Subgroups — C₂×#26018220 #36464186

Order 22784.

462 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018220	C₂	1	4	yes	maximal, direct product factor
4	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₇₁₂	C₄	1	1	yes	characteristic
6	C₂×C₄×D₃₅₆	—	2	1	no	—
7	C₄×D₇₁₂	—	2	1	no	—
8	C₄×D₇₁₂	—	2	1	no	—
9	C₄×D₇₁₂	C₄	1	2	yes	—
10	C₄×D₇₁₂	C₂²	1	2	yes	—
11	C₂×#15667946	C₄	1	1	yes	characteristic
12	C₂×C₄⋊#10623494	—	2	1	no	—
13	C₂×#15667934	—	2	1	no	—
14	#19151349	—	2	1	no	—
15	#19151349	C₂²	1	2	yes	complemented
16	C₂²×C₁₆×C₈₉	—	2	1	no	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#346×C₈₉	—	2	1	no	—
19	#346×C₈₉	C₂²	1	2	yes	complemented
20	C₂²×D₃₅₆	—	2	1	no	—
21	C₂²×C₄×D₈₉	—	4	1	no	—
22	C₂×D₇₁₂	—	2	1	no	—
23	C₂×D₇₁₂	—	2	2	no	—
24	C₂×D₇₁₂	C₈	1	4	yes	complemented
25	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
26	C₂²×#10623494	—	4	1	no	—
27	C₄×D₃₅₆	—	2	2	no	—
28	C₄×D₃₅₆	—	2	2	no	—
29	C₄×D₃₅₆	—	4	1	no	—
30	C₄×D₃₅₆	—	4	1	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×#13083551	—	4	1	no	—
33	#15667946	—	2	1	no	—
34	#15667946	C₂×C₄	1	2	yes	—
35	C₄⋊#10623494	—	2	2	no	—
36	C₄⋊#10623494	—	4	1	no	—
37	#15667934	—	2	2	no	—
38	#15667934	—	2	2	no	—
39	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₈₉	—	2	4	no	—
43	C₂×C₁₆×C₈₉	—	2	2	no	—
44	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂³×D₈₉	—	4	1	no	—
49	C₂×D₃₅₆	—	2	4	no	—
50	C₂×D₃₅₆	—	2	2	no	—
51	C₂×D₃₅₆	—	4	1	no	—
52	C₂×D₃₅₆	—	4	2	no	—
53	C₂²×Dic₈₉	—	4	1	no	—
54	C₂×C₄×D₈₉	—	4	2	no	—
55	C₂×C₄×D₈₉	—	4	2	no	—
56	C₂×C₄×D₈₉	—	8	4	no	—
57	D₇₁₂	—	2	4	no	—
58	D₇₁₂	—	2	2	no	—
59	D₇₁₂	C₂×C₈	1	4	yes	complemented
60	C₂×#10623494	—	4	2	no	—
61	C₂×#10623494	—	8	2	no	—
62	C₄⋊Dic₈₉	—	2	2	no	—
63	C₄⋊Dic₈₉	—	4	1	no	—
64	#13083551	—	4	2	no	—
65	#13083551	—	4	2	no	—
66	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₈₉	D₈	1	1	yes	characteristic
78	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
79	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₁₆×C₈₉	—	2	4	no	—
81	C₂³×C₈₉	#102	1	1	yes	characteristic
82	C₂²×D₈₉	—	4	4	no	—
83	C₂²×D₈₉	—	4	2	no	—
84	C₂²×D₈₉	—	8	2	no	—
85	C₂²×D₈₉	—	8	2	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
89	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#104	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×Dic₈₉	—	4	2	no	—
97	C₂×Dic₈₉	—	8	2	no	—
98	D₃₅₆	—	2	4	no	—
99	D₃₅₆	—	4	4	no	—
100	D₃₅₆	—	4	2	no	—
101	C₄×D₈₉	—	8	4	no	—
102	C₄×D₈₉	—	8	4	no	—
103	C₈×C₈₉	—	2	1	no	—
104	C₈×C₈₉	—	2	1	no	—
105	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
106	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
107	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
