Subgroups — C₂×#19152347 #26034381

Order 11392.

1024 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152347	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15668010	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15668007	C₂	1	2	yes	maximal, complemented
4	C₂×#15668006	C₂	1	2	yes	maximal, complemented
5	C₂×#15668002	C₂	1	2	yes	maximal, complemented
6	C₂×#15668001	C₂	1	1	yes	maximal, characteristic
7	C₂×#15667997	C₂	1	2	yes	maximal
8	C₂×#15667990	C₂	1	2	yes	maximal, complemented
9	C₂×#15667988	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
12	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
14	#15668010	C₂²	1	8	yes	complemented
15	#15668007	C₂²	1	16	yes	complemented
16	#15668006	C₂²	1	16	yes	complemented
17	#15668002	C₂²	1	16	yes	complemented
18	#15668001	C₂²	1	8	yes	—
19	#15667997	C₂²	1	16	yes	—
20	#15667990	C₂²	1	16	yes	complemented
21	#15667988	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
23	C₂×#13083553	C₂²	1	2	yes	—
24	C₂×#13083553	C₂²	1	2	yes	—
25	C₂×#13083553	C₂²	1	2	yes	—
26	C₂×#13083553	C₂²	1	2	yes	—
27	C₂×#13083553	C₂²	1	2	yes	—
28	C₂×#13083551	C₂²	1	2	yes	—
29	C₂×#13083551	C₂²	1	2	yes	—
30	C₂×#13083551	C₂²	1	2	yes	—
31	C₂×#13083551	C₂²	1	2	yes	—
32	C₂×#13083551	C₂²	1	2	yes	—
33	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊Q₈×C₈₉	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
39	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
40	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
41	C₂³×D₈₉	C₂³	1	2	yes	—
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₂×Dic₁₇₈	C₂³	1	4	yes	—
44	C₂×D₃₅₆	—	4	2	no	—
45	C₂×D₃₅₆	C₂³	1	4	yes	—
46	C₂²×Dic₈₉	C₂³	1	2	yes	—
47	C₂²×Dic₈₉	C₂³	1	2	yes	—
48	C₂²×Dic₈₉	C₂³	1	2	yes	—
49	C₂×C₄×D₈₉	—	4	4	no	—
50	C₂×C₄×D₈₉	C₂³	1	8	yes	—
51	C₄⋊Dic₈₉	C₂³	1	8	yes	—
52	#13083553	C₂³	1	8	yes	—
53	#13083553	C₂³	1	8	yes	—
54	#13083553	C₂³	1	8	yes	—
55	#13083553	C₂³	1	8	yes	—
56	#13083553	C₂³	1	8	yes	—
57	#13083551	C₂³	1	8	yes	—
58	#13083551	C₂³	1	8	yes	—
59	#13083551	C₂³	1	8	yes	—
60	#13083551	C₂³	1	8	yes	—
61	#13083551	C₂³	1	8	yes	—
62	C₄×Dic₈₉	C₂³	1	4	yes	—
63	C₄×Dic₈₉	C₂³	1	4	yes	—
64	C₂×Q₈×C₈₉	—	4	2	no	—
65	C₂×Q₈×C₈₉	—	4	2	no	—
66	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
67	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
68	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
74	C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
75	C₄²×C₈₉	C₂³	1	4	yes	—
76	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
77	C₂²×D₈₉	—	2	4	no	—
78	C₂²×D₈₉	—	2	8	no	—
79	C₂²×D₈₉	C₂⁴	1	4	yes	—
80	Q₈×C₈₉	—	4	4	no	—
81	Q₈×C₈₉	—	4	4	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	4	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	4	no	—
