Subgroups — #26023341

Order 11392.

496 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668012	C₂	1	1	yes	maximal, characteristic
2	#19151502	C₂	1	4	yes	maximal
3	#19151496	C₂	1	2	yes	maximal
4	#19151494	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₈₉	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₈₉	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
10	#15668012	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083553	C₂²	1	4	yes	—
16	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
17	C₂×C₄×Dic₈₉	C₂²	1	4	yes	—
18	#15667919	C₂²	1	4	yes	—
19	#15667919	C₂²	1	2	yes	—
20	#15667919	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	#95×C₈₉	C₂²	1	2	yes	—
26	C₂×Dic₁₇₈	—	4	2	no	—
27	C₂×Dic₁₇₈	—	4	2	no	—
28	C₂×Dic₁₇₈	C₂×C₄	1	8	yes	—
29	C₂²×Dic₈₉	C₂³	1	2	yes	—
30	C₂²×Dic₈₉	C₂³	1	4	yes	—
31	C₂²×Dic₈₉	C₂³	1	2	yes	—
32	C₄⋊Dic₈₉	—	2	2	no	—
33	C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
34	C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
35	C₄⋊Dic₈₉	Q₈	1	4	yes	complemented
36	#13083553	—	4	4	no	—
37	C₄×Dic₈₉	—	2	4	no	—
38	C₄×Dic₈₉	—	4	4	no	—
39	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₈₉	—	2	2	no	—
45	C₄⋊C₄×C₈₉	Q₈	1	4	yes	complemented
46	C₄²×C₈₉	—	2	4	no	—
47	C₄²×C₈₉	C₂×C₄	1	4	yes	—
48	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	2	2	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	2	2	no	—
56	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
61	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₈₉	#41	1	2	yes	—
63	C₂×C₄×C₈₉	#41	1	2	yes	—
64	C₂×Dic₈₉	—	2	2	no	—
65	C₂×Dic₈₉	—	2	2	no	—
66	C₂×Dic₈₉	—	2	4	no	—
67	C₂×Dic₈₉	—	2	4	no	—
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	2	2	no	—
71	C₂×Dic₈₉	C₂²×C₄	1	8	yes	—
72	C₂×Dic₈₉	C₂×Q₈	1	4	yes	—
73	Dic₁₇₈	—	4	8	no	—
74	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
78	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
79	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
81	C₄×C₈₉	—	2	2	no	—
82	C₄×C₈₉	—	2	4	no	—
83	C₄×C₈₉	—	4	2	no	—
84	C₄×C₈₉	C₄×Q₈	1	4	yes	—
85	C₄×C₈₉	C₄×Q₈	1	4	yes	—
86	Dic₈₉	—	2	8	no	—
87	Dic₈₉	—	2	4	no	—
88	Dic₈₉	—	4	2	no	—
89	C₂×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
90	C₂×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₈₉	#518	1	1	yes	characteristic
92	C₂×C₈₉	#554	1	1	yes	characteristic
93	C₂×C₈₉	#554	1	1	yes	characteristic
94	C₂×C₈₉	#555	1	1	yes	characteristic
95	C₂×C₈₉	Q₈²	1	1	yes	characteristic
96	#2369	—	89	1	no	maximal, Sylow
97	C₈₉	#2369	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊Q₈	—	89	1	no	—
99	#391	—	89	4	no	—
100	#389	—	89	2	no	—
101	#389	—	89	2	no	—
102	C₄×C₄⋊C₄	—	89	2	no	—
103	C₄×C₄⋊C₄	—	89	2	no	—
104	C₄×C₄⋊C₄	—	89	1	no	—
105	C₄×C₄⋊C₄	D₈₉	1	1	yes	characteristic
106	C₂²×Q₈	—	89	2	no	—
107	C₄⋊Q₈	—	89	8	no	—
108	C₂×C₄⋊C₄	—	89	2	no	—
109	C₂×C₄⋊C₄	—	89	2	no	—
110	C₂×C₄⋊C₄	—	89	1	no	—
111	C₂×C₄⋊C₄	—	89	4	no	—
112	C₂×C₄⋊C₄	—	89	1	no	—
113	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
115	C₂×C₄²	—	89	4	no	—
116	C₂×C₄²	—	89	2	no	—
117	C₂×C₄²	C₂×D₈₉	1	2	yes	—
118	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
119	#95	—	89	4	no	—
120	#95	—	89	4	no	—
121	#95	—	89	2	no	—
122	#95	C₂×D₈₉	1	2	yes	—
123	C₂×Q₈	—	89	8	no	—
124	C₂×Q₈	—	356	2	no	—
125	C₂×Q₈	—	356	2	no	—
126	C₂²×C₄	—	89	2	no	—
127	C₂²×C₄	—	89	4	no	—
128	C₂²×C₄	—	89	2	no	—
129	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
130	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
132	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	89	8	no	—
136	C₄⋊C₄	—	89	8	no	—
137	C₄⋊C₄	—	89	4	no	—
138	C₄⋊C₄	—	356	4	no	—
139	C₄⋊C₄	—	178	2	no	—
140	C₄⋊C₄	Dic₁₇₈	1	4	yes	complemented
141	C₄²	—	2	4	no	—
142	C₄²	—	356	4	no	—
143	C₄²	—	178	4	no	—
144	C₄²	C₄×D₈₉	1	4	yes	—
145	C₂³	C₂³×D₈₉	1	1	yes	characteristic
146	Q₈	—	356	8	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	89	8	no	—
155	C₂×C₄	—	89	4	no	—
156	C₂×C₄	—	178	2	no	—
157	C₂×C₄	—	178	2	no	—
158	C₂×C₄	—	178	2	no	—
159	C₂×C₄	—	178	2	no	—
160	C₂×C₄	—	178	2	no	—
161	C₂×C₄	—	178	4	no	—
162	C₂×C₄	—	178	4	no	—
163	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
164	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
166	C₂×C₄	Q₈×D₈₉	1	2	yes	—
167	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
168	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
169	C₂×C₄	#13083573	1	2	yes	—
170	C₂×C₄	#13083574	1	2	yes	—
171	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
172	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
173	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
174	C₂²	C₂×#13083573	1	1	yes	characteristic
175	C₂²	C₂×#13083574	1	1	yes	characteristic
176	C₂²	#15668058	1	1	yes	characteristic
177	C₂²	#15668058	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	356	2	no	—
182	C₄	—	178	4	no	—
183	C₄	—	178	8	no	—
184	C₄	#15667977	1	4	yes	—
185	C₄	C₄×Dic₁₇₈	1	4	yes	—
186	C₂	C₂×#15667977	1	1	yes	characteristic
187	C₂	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
188	C₂	#19152072	1	1	yes	characteristic
189	C₂	#19152305	1	1	yes	characteristic
190	C₂	#19152308	1	1	yes	characteristic
191	C₂	#19152313	1	1	yes	characteristic
192	C₂	Q₈×Dic₁₇₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₈₉	C₂
socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#2369
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #15668012	C₄	semidirect
35. C₄⋊Dic₈₉	Q₈	semidirect
45. C₄⋊C₄×C₈₉	Q₈	semidirect
97. C₈₉	#2369	semidirect
140. C₄⋊C₄	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668012	2
#19151502	2
#19151496	2
#19151494	2
C₄⋊C₄×Dic₈₉	2
C₄×C₄⋊Dic₈₉	2
C₄×C₄⋊C₄×C₈₉	2
#2369	89