108	#10623494	—	8	2	no	—
109	C₂²×C₈₉	D₄⋊C₈	1	2	yes	—
110	C₂²×C₈₉	#360	1	2	yes	—
111	C₂²×C₈₉	#361	1	2	yes	—
112	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
113	C₂×D₈₉	—	4	4	no	—
114	C₂×D₈₉	—	8	8	no	—
115	C₂×D₈₉	—	8	2	no	—
116	C₂×D₈₉	—	8	2	no	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	—	2	1	no	—
120	C₄×C₈₉	—	2	1	no	—
121	C₄×C₈₉	D₄⋊C₈	1	2	yes	—
122	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
124	Dic₈₉	—	8	2	no	—
125	C₂×#1333	—	89	1	no	maximal, Sylow
126	C₂×C₈₉	#1333	1	4	yes	complemented
127	C₂×C₈₉	C₂×D₄⋊C₈	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#360	1	1	yes	characteristic
129	C₂×C₈₉	C₂×#361	1	1	yes	characteristic
130	D₈₉	—	8	4	no	—
131	C₂×C₄×D₈	—	89	1	no	—
132	C₂×#346	D₈₉	1	1	yes	characteristic, complemented
133	C₂×#334	—	89	1	no	—
134	#1333	—	89	4	no	—
135	C₈₉	C₂×#1333	1	1	yes	characteristic, complemented, Sylow
136	C₂²×D₈	—	89	1	no	—
137	C₂×C₄×D₄	—	178	1	no	—
138	C₂²×C₁₆	—	2	1	no	—
139	C₄×D₈	—	89	2	no	—
140	C₄×D₈	—	89	2	no	—
141	C₄×D₈	—	178	1	no	—
142	C₄×D₈	—	178	1	no	—
143	C₂×#107	—	89	1	no	—
144	C₂×C₄⋊C₈	—	178	1	no	—
145	C₂×#102	—	178	1	no	—
146	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
147	#346	—	2	1	no	—
148	#346	C₂×D₈₉	1	2	yes	complemented
149	#334	—	89	2	no	—
150	#334	—	178	1	no	—
151	C₂²×D₄	—	178	1	no	—
152	C₂³×C₄	—	356	1	no	—
153	C₂×D₈	—	89	4	no	—
154	C₂×D₈	—	89	2	no	—
155	C₂×D₈	—	178	1	no	—
156	C₂×D₈	—	178	2	no	—
157	C₂²×C₈	—	356	1	no	—
158	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
160	C₄×D₄	—	356	1	no	—
161	C₄×D₄	—	356	1	no	—
162	C₄×D₄	—	178	2	no	—
163	C₄×D₄	—	178	2	no	—
164	C₂×C₄⋊C₄	—	178	1	no	—
165	C₂×C₂²⋊C₄	—	356	1	no	—
166	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
167	C₂×C₁₆	—	2	4	no	—
168	C₂×C₁₆	—	2	2	no	—
169	#107	—	89	2	no	—
170	#107	—	178	1	no	—
171	C₄⋊C₈	—	356	1	no	—
172	C₄⋊C₈	—	178	2	no	—
173	#102	—	178	2	no	—
174	#102	—	178	2	no	—
175	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
176	C₄×C₈	D₃₅₆	1	1	yes	characteristic
177	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
178	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
179	C₂⁴	—	356	1	no	—
180	C₂×D₄	—	356	1	no	—
181	C₂×D₄	—	356	2	no	—
182	C₂×D₄	—	178	4	no	—
183	C₂×D₄	—	178	2	no	—
184	C₂²×C₄	—	356	1	no	—
185	C₂²×C₄	—	356	2	no	—
186	C₂²×C₄	—	356	2	no	—
187	C₂²×C₄	—	712	4	no	—
188	C₂²×C₄	#13083551	1	1	yes	characteristic
189	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
190	C₂²×C₄	#13083563	1	1	yes	characteristic
191	D₈	—	89	4	no	—
192	D₈	—	178	2	no	—
193	D₈	—	178	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	356	2	no	—
196	C₂×C₈	—	712	2	no	—
197	C₂×C₈	#13083551	1	1	yes	characteristic
