Subgroups — C₂×#26022007 #36707406

Order 22784.

798 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667952	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26022007	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151416	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10623494	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₈₉	C₄	1	1	yes	characteristic
8	C₂×#15667952	—	2	4	no	—
9	C₂×#15667952	C₂²	1	4	yes	—
10	C₂²×#13083549	C₂²	1	2	yes	—
11	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
12	C₂×C₈×Dic₈₉	C₄	1	4	yes	—
13	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
14	C₂×C₈×Dic₈₉	C₂²	1	4	yes	—
15	#19151416	C₂²	1	16	yes	complemented
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₈₉	—	2	4	no	—
18	C₂×M₃₂×C₈₉	C₂²	1	4	yes	—
19	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂³×Dic₈₉	C₈	1	2	yes	—
21	C₂²×#10623494	—	2	4	no	—
22	C₂²×#10623494	C₂×C₄	1	2	yes	—
23	C₂²×#10623494	C₂×C₄	1	4	yes	—
24	C₂×C₄×Dic₈₉	C₈	1	4	yes	—
25	C₂×C₄×Dic₈₉	C₈	1	1	yes	characteristic
26	C₂×C₄×Dic₈₉	C₈	1	1	yes	characteristic
27	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
28	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
29	#15667952	—	2	8	no	—
30	C₂×#13083549	—	2	4	no	—
31	C₂×#13083549	C₂³	1	4	yes	—
32	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
33	C₈×Dic₈₉	C₂×C₄	1	8	yes	—
34	C₈×Dic₈₉	C₂³	1	4	yes	—
35	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₈₉	D₄	1	4	yes	—
39	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	M₃₂×C₈₉	—	2	8	no	—
43	C₂×C₁₆×C₈₉	—	2	4	no	—
44	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
45	C₂²×Dic₈₉	—	2	8	no	—
46	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
47	C₂²×Dic₈₉	C₂×C₈	1	8	yes	—
48	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
49	C₂×#10623494	—	2	8	no	—
50	C₂×#10623494	—	2	4	no	—
51	C₂×#10623494	C₂²×C₄	1	4	yes	—
52	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
53	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
54	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
55	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
56	#13083549	—	2	4	no	—
57	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
62	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
65	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₁₆×C₈₉	—	2	4	no	—
74	C₂³×C₈₉	—	2	4	no	—
75	C₂³×C₈₉	C₄⋊C₈	1	4	yes	—
76	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
77	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₄⋊C₈	1	8	yes	—
79	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
85	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₈₉	—	2	4	no	—
88	C₂×Dic₈₉	—	2	16	no	—
89	C₂×Dic₈₉	M₃₂	1	8	yes	complemented
90	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
91	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
92	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₈×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
94	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
96	#10623494	—	2	4	no	—
97	C₂²×C₈₉	—	2	4	no	—
98	C₂²×C₈₉	—	2	8	no	—
99	C₂²×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
100	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
101	C₂²×C₈₉	C₂×M₃₂	1	4	yes	—
102	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
104	C₄×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
105	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
107	Dic₈₉	—	2	4	no	—
108	Dic₈₉	C₂×M₃₂	1	8	yes	complemented
109	C₂×#2150	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	—	2	4	no	—
111	C₂×C₈₉	#2150	1	4	yes	complemented
112	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
113	C₂×C₈₉	C₂²×M₃₂	1	2	yes	—
114	C₂²×M₃₂	—	89	1	no	—
115	C₂²×M₃₂	D₈₉	1	1	yes	characteristic
116	C₂²×C₄×C₈	—	89	1	no	—
117	#2150	—	89	8	no	—
118	C₂×C₄⋊C₁₆	—	89	4	no	—
119	C₈₉	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₈	—	89	1	no	—
121	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
122	C₂²×C₄²	—	89	1	no	—
123	C₂×M₃₂	—	2	4	no	—
124	C₂×M₃₂	—	89	4	no	—
125	C₂×M₃₂	—	178	4	no	—
