Subgroups — #36453045

Order 22784.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019004	C₂	1	2	yes	maximal
2	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
3	#26018535	C₂	1	1	yes	maximal, characteristic
4	#26018103	C₂	1	2	yes	maximal
5	C₄×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	1	yes	characteristic
7	#19151651	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	#19151351	—	2	2	no	—
10	#19151351	—	2	1	no	—
11	#19151351	—	2	1	no	—
12	#19151349	—	2	2	no	—
13	#19151319	—	2	1	no	—
14	C₈×#10623494	—	2	1	no	—
15	C₂×#108×C₈₉	C₄	1	1	yes	characteristic
16	C₂×#108×C₈₉	C₄	1	1	yes	characteristic
17	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#341×C₈₉	C₄	1	2	yes	—
20	C₂²×#10623494	—	2	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	1	no	—
24	C₄⋊#10623494	D₄	1	1	yes	characteristic
25	C₄⋊#10623494	D₄	1	1	yes	characteristic
26	C₂²⋊#10623494	—	2	1	no	—
27	C₂²⋊#10623494	Q₈	1	2	yes	—
28	C₄×#10623494	—	2	1	no	—
29	C₄×#10623494	—	2	1	no	—
30	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
35	#108×C₈₉	—	2	2	no	—
36	#108×C₈₉	C₈	1	4	yes	complemented
37	#97×C₈₉	—	2	2	no	—
38	C₄×C₈×C₈₉	—	2	2	no	—
39	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
43	C₂×#10623494	—	2	1	no	—
44	C₂×#10623494	—	2	1	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	8	2	no	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
51	M₁₆×C₈₉	—	2	2	no	—
52	M₁₆×C₈₉	—	2	2	no	—
53	M₁₆×C₈₉	—	2	2	no	—
54	M₁₆×C₈₉	C₂×C₈	1	4	yes	—
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₂×C₈×C₈₉	—	2	1	no	—
57	C₂×C₈×C₈₉	—	2	1	no	—
58	C₂×C₈×C₈₉	—	2	2	no	—
59	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
65	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₈₉	#41	1	1	yes	characteristic
69	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	4	1	no	—
73	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
83	C₈×C₈₉	—	2	1	no	—
84	C₈×C₈₉	—	2	1	no	—
85	C₈×C₈₉	—	2	4	no	—
86	C₈×C₈₉	—	4	2	no	—
87	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
88	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
89	#10623494	—	4	2	no	—
90	#10623494	—	4	1	no	—
91	#10623494	—	4	1	no	—
92	#10623494	—	8	2	no	—
93	C₂²×C₈₉	—	2	1	no	—
94	C₂²×C₈₉	—	2	1	no	—
95	C₂²×C₈₉	#389	1	1	yes	characteristic
96	C₂²×C₈₉	C₈×Q₈	1	1	yes	characteristic
97	C₂²×C₈₉	#446	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	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₈₉	#424	1	1	yes	characteristic
103	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
104	C₄×C₈₉	#436	1	1	yes	characteristic
105	C₄×C₈₉	#472	1	1	yes	characteristic
106	C₄×C₈₉	#473	1	1	yes	characteristic
107	#9608	—	89	1	no	maximal, Sylow
108	C₂×C₈₉	—	2	1	no	—
109	C₂×C₈₉	—	2	1	no	—
110	C₂×C₈₉	#1939	1	1	yes	characteristic
111	C₂×C₈₉	#1952	1	1	yes	characteristic
112	C₂×C₈₉	#1953	1	1	yes	characteristic
113	C₄×#108	D₈₉	1	1	yes	characteristic
114	#1565	—	89	2	no	—
115	C₂×#335	—	89	1	no	—
116	#1450	—	89	1	no	—
117	#1283	—	89	2	no	—
118	C₈₉	#9608	1	1	yes	characteristic, complemented, Sylow
119	#431	—	89	1	no	—
120	#431	—	89	1	no	—
121	C₂×#108	Dic₈₉	1	1	yes	characteristic
122	C₂×#108	Dic₈₉	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	89	2	no	—
124	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
126	#341	Dic₈₉	1	2	yes	—
127	#335	—	178	2	no	—
128	#335	—	178	1	no	—
