Subgroups — C₂×#26019058 #36523798

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151722	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26019058	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151328	C₂	1	2	yes	maximal
5	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
6	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
9	#19151722	—	2	4	no	—
10	#19151722	C₂²	1	8	yes	—
11	C₄×Dic₃₅₆	C₄	1	4	yes	—
12	C₄×Dic₃₅₆	C₂²	1	4	yes	—
13	C₂×#15667946	C₄	1	1	yes	characteristic
14	C₂×#15667941	—	2	2	no	—
15	C₂×#15667937	C₄	1	2	yes	—
16	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₈₉	—	2	2	no	—
19	#19151328	C₂²	1	8	yes	—
20	C₈×#10623494	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#335×C₈₉	C₂²	1	4	yes	—
24	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
25	C₂×Dic₃₅₆	C₈	1	8	yes	complemented
26	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
27	C₂²×#10623494	—	2	1	no	—
28	C₂²×#10623494	D₄	1	1	yes	characteristic
29	C₂²×#10623494	D₄	1	1	yes	characteristic
30	C₄×Dic₁₇₈	—	2	2	no	—
31	C₄×Dic₁₇₈	—	2	2	no	—
32	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₇₈	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
35	C₂×#13083553	—	2	2	no	—
36	C₂×C₄×Dic₈₉	—	2	2	no	—
37	#15667946	C₂×C₄	1	4	yes	—
38	#15667941	—	2	8	no	—
39	#15667937	C₂×C₄	1	8	yes	—
40	C₄×#10623494	—	2	1	no	—
41	C₄×#10623494	D₄	1	2	yes	—
42	C₄×#10623494	C₂³	1	2	yes	—
43	C₄×#10623494	C₂³	1	2	yes	—
44	C₈×Dic₈₉	—	2	8	no	—
45	C₂²×C₈×C₈₉	—	2	2	no	—
46	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₈₉	—	2	2	no	—
50	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
51	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₈₉	C₂³	1	2	yes	—
53	C₂×Dic₁₇₈	—	2	2	no	—
54	C₂×Dic₁₇₈	—	2	4	no	—
55	C₂×Dic₁₇₈	C₂×C₈	1	8	yes	—
56	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
57	C₂²×Dic₈₉	—	2	2	no	—
58	C₂²×Dic₈₉	—	2	2	no	—
59	Dic₃₅₆	C₂×C₈	1	16	yes	complemented
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	2	2	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	Q₁₆	1	4	yes	complemented
64	C₂×#10623494	C₂×D₄	1	2	yes	—
65	C₂×#10623494	C₂×D₄	1	2	yes	—
66	C₄⋊Dic₈₉	—	2	2	no	—
67	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
68	#13083553	—	2	4	no	—
69	#13083553	—	2	4	no	—
70	C₄×Dic₈₉	—	2	4	no	—
71	C₄×Dic₈₉	—	2	4	no	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	—	2	4	no	—
77	C₂×C₈×C₈₉	—	4	4	no	—
78	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	#41	1	1	yes	characteristic
84	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
85	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₈₉	#131	1	2	yes	—
92	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₈₉	#135	1	2	yes	—
94	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
97	C₂×Dic₈₉	—	2	4	no	—
98	C₂×Dic₈₉	—	2	8	no	—
99	C₂×Dic₈₉	—	2	4	no	—
100	C₂×Dic₈₉	—	4	4	no	—
101	Dic₁₇₈	—	2	4	no	—
102	Dic₁₇₈	—	2	8	no	—
103	Dic₁₇₈	C₂²×C₈	1	8	yes	—
104	C₈×C₈₉	—	2	2	no	—
105	C₈×C₈₉	—	4	4	no	—
106	C₈×C₈₉	C₂²×C₈	1	4	yes	—
107	#10623494	—	2	4	no	—
108	#10623494	—	2	2	no	—
109	#10623494	C₂×Q₁₆	1	4	yes	complemented
110	C₂²×C₈₉	C₈×D₄	1	2	yes	—
111	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
112	C₂²×C₈₉	#443	1	2	yes	—
113	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₈₉	—	2	1	no	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	C₈×D₄	1	2	yes	—
117	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
118	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
119	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
120	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
121	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
122	Dic₈₉	—	2	8	no	—
123	Dic₈₉	—	4	4	no	—
124	C₂×C₈×Q₁₆	—	89	1	no	maximal, Sylow
125	C₂×C₈₉	C₈×Q₁₆	1	4	yes	complemented
126	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
127	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
129	C₂×C₈×Q₈	—	89	2	no	—
130	C₂×C₄×Q₁₆	—	89	1	no	—
131	C₈×Q₁₆	—	89	8	no	—
132	C₂×#335	D₈₉	1	1	yes	characteristic
133	C₂×Q₈⋊C₈	—	89	2	no	—
134	C₂×C₈²	—	89	1	no	—
135	C₈₉	C₂×C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₁₆	—	89	1	no	—
