Subgroups — C₂×#26018356 #36469340

Order 22784.

554 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667948	C₂	1	2	yes	maximal
2	#26018356	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083558	—	2	2	no	—
5	C₂×#15667948	—	2	2	no	—
6	C₂×#15667948	—	2	2	no	—
7	C₂×#15667948	—	2	2	no	—
8	C₂×#15667948	C₄	1	4	yes	—
9	C₂×#15667948	C₄	1	4	yes	—
10	C₂×#15667948	C₂²	1	4	yes	—
11	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₈₉	—	2	2	no	—
13	C₂×M₃₂×C₈₉	—	2	2	no	—
14	C₂×M₃₂×C₈₉	C₄	1	2	yes	—
15	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₈₉	C₄	1	2	yes	—
17	C₂×#13083558	—	2	4	no	—
18	C₂×#13083558	—	2	4	no	—
19	C₂×#13083558	—	2	4	no	—
20	C₂×#13083558	—	4	2	no	—
21	C₂×#13083558	—	4	2	no	—
22	C₂×#13083558	—	4	2	no	—
23	C₂²×#10623494	—	4	2	no	—
24	#15667948	—	2	4	no	—
25	#15667948	—	2	4	no	—
26	#15667948	—	2	4	no	—
27	#15667948	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	D₄	1	2	yes	—
35	C₂²×C₈×C₈₉	D₄	1	2	yes	—
36	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	M₃₂×C₈₉	—	2	4	no	—
40	M₃₂×C₈₉	—	2	4	no	—
41	C₂×C₁₆×C₈₉	—	2	2	no	—
42	C₂×C₁₆×C₈₉	—	2	2	no	—
43	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
44	#13083558	—	2	8	no	—
45	#13083558	—	4	4	no	—
46	#13083558	—	4	4	no	—
47	#13083558	—	4	4	no	—
48	C₂×#10623494	—	4	4	no	—
49	C₂×#10623494	—	4	4	no	—
50	C₂×#10623494	—	4	4	no	—
51	C₂⁴×C₈₉	Q₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
63	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₈₉	C₄²	1	2	yes	—
65	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
80	C₁₆×C₈₉	—	2	4	no	—
81	C₂³×C₈₉	—	4	2	no	—
82	C₂³×C₈₉	#103	1	2	yes	—
83	C₂³×C₈₉	#103	1	2	yes	—
84	C₂³×C₈₉	#107	1	1	yes	characteristic
85	C₂³×C₈₉	#107	1	1	yes	characteristic
86	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#95	1	2	yes	—
89	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#102	1	2	yes	—
91	C₂×C₄×C₈₉	#102	1	2	yes	—
92	C₂×C₄×C₈₉	#102	1	2	yes	—
93	C₂×C₄×C₈₉	#102	1	2	yes	—
94	C₂×C₄×C₈₉	#103	1	2	yes	—
95	C₂×C₄×C₈₉	#103	1	2	yes	—
96	C₂×C₄×C₈₉	#106	1	2	yes	—
97	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
105	C₈×C₈₉	#95	1	1	yes	characteristic
106	C₈×C₈₉	#95	1	1	yes	characteristic
107	C₈×C₈₉	#95	1	2	yes	—
108	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
109	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
111	#10623494	—	4	8	no	—
112	C₂²×C₈₉	—	2	4	no	—
113	C₂²×C₈₉	—	2	2	no	—
114	C₂²×C₈₉	—	4	2	no	—
115	C₂²×C₈₉	—	4	1	no	—
116	C₂²×C₈₉	—	4	1	no	—
117	C₂²×C₈₉	#340	1	2	yes	—
118	C₂²×C₈₉	#340	1	1	yes	characteristic
119	C₂²×C₈₉	#340	1	1	yes	characteristic
120	C₂²×C₈₉	C₂×#103	1	2	yes	—
121	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
122	C₄×C₈₉	#340	1	2	yes	—
123	C₄×C₈₉	#340	1	1	yes	characteristic
124	C₄×C₈₉	#340	1	1	yes	characteristic
125	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
126	C₄×C₈₉	C₂×#102	1	2	yes	—
127	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
128	C₂×#1385	—	89	1	no	maximal, Sylow
129	C₂×C₈₉	—	2	2	no	—
130	C₂×C₈₉	—	2	1	no	—
131	C₂×C₈₉	—	2	2	no	—
132	C₂×C₈₉	—	2	1	no	—
133	C₂×C₈₉	#1385	1	2	yes	complemented
134	C₂×C₈₉	C₂×#340	1	1	yes	characteristic
135	C₂²×M₃₂	D₈₉	1	1	yes	characteristic
136	C₂²×#108	—	89	2	no	—
137	#1385	—	89	4	no	—
138	C₈₉	C₂×#1385	1	1	yes	characteristic, complemented, Sylow
139	C₂²×M₁₆	—	178	2	no	—
140	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
141	C₂×M₃₂	—	2	2	no	—
142	C₂×M₃₂	—	2	2	no	—
143	C₂×M₃₂	Dic₈₉	1	2	yes	—
144	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
145	C₂²×C₁₆	Dic₈₉	1	2	yes	—
146	C₂×#108	—	89	4	no	—
147	C₂×#108	—	89	4	no	—
148	C₂×#108	—	89	4	no	—
149	C₂×#108	—	178	2	no	—
150	C₂×#108	—	178	2	no	—
151	C₂×#108	—	178	2	no	—
152	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
153	C₂×M₁₆	—	356	2	no	—
154	C₂×M₁₆	—	356	2	no	—
155	C₂×M₁₆	—	356	2	no	—
156	C₂×M₁₆	—	178	4	no	—
157	C₂×M₁₆	—	178	4	no	—
158	C₂×M₁₆	—	178	4	no	—
159	C₂²×C₈	—	356	2	no	—
160	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
161	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₈₉	1	2	yes	—
