Subgroups — #36472683

Order 22784.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151978	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151413	C₂	1	4	yes	maximal
3	C₄×#15667952	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₆	—	2	1	no	—
6	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
7	C₂×#15668012	—	2	1	no	—
8	C₂×#15667952	C₂²	1	2	yes	—
9	#19151978	—	2	1	no	—
10	#19151978	—	2	1	no	—
11	#19151978	C₄	1	2	yes	—
12	#19151978	C₄	1	2	yes	—
13	#19151413	—	2	8	no	—
14	#19151374	—	2	1	no	—
15	C₄×#13083549	—	2	1	no	—
16	C₂×#108×C₈₉	—	2	1	no	—
17	C₂×#108×C₈₉	C₂²	1	2	yes	—
18	#423×C₈₉	—	2	1	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₃₅₆	—	2	2	no	—
22	C₂×Dic₃₅₆	—	2	2	no	—
23	C₂×Dic₃₅₆	—	2	4	no	—
24	C₂×Dic₃₅₆	—	2	2	no	—
25	C₂×Dic₃₅₆	—	2	4	no	—
26	C₂×Dic₃₅₆	—	4	2	no	—
27	C₂×Dic₃₅₆	—	4	2	no	—
28	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
29	#15668012	—	2	2	no	—
30	#15668012	—	2	2	no	—
31	#15668012	—	4	2	no	—
32	C₂×C₄⋊Dic₈₉	—	4	2	no	—
33	#15667952	—	2	2	no	—
34	#15667952	D₄	1	4	yes	—
35	C₂×#13083549	—	2	2	no	—
36	C₂×M₁₆×C₈₉	—	2	2	no	—
37	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	#108×C₈₉	—	2	4	no	—
41	#108×C₈₉	—	4	2	no	—
42	C₄⋊C₈×C₈₉	—	4	2	no	—
43	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂×Dic₁₇₈	—	2	4	no	—
48	C₂×Dic₁₇₈	—	4	2	no	—
49	C₂×Dic₁₇₈	—	4	2	no	—
50	C₂×Dic₁₇₈	—	4	2	no	—
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×Dic₁₇₈	—	4	2	no	—
53	C₂²×Dic₈₉	—	4	2	no	—
54	Dic₃₅₆	—	2	8	no	—
55	Dic₃₅₆	—	4	4	no	—
56	Dic₃₅₆	—	4	4	no	—
57	Dic₃₅₆	—	4	4	no	—
58	C₄⋊Dic₈₉	—	4	4	no	—
59	C₄⋊Dic₈₉	—	4	4	no	—
60	#13083549	—	2	4	no	—
61	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	M₁₆×C₈₉	—	4	2	no	—
64	M₁₆×C₈₉	—	4	2	no	—
65	C₂×C₈×C₈₉	—	4	2	no	—
66	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	D₈	1	2	yes	—
69	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	#41	1	2	yes	—
74	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₈₉	D₈	1	2	yes	—
77	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₉	—	2	4	no	—
79	C₂×C₄×C₈₉	#102	1	2	yes	—
80	C₂×C₄×C₈₉	#102	1	2	yes	—
81	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×Dic₈₉	—	4	4	no	—
88	C₂×Dic₈₉	—	8	2	no	—
89	C₂×Dic₈₉	—	8	2	no	—
90	Dic₁₇₈	—	4	4	no	—
91	Dic₁₇₈	—	4	4	no	—
92	Dic₁₇₈	—	8	2	no	—
93	Dic₁₇₈	—	8	2	no	—
94	C₈×C₈₉	—	8	2	no	—
95	C₈×C₈₉	#102	1	2	yes	—
96	C₈×C₈₉	#102	1	2	yes	—
97	C₈×C₈₉	C₄×D₄	1	2	yes	—
98	C₈×C₈₉	#121	1	2	yes	—
99	C₂²×C₈₉	—	2	2	no	—
100	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
101	C₂²×C₈₉	#486	1	1	yes	characteristic
102	C₂²×C₈₉	#492	1	1	yes	characteristic
103	C₄×C₈₉	—	2	2	no	—
104	C₄×C₈₉	#362	1	4	yes	complemented
105	C₄×C₈₉	#390	1	1	yes	characteristic
106	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
107	C₄×C₈₉	#486	1	1	yes	characteristic
108	C₄×C₈₉	#493	1	1	yes	characteristic
109	Dic₈₉	—	8	4	no	—
110	#13212	—	89	1	no	maximal, Sylow
111	C₂×C₈₉	—	2	1	no	—
112	C₂×C₈₉	—	2	1	no	—
113	C₂×C₈₉	#1957	1	1	yes	characteristic
114	C₂×C₈₉	C₂×#362	1	2	yes	—
115	C₂×#494	—	89	1	no	—
116	C₂×#362	—	89	4	no	—
117	C₄×M₃₂	—	89	1	no	—
118	#1848	D₈₉	1	1	yes	characteristic
119	C₈₉	#13212	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₁₆	—	89	2	no	—
121	C₂²×Q₁₆	—	178	1	no	—
122	C₂×C₄⋊Q₈	—	178	1	no	—
123	C₂×M₃₂	—	89	2	no	—
124	#494	—	89	2	no	—
125	#494	—	89	2	no	—
126	#494	—	178	1	no	—
127	#494	—	178	1	no	—
128	C₂×#108	—	2	1	no	—
129	C₂×#108	C₂×D₈₉	1	2	yes	—