87	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
88	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
89	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
90	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
91	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
92	C₂×Dic₈₉	—	2	4	no	—
93	C₂×Dic₈₉	—	2	4	no	—
94	C₂×Dic₈₉	—	2	4	no	—
95	C₂×Dic₈₉	C₂⁴	1	4	yes	—
96	C₂×Dic₈₉	C₂⁴	1	4	yes	—
97	C₂×Dic₈₉	C₂⁴	1	4	yes	—
98	D₃₅₆	—	4	4	no	—
99	C₄×D₈₉	—	4	8	no	—
100	Dic₁₇₈	—	4	4	no	—
101	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₈₉	2₋¹⁺⁴	1	4	yes	—
103	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
104	C₂×D₈₉	—	2	4	no	—
105	C₂×D₈₉	—	2	8	no	—
106	C₂×D₈₉	—	4	8	no	—
107	C₄×C₈₉	—	2	2	no	—
108	C₄×C₈₉	—	2	4	no	—
109	C₄×C₈₉	—	2	2	no	—
110	C₄×C₈₉	—	2	2	no	—
111	C₄×C₈₉	—	2	4	no	—
112	Dic₈₉	—	2	4	no	—
113	Dic₈₉	—	2	4	no	—
114	Dic₈₉	—	2	4	no	—
115	C₂×C₈₉	#563	1	4	yes	complemented
116	C₂×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
118	D₈₉	—	4	4	no	—
119	C₂×#563	—	89	1	no	maximal, Sylow
120	C₈₉	C₂×#563	1	1	yes	characteristic, complemented, Sylow
121	#563	—	89	16	no	—
122	C₂×C₄⋊Q₈	—	89	1	no	—
123	C₂×C₄⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
124	C₂×#126	—	89	2	no	—
125	C₂×#126	—	89	2	no	—
126	C₂×#125	—	89	2	no	—
127	C₂×#124	—	89	1	no	—
128	C₂×#123	—	89	2	no	—
129	C₂×C₂²⋊Q₈	—	89	2	no	—
130	C₂×C₂²⋊Q₈	—	89	2	no	—
131	C₂²×Q₈	—	89	1	no	—
132	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
133	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
134	C₂²×D₄	—	89	1	no	—
135	C₂³×C₄	—	89	2	no	—
136	C₄⋊Q₈	—	89	8	no	—
137	C₄⋊Q₈	C₂×D₈₉	1	8	yes	complemented
138	#126	—	89	16	no	—
139	#126	—	89	16	no	—
140	#125	—	89	16	no	—
141	#124	—	89	8	no	—
142	#123	—	89	16	no	—
143	C₂²⋊Q₈	—	89	16	no	—
144	C₂²⋊Q₈	—	89	16	no	—
145	C₂×C₄⋊C₄	—	89	2	no	—
146	C₂×C₄⋊C₄	—	89	2	no	—
147	C₂×C₄⋊C₄	—	89	2	no	—
148	C₂×C₄⋊C₄	—	89	2	no	—
149	C₂×C₄⋊C₄	—	89	2	no	—
150	C₂×C₄⋊C₄	—	89	2	no	—
151	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
152	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
153	C₂×C₂²⋊C₄	—	89	2	no	—
154	C₂×C₂²⋊C₄	—	89	2	no	—
155	C₂×C₂²⋊C₄	—	89	2	no	—
156	C₂×C₂²⋊C₄	—	89	2	no	—
157	C₂×C₂²⋊C₄	—	89	2	no	—
158	C₂×C₄²	—	89	1	no	—
159	C₂×C₄²	—	89	1	no	—
160	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
161	C₂⁴	—	89	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	89	4	no	—
165	C₂×Q₈	—	356	2	no	—
166	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
167	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
168	C₂×D₄	—	89	4	no	—
169	C₂×D₄	—	356	2	no	—
170	C₂²×C₄	—	89	2	no	—
171	C₂²×C₄	—	89	2	no	—
172	C₂²×C₄	—	89	2	no	—
173	C₂²×C₄	—	89	8	no	—
174	C₂²×C₄	—	356	4	no	—