163	C₂²×C₈	C₄×D₈₉	1	2	yes	—
164	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
165	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
166	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
169	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
170	M₃₂	—	2	4	no	—
171	M₃₂	—	2	4	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
175	#108	—	89	8	no	—
176	#108	—	178	4	no	—
177	#108	—	178	4	no	—
178	#108	—	178	4	no	—
179	C₂⁴	Dic₃₅₆	1	1	yes	characteristic
180	C₂²×C₄	#13083551	1	2	yes	—
181	C₂²×C₄	#13083551	1	2	yes	—
182	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
184	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
185	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
186	C₂²×C₄	#13083563	1	2	yes	—
187	C₂²×C₄	#13083563	1	2	yes	—
188	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
190	M₁₆	—	356	4	no	—
191	M₁₆	—	356	4	no	—
192	M₁₆	—	356	4	no	—
193	M₁₆	—	178	8	no	—
194	C₂×C₈	—	356	4	no	—
195	C₂×C₈	—	356	4	no	—
196	C₂×C₈	—	356	4	no	—
197	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
198	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
199	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
200	C₂×C₈	#13083551	1	2	yes	—
201	C₂×C₈	#13083551	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
205	C₂×C₈	#13083553	1	2	yes	—
206	C₂×C₈	#13083553	1	2	yes	—
207	C₂×C₈	#13083553	1	2	yes	—
208	C₂×C₈	#13083553	1	2	yes	—
209	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
210	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
212	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
214	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
215	C₁₆	—	2	4	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#15667937	1	2	yes	—
218	C₂³	#15667937	1	2	yes	—
219	C₂³	#15667946	1	1	yes	characteristic
220	C₂³	#15667946	1	1	yes	characteristic
221	C₂³	C₂×Dic₃₅₆	1	1	yes	characteristic
222	C₂×C₄	#15667919	1	2	yes	—
223	C₂×C₄	#15667919	1	1	yes	characteristic
224	C₂×C₄	#15667919	1	1	yes	characteristic
225	C₂×C₄	#15667934	1	2	yes	—
226	C₂×C₄	#15667934	1	2	yes	—
227	C₂×C₄	#15667934	1	2	yes	—
228	C₂×C₄	#15667934	1	2	yes	—
229	C₂×C₄	#15667937	1	2	yes	—
230	C₂×C₄	#15667937	1	2	yes	—
231	C₂×C₄	#15667944	1	2	yes	—
232	C₂×C₄	#15667944	1	1	yes	characteristic
233	C₂×C₄	#15667944	1	1	yes	characteristic
234	C₂×C₄	#15667946	1	1	yes	characteristic
235	C₂×C₄	#15667946	1	1	yes	characteristic
236	C₂×C₄	C₂×#13083551	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
238	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
239	C₂×C₄	C₂×#13083563	1	2	yes	—
240	C₈	—	356	8	no	—
241	C₈	#15667919	1	1	yes	characteristic
242	C₈	#15667919	1	1	yes	characteristic
243	C₈	#15667919	1	2	yes	—
244	C₈	C₂×C₄×Dic₈₉	1	1	yes	characteristic
245	C₈	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
246	C₈	C₂×#13083553	1	2	yes	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	#19151361	1	2	yes	—
253	C₂²	#19151361	1	1	yes	characteristic
254	C₂²	#19151361	1	1	yes	characteristic
255	C₂²	C₂×#15667937	1	2	yes	—
256	C₂²	C₂×#15667946	1	1	yes	characteristic
257	C₄	#19151361	1	2	yes	—
258	C₄	#19151361	1	1	yes	characteristic
259	C₄	#19151361	1	1	yes	characteristic
260	C₄	C₂×#15667919	1	1	yes	characteristic
261	C₄	C₂×#15667934	1	2	yes	—
262	C₄	C₂×#15667944	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	2	no	—
267	C₂	#26018356	1	2	yes	direct product factor
268	C₂	C₂×#19151361	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667944
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂²×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂⁴×C₈₉	Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂²×M₃₂	D₈₉
hypocenter	C₈₉	C₂×#1385
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #26018356	C₂	direct (N × H)
133. C₂×C₈₉	#1385	semidirect
138. C₈₉	C₂×#1385	semidirect
267. C₂	#26018356	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×#15667948	2
#26018356	2
C₂²×M₃₂×C₈₉	2
C₂×#1385	89

Subgroups — C2×#26018356 #36469340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018356 #36469340