137	C₂×C₄×Q₈	—	89	2	no	—
138	C₈×Q₈	—	89	8	no	—
139	C₈×Q₈	—	178	4	no	—
140	C₄×Q₁₆	—	89	4	no	—
141	C₄×Q₁₆	—	89	4	no	—
142	C₂×#107	—	89	1	no	—
143	C₂×C₄⋊C₈	—	178	2	no	—
144	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
145	C₂×#103	—	89	2	no	—
146	C₂×C₄×C₈	—	89	1	no	—
147	C₂×C₄×C₈	—	89	1	no	—
148	C₂×C₄×C₈	—	178	2	no	—
149	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
150	#335	C₂×D₈₉	1	4	yes	—
151	Q₈⋊C₈	—	89	8	no	—
152	C₈²	—	89	4	no	—
153	C₂²×Q₈	—	89	2	no	—
154	C₂×Q₁₆	—	89	8	no	—
155	C₂×Q₁₆	—	89	4	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	89	1	no	—
158	C₂²×C₈	—	89	1	no	—
159	C₂²×C₈	—	178	1	no	—
160	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
162	C₄×Q₈	—	89	4	no	—
163	C₄×Q₈	—	89	4	no	—
164	C₄×Q₈	—	178	2	no	—
165	C₄×Q₈	—	178	2	no	—
166	C₂×C₄⋊C₄	—	89	2	no	—
167	C₂×C₄⋊C₄	—	178	2	no	—
168	C₂×C₄²	—	178	2	no	—
169	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
170	#107	—	89	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	178	8	no	—
173	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
174	#103	—	89	8	no	—
175	C₄×C₈	—	89	2	no	—
176	C₄×C₈	—	89	2	no	—
177	C₄×C₈	—	89	2	no	—
178	C₄×C₈	—	178	8	no	—
179	C₄×C₈	—	178	1	no	—
180	C₄×C₈	C₄×D₈₉	1	2	yes	—
181	C₄×C₈	C₂²×D₈₉	1	2	yes	—
182	C₂×Q₈	—	89	8	no	—
183	C₂×Q₈	—	89	4	no	—
184	C₂×Q₈	—	178	2	no	—
185	C₂×Q₈	—	178	4	no	—
186	C₂²×C₄	—	178	2	no	—
187	C₂²×C₄	—	178	2	no	—
188	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
190	C₂²×C₄	#13083575	1	1	yes	characteristic
191	Q₁₆	—	89	16	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	89	4	no	—
196	C₂×C₈	—	89	2	no	—
197	C₂×C₈	—	89	2	no	—
198	C₂×C₈	—	178	4	no	—
199	C₂×C₈	—	178	2	no	—
200	C₂×C₈	—	178	2	no	—
201	C₂×C₈	C₈×D₈₉	1	4	yes	—
202	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
204	C₄⋊C₄	—	89	4	no	—
205	C₄⋊C₄	—	178	2	no	—
206	C₄⋊C₄	—	178	4	no	—
207	C₄⋊C₄	—	178	4	no	—
208	C₄²	—	178	4	no	—
209	C₄²	—	178	4	no	—
210	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₄²	D₄×D₈₉	1	1	yes	characteristic
212	C₄²	#13083575	1	1	yes	characteristic
213	C₄²	C₂³×D₈₉	1	1	yes	characteristic
214	C₂³	#15667974	1	1	yes	characteristic
215	Q₈	—	89	8	no	—
216	Q₈	—	178	8	no	—
217	Q₈	—	178	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	356	4	no	—
220	C₂×C₄	—	178	4	no	—
221	C₂×C₄	—	178	8	no	—
222	C₂×C₄	—	178	4	no	—
223	C₂×C₄	#15667974	1	1	yes	characteristic
224	C₂×C₄	#15667974	1	1	yes	characteristic
225	C₂×C₄	#15667974	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
227	C₂×C₄	#15668020	1	2	yes	—
228	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
229	C₂×C₄	#15668031	1	2	yes	—
230	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
232	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	4	4	no	—
235	C₈	—	89	4	no	—
236	C₈	—	178	4	no	—
237	C₈	—	178	2	no	—
238	C₈	C₂×C₈×D₈₉	1	4	yes	—
239	C₂²	#19151663	1	2	yes	—
240	C₂²	#19151692	1	2	yes	—
241	C₂²	#19151714	1	2	yes	—
242	C₂²	C₂×#15667974	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	356	4	no	—
246	C₄	—	178	8	no	—
247	C₄	#19151663	1	2	yes	—
248	C₄	#19151692	1	2	yes	—
249	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
250	C₄	C₂×#15668020	1	1	yes	characteristic
251	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
252	C₄	C₂×#15668031	1	1	yes	characteristic
253	C₂	#26019058	1	4	yes	direct product factor
254	C₂	C₂×#19151663	1	1	yes	characteristic
255	C₂	C₂×#19151692	1	1	yes	characteristic
256	C₂	C₂×#19151714	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	C₂×C₈×Q₁₆
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #26019058	C₂	direct (N × H)
25. C₂×Dic₃₅₆	C₈	semidirect
59. Dic₃₅₆	C₂×C₈	semidirect
63. C₂×#10623494	Q₁₆	semidirect
109. #10623494	C₂×Q₁₆	semidirect
125. C₂×C₈₉	C₈×Q₁₆	semidirect
135. C₈₉	C₂×C₈×Q₁₆	semidirect
253. C₂	#26019058	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19151722	2
C₂×C₄×Dic₃₅₆	2
#26019058	2
C₂×#19151328	2
C₂×C₈×#10623494	2
C₂×#335×C₈₉	2
C₂×C₈×Q₁₆	89

Subgroups — C2×#26019058 #36523798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019058 #36523798