198	C₂×C₈	#13083551	1	1	yes	characteristic
199	C₂×C₈	C₈×D₈₉	1	2	yes	—
200	C₂×C₈	#13083557	1	2	yes	—
201	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
203	C₄⋊C₄	—	356	1	no	—
204	C₄⋊C₄	—	178	2	no	—
205	C₂²⋊C₄	—	356	2	no	—
206	C₂²⋊C₄	—	356	2	no	—
207	C₄²	#13083551	1	1	yes	characteristic
208	C₄²	D₇₁₂	1	1	yes	characteristic
209	C₄²	#13083563	1	1	yes	characteristic
210	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
211	C₁₆	—	2	4	no	—
212	C₂³	—	356	4	no	—
213	C₂³	—	356	2	no	—
214	C₂³	—	712	2	no	—
215	C₂³	—	712	2	no	—
216	C₂³	#15667934	1	1	yes	characteristic
217	D₄	—	356	4	no	—
218	D₄	—	356	2	no	—
219	D₄	—	178	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	356	2	no	—
223	C₂×C₄	—	712	4	no	—
224	C₂×C₄	—	712	4	no	—
225	C₂×C₄	—	712	2	no	—
226	C₂×C₄	#15667924	1	2	yes	—
227	C₂×C₄	#15667934	1	1	yes	characteristic
228	C₂×C₄	#15667934	1	1	yes	characteristic
229	C₂×C₄	#15667934	1	1	yes	characteristic
230	C₂×C₄	#15667940	1	2	yes	—
231	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
232	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
233	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	712	2	no	—
237	C₈	#15667924	1	2	yes	—
238	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
239	C₈	C₂×#13083557	1	1	yes	characteristic
240	C₂²	—	356	4	no	—
241	C₂²	—	712	2	no	—
242	C₂²	—	712	2	no	—
243	C₂²	—	712	8	no	—
244	C₂²	#19151327	1	2	yes	—
245	C₂²	#19151409	1	2	yes	—
246	C₂²	#19151412	1	2	yes	—
247	C₂²	C₂×#15667934	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	712	2	no	—
253	C₄	#19151327	1	2	yes	—
254	C₄	C₂×#15667924	1	1	yes	characteristic
255	C₄	C₂×#15667940	1	1	yes	characteristic
256	C₂	—	712	4	no	—
257	C₂	#26018220	1	4	yes	direct product factor
258	C₂	C₂×#19151327	1	1	yes	characteristic
259	C₂	C₂×#19151409	1	1	yes	characteristic
260	C₂	C₂×#19151412	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667934
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂³×C₈₉	#102
prime radical	C₂²×D₇₁₂	C₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	C₂×#1333
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×#19151349	C₂	semidirect
3. #26018220	C₂	direct (N × H)
4. C₂×#346×C₈₉	C₂	semidirect
15. #19151349	C₂²	semidirect
19. #346×C₈₉	C₂²	semidirect
24. C₂×D₇₁₂	C₈	semidirect
59. D₇₁₂	C₂×C₈	semidirect
126. C₂×C₈₉	#1333	semidirect
132. C₂×#346	D₈₉	semidirect
135. C₈₉	C₂×#1333	semidirect
148. #346	C₂×D₈₉	semidirect
257. C₂	#26018220	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×C₄×D₇₁₂	2
C₂×#19151349	2
#26018220	2
C₂×#346×C₈₉	2
C₂×#1333	89

Subgroups — C2×#26018220 #36464186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018220 #36464186