄×C₈₉	#419	1	1	yes	characteristic
106	C₄×C₈₉	#421	1	1	yes	characteristic
107	#11642	—	89	1	no	maximal, Sylow
108	C₂×C₈₉	—	2	2	no	—
109	C₂×C₈₉	#1479	1	1	yes	characteristic
110	C₂×C₈₉	C₂×#349	1	1	yes	characteristic
111	C₂×C₈₉	C₂×#350	1	1	yes	characteristic
112	C₂×#446	D₈₉	1	1	yes	characteristic
113	#2163	—	89	1	no	—
114	#2162	—	89	1	no	—
115	#1338	—	89	4	no	—
116	C₈₉	#11642	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄×Q₈	Dic₈₉	1	1	yes	characteristic
118	#446	—	2	1	no	—
119	#446	—	2	1	no	—
120	#446	Dic₈₉	1	2	yes	—
121	#446	C₂×D₈₉	1	2	yes	—
122	C₂×C₄⋊C₈	—	2	1	no	—
123	C₂×C₄⋊C₈	Dic₈₉	1	1	yes	characteristic
124	C₂×#97	—	2	1	no	—
125	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
126	C₄⋊C₁₆	—	89	2	no	—
127	C₄⋊C₁₆	—	178	1	no	—
128	C₂²⋊C₁₆	—	178	1	no	—
129	C₂²⋊C₁₆	—	178	1	no	—
130	#346	—	89	2	no	—
131	C₄×C₁₆	—	178	1	no	—
132	C₂²×Q₈	#10623494	1	1	yes	characteristic
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	#10623494	1	2	yes	—
139	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	#10623494	1	1	yes	characteristic
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
144	C₂×C₁₆	—	178	2	no	—
145	C₂×C₁₆	—	178	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	1	no	—
149	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
150	#97	—	2	2	no	—
151	#97	—	2	2	no	—
152	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	C₂×#10623494	1	2	yes	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
162	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
163	C₂²×C₄	#13083558	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	C₂×#10623494	1	2	yes	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
179	C₄²	C₂×#10623494	1	1	yes	characteristic
180	C₄²	#13083558	1	1	yes	characteristic
181	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
182	C₁₆	—	356	2	no	—
183	C₁₆	—	356	2	no	—
184	C₂³	C₂²⋊#10623494	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
199	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
200	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	#15668043	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#19151382	1	2	yes	—
209	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	#19151380	1	2	yes	—
217	C₄	#19151607	1	1	yes	characteristic
218	C₄	#19151615	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#26018647	1	1	yes	characteristic
221	C₂	C₂×#19151380	1	1	yes	characteristic
222	C₂	C₂×#19151382	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×#446×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₉	C₂²⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×#446	D₈₉
hypocenter	C₈₉	#11642
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #26018234	C₂	semidirect
116. C₈₉	#11642	semidirect

Maximal subgroups

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

subgroup	index
#26022055	2
#26022051	2
#26018234	2
C₂×#446×C₈₉	2
#11642	89