130	#423	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
132	#362	—	178	8	no	—
133	#346	—	178	1	no	—
134	C₄×C₁₆	—	178	1	no	—
135	C₂²×Q₈	—	178	2	no	—
136	C₂×Q₁₆	—	89	4	no	—
137	C₂×Q₁₆	—	356	2	no	—
138	C₂×Q₁₆	—	356	2	no	—
139	C₂×Q₁₆	—	178	2	no	—
140	C₂×Q₁₆	—	178	2	no	—
141	C₂×Q₁₆	—	178	4	no	—
142	C₂×Q₁₆	—	178	2	no	—
143	C₂×Q₁₆	—	178	4	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
147	C₄⋊Q₈	—	356	2	no	—
148	C₄⋊Q₈	—	178	2	no	—
149	C₄⋊Q₈	—	178	2	no	—
150	C₂×C₄⋊C₄	—	356	2	no	—
151	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
152	M₃₂	—	89	4	no	—
153	M₃₂	—	178	2	no	—
154	C₂×C₁₆	—	178	2	no	—
155	#108	—	2	4	no	—
156	#108	—	4	2	no	—
157	C₄⋊C₈	—	4	2	no	—
158	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
159	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
160	C₄×C₈	D₃₅₆	1	1	yes	characteristic
161	C₄×C₈	D₃₅₆	1	1	yes	characteristic
162	C₂×Q₈	—	356	2	no	—
163	C₂×Q₈	—	356	2	no	—
164	C₂×Q₈	—	356	2	no	—
165	C₂×Q₈	—	356	2	no	—
166	C₂×Q₈	—	356	2	no	—
167	C₂×Q₈	—	178	4	no	—
168	C₂²×C₄	—	356	2	no	—
169	C₂²×C₄	#13083561	1	2	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
171	Q₁₆	—	356	4	no	—
172	Q₁₆	—	356	4	no	—
173	Q₁₆	—	356	4	no	—
174	Q₁₆	—	178	8	no	—
175	M₁₆	—	4	2	no	—
176	M₁₆	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	#13083551	1	1	yes	characteristic
179	C₂×C₈	#13083551	1	1	yes	characteristic
180	C₂×C₈	#13083559	1	2	yes	—
181	C₂×C₈	#13083561	1	2	yes	—
182	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
183	C₂×C₈	D₄×D₈₉	1	2	yes	—
184	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
185	C₂×C₈	#13083575	1	2	yes	—
186	C₄⋊C₄	—	356	4	no	—
187	C₄⋊C₄	—	356	4	no	—
188	C₄²	#13083551	1	1	yes	characteristic
189	C₄²	#13083551	1	1	yes	characteristic
190	C₄²	#13083559	1	2	yes	—
191	C₁₆	—	178	4	no	—
192	C₂³	C₂×#13083561	1	1	yes	characteristic
193	Q₈	—	356	4	no	—
194	Q₈	—	356	4	no	—
195	Q₈	—	712	2	no	—
196	Q₈	—	712	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	356	4	no	—
199	C₂×C₄	—	712	2	no	—
200	C₂×C₄	—	712	2	no	—
201	C₂×C₄	#15667932	1	2	yes	—
202	C₂×C₄	#15667932	1	2	yes	—
203	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
204	C₂×C₄	#15668001	1	1	yes	characteristic
205	C₂×C₄	#15668008	1	1	yes	characteristic
206	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
207	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
208	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
209	C₈	—	8	2	no	—
210	C₈	#15667932	1	2	yes	—
211	C₈	#15667932	1	2	yes	—
212	C₈	#15667974	1	2	yes	—
213	C₈	#15667987	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#15667932	1	1	yes	characteristic
216	C₂²	#19151946	1	1	yes	characteristic
217	C₂²	#19151972	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	712	4	no	—
220	C₄	#19151413	1	4	yes	complemented
221	C₄	#19151499	1	1	yes	characteristic
222	C₄	C₂×#15667932	1	1	yes	characteristic
223	C₄	#19151946	1	1	yes	characteristic
224	C₄	#19151974	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#26021057	1	1	yes	characteristic
228	C₂	C₂×#19151413	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#15667932
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#13212
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
104. C₄×C₈₉	#362	semidirect
119. C₈₉	#13212	semidirect
220. C₄	#19151413	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151978	2
C₂×#19151413	2
C₄×#15667952	2
C₈₉×#1848	2
#13212	89