126	C₂×M₃₂	C₂×D₈₉	1	4	yes	—
127	C₂²×C₁₆	—	89	2	no	—
128	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
129	C₂×C₄×C₈	—	89	2	no	—
130	C₂×C₄×C₈	—	89	4	no	—
131	C₂×C₄×C₈	—	89	4	no	—
132	C₂×C₄×C₈	—	89	2	no	—
133	C₄⋊C₁₆	—	89	16	no	—
134	C₂³×C₄	—	89	2	no	—
135	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
136	C₂²×C₈	—	89	4	no	—
137	C₂²×C₈	—	89	2	no	—
138	C₂²×C₈	—	178	4	no	—
139	C₂²×C₈	Dic₁₇₈	1	4	yes	—
140	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
141	C₂²×C₈	C₄×D₈₉	1	2	yes	—
142	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
143	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
144	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
145	C₂×C₄²	—	89	1	no	—
146	C₂×C₄²	—	89	4	no	—
147	C₂×C₄²	—	89	1	no	—
148	C₂×C₄²	—	89	2	no	—
149	C₂×C₄²	—	89	4	no	—
150	M₃₂	—	2	8	no	—
151	M₃₂	—	178	8	no	—
152	C₂×C₁₆	—	2	4	no	—
153	C₂×C₁₆	—	89	4	no	—
154	C₂×C₁₆	—	178	4	no	—
155	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
156	C₄×C₈	—	89	8	no	—
157	C₄×C₈	—	89	4	no	—
158	C₄×C₈	—	89	4	no	—
159	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
160	C₂²×C₄	—	89	2	no	—
161	C₂²×C₄	—	89	8	no	—
162	C₂²×C₄	—	89	2	no	—
163	C₂²×C₄	—	178	8	no	—
164	C₂²×C₄	#13083553	1	4	yes	—
165	C₂²×C₄	C₈×D₈₉	1	2	yes	—
166	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
167	C₂²×C₄	#13083557	1	4	yes	—
168	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
170	C₂×C₈	—	89	4	no	—
171	C₂×C₈	—	178	8	no	—
172	C₂×C₈	—	178	4	no	—
173	C₂×C₈	#13083553	1	4	yes	—
174	C₂×C₈	#13083553	1	8	yes	—
175	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
176	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
178	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
179	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
180	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
181	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
182	C₄²	—	89	4	no	—
183	C₄²	—	89	4	no	—
184	C₄²	—	89	4	no	—
185	C₄²	—	89	4	no	—
186	C₁₆	—	2	4	no	—
187	C₁₆	—	178	4	no	—
188	C₂³	—	2	4	no	—
189	C₂³	#15667941	1	4	yes	—
190	C₂³	C₂×C₈×D₈₉	1	2	yes	—
191	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
192	C₂×C₄	—	89	8	no	—
193	C₂×C₄	—	89	4	no	—
194	C₂×C₄	—	89	4	no	—
195	C₂×C₄	—	178	16	no	—
196	C₂×C₄	—	178	4	no	—
197	C₂×C₄	#15667941	1	8	yes	—
198	C₂×C₄	#15667941	1	4	yes	—
199	C₂×C₄	C₂×#13083553	1	4	yes	—
200	C₂×C₄	C₂×#13083553	1	2	yes	—
201	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂×#13083557	1	4	yes	—
204	C₂×C₄	C₂×#13083557	1	2	yes	—
205	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
206	C₈	—	178	4	no	—
207	C₈	C₂×#13083553	1	4	yes	—
208	C₈	C₂×#13083553	1	2	yes	—
209	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
210	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
211	C₂²	—	2	8	no	—
212	C₂²	—	2	4	no	—
213	C₂²	C₂×#15667941	1	4	yes	—
214	C₂²	C₂×#15667941	1	2	yes	—
215	C₂²	M₃₂×D₈₉	1	4	yes	—
216	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
217	C₄	—	89	8	no	—
218	C₄	—	178	4	no	—
219	C₄	C₂×#15667941	1	4	yes	—
220	C₄	C₂×#15667941	1	2	yes	—
221	C₄	C₂²×#13083553	1	1	yes	characteristic
222	C₄	C₂²×#13083557	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	#26022007	1	4	yes	direct product factor
225	C₂	C₂²×#15667941	1	1	yes	characteristic
226	C₂	C₂×M₃₂×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂²×M₃₂×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×C₈
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂²×M₃₂	D₈₉
hypocenter	C₈₉	C₂×#2150
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #26022007	C₂	direct (N × H)
4. C₂×#19151416	C₂	semidirect
15. #19151416	C₂²	semidirect
89. C₂×Dic₈₉	M₃₂	semidirect
108. Dic₈₉	C₂×M₃₂	semidirect
111. C₂×C₈₉	#2150	semidirect
119. C₈₉	C₂×#2150	semidirect
224. C₂	#26022007	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15667952	2
C₂²×C₈×Dic₈₉	2
#26022007	2
C₂×#19151416	2
C₂²×M₃₂×C₈₉	2
C₂×#2150	89

Subgroups — C2×#26022007 #36707406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022007 #36707406