175	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
178	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
180	C₄⋊C₄	—	89	8	no	—
181	C₄⋊C₄	—	89	8	no	—
182	C₄⋊C₄	—	89	8	no	—
183	C₄⋊C₄	—	89	8	no	—
184	C₄⋊C₄	—	89	8	no	—
185	C₄⋊C₄	—	89	8	no	—
186	C₄⋊C₄	C₂²×D₈₉	1	8	yes	—
187	C₄⋊C₄	C₂²×D₈₉	1	8	yes	—
188	C₂²⋊C₄	—	89	8	no	—
189	C₂²⋊C₄	—	89	8	no	—
190	C₂²⋊C₄	—	89	8	no	—
191	C₂²⋊C₄	—	89	8	no	—
192	C₂²⋊C₄	—	89	8	no	—
193	C₄²	—	89	4	no	—
194	C₄²	—	89	4	no	—
195	C₄²	C₂²×D₈₉	1	4	yes	—
196	C₂³	—	89	4	no	—
197	C₂³	—	178	4	no	—
198	C₂³	—	178	8	no	—
199	C₂³	C₂³×D₈₉	1	1	yes	characteristic
200	Q₈	—	4	4	no	—
201	Q₈	—	4	4	no	—
202	Q₈	—	356	4	no	—
203	D₄	—	356	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	89	4	no	—
210	C₂×C₄	—	89	4	no	—
211	C₂×C₄	—	89	4	no	—
212	C₂×C₄	—	356	8	no	—
213	C₂×C₄	—	178	4	no	—
214	C₂×C₄	—	178	4	no	—
215	C₂×C₄	—	178	4	no	—
216	C₂×C₄	C₂³×D₈₉	1	2	yes	—
217	C₂×C₄	C₂³×D₈₉	1	2	yes	—
218	C₂×C₄	C₂³×D₈₉	1	4	yes	—
219	C₂×C₄	C₂³×D₈₉	1	2	yes	—
220	C₂×C₄	C₂³×D₈₉	1	4	yes	—
221	C₂²	—	356	8	no	—
222	C₂²	—	178	4	no	—
223	C₂²	—	178	8	no	—
224	C₂²	#15668057	1	2	yes	—
225	C₂²	#15668060	1	4	yes	—
226	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	178	4	no	—
233	C₄	—	178	4	no	—
234	C₄	—	178	4	no	—
235	C₂	—	356	4	no	—
236	C₂	#19152347	1	4	yes	direct product factor
237	C₂	C₂×#15668057	1	1	yes	characteristic
238	C₂	C₂×#15668060	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₈₉
commutator, cosocle	C₂²×C₈₉	C₂⁵
Frattini	C₂²	C₂⁴×D₈₉
Fitting	C₂×C₄⋊Q₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂⁴
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	C₂×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	C₂×#563
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #19152347	C₂	direct (N × H)
2. C₂×#15668010	C₂	semidirect
3. C₂×#15668007	C₂	semidirect
4. C₂×#15668006	C₂	semidirect
5. C₂×#15668002	C₂	semidirect
8. C₂×#15667990	C₂	semidirect
9. C₂×#15667988	C₂	semidirect
10. C₂×C₄⋊Q₈×C₈₉	C₂	semidirect
14. #15668010	C₂²	semidirect
15. #15668007	C₂²	semidirect
16. #15668006	C₂²	semidirect
17. #15668002	C₂²	semidirect
20. #15667990	C₂²	semidirect
21. #15667988	C₂²	semidirect
37. C₄⋊Q₈×C₈₉	C₂²	semidirect
115. C₂×C₈₉	#563	semidirect
120. C₈₉	C₂×#563	semidirect
123. C₂×C₄⋊Q₈	D₈₉	semidirect
137. C₄⋊Q₈	C₂×D₈₉	semidirect
236. C₂	#19152347	direct (N × H)

Maximal subgroups

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

subgroup	index
#19152347	2
C₂×#15668010	2
C₂×#15668007	2
C₂×#15668006	2
C₂×#15668002	2
C₂×#15668001	2
C₂×#15667997	2
C₂×#15667990	2
C₂×#15667988	2
C₂×C₄⋊Q₈×C₈₉	2
C₂×#563	89

Subgroups — C2×#19152347 #26034381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19152347 #26034381