129	#335	—	178	1	no	—
130	#334	—	178	2	no	—
131	#322	—	178	1	no	—
132	C₈²	—	178	1	no	—
133	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
134	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
135	C₂²×C₈	—	178	2	no	—
136	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
139	#108	—	2	2	no	—
140	#108	#10623494	1	4	yes	complemented
141	C₄⋊C₈	—	89	1	no	—
142	C₄⋊C₈	—	89	1	no	—
143	C₄⋊C₈	—	178	2	no	—
144	C₄⋊C₈	—	178	2	no	—
145	C₄⋊C₈	—	178	1	no	—
146	C₂²⋊C₈	—	89	2	no	—
147	C₂²⋊C₈	—	178	1	no	—
148	#97	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	178	1	no	—
151	C₄×C₈	—	178	1	no	—
152	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
153	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
154	C₄×C₈	D₃₅₆	1	1	yes	characteristic
155	C₄×C₈	D₃₅₆	1	1	yes	characteristic
156	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
157	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
158	C₂²×C₄	#13083575	1	1	yes	characteristic
159	M₁₆	—	2	2	no	—
160	M₁₆	—	2	2	no	—
161	M₁₆	—	2	2	no	—
162	M₁₆	C₂×#10623494	1	4	yes	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	356	2	no	—
168	C₂×C₈	—	178	1	no	—
169	C₂×C₈	—	178	1	no	—
170	C₂×C₈	—	178	2	no	—
171	C₂×C₈	—	712	2	no	—
172	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
173	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
174	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
175	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
176	C₂×C₈	C₂×#10623494	1	2	yes	—
177	C₂×C₈	#13083558	1	2	yes	—
178	C₄²	D₄×D₈₉	1	1	yes	characteristic
179	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
181	C₄²	#13083573	1	1	yes	characteristic
182	C₂³	Q₈×Dic₈₉	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
188	C₂×C₄	#15667987	1	1	yes	characteristic
189	C₂×C₄	#15667992	1	1	yes	characteristic
190	C₂×C₄	#15668003	1	1	yes	characteristic
191	C₂×C₄	#15668011	1	1	yes	characteristic
192	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
193	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
194	C₂×C₄	#15668019	1	1	yes	characteristic
195	C₂×C₄	#15668019	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	356	2	no	—
201	C₈	—	356	1	no	—
202	C₈	—	356	1	no	—
203	C₈	—	712	2	no	—
204	C₈	C₄⋊#10623494	1	2	yes	—
205	C₈	C₄⋊#10623494	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#19151497	1	1	yes	characteristic
209	C₂²	Q₈×#10623494	1	1	yes	characteristic
210	C₂²	#19151729	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×C₄⋊#10623494	1	1	yes	characteristic
215	C₄	#19151625	1	1	yes	characteristic
216	C₄	D₄×#10623494	1	1	yes	characteristic
217	C₄	#19151676	1	1	yes	characteristic
218	C₄	#19151869	1	1	yes	characteristic
219	C₄	#19151874	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#26020971	1	1	yes	characteristic
223	C₂	#26021038	1	1	yes	characteristic
224	C₂	#26021043	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667987
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#108×C₈₉	C₂
socle	C₂²×C₈₉	#389
prime radical	C₂³×C₈₉	C₄×Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#108	D₈₉
hypocenter	C₈₉	#9608
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
36. #108×C₈₉	C₈	semidirect
118. C₈₉	#9608	semidirect
140. #108	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26019004	2
C₂×#19151351	2
#26018535	2
#26018103	2
C₄×#108×C₈₉	2
#9608	89