Subgroups — C₂×#26019162 #36524193

Order 22784.

654 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151728	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26019162	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151330	C₂	1	2	yes	maximal
6	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
9	#19151728	—	2	4	no	—
10	#19151728	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×C₄⋊#10623494	C₂²	1	2	yes	—
15	C₂×#15667941	—	2	2	no	—
16	C₂×#15667937	C₄	1	2	yes	—
17	C₂×#15667922	—	2	2	no	—
18	#19151349	C₂²	1	4	yes	—
19	#19151330	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#322×C₈₉	C₂²	1	4	yes	—
24	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
25	C₂×Dic₃₅₆	—	2	4	no	—
26	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
27	C₂²×#10623494	—	2	2	no	—
28	C₄×Dic₁₇₈	—	2	2	no	—
29	C₄×Dic₁₇₈	—	2	2	no	—
30	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
31	C₄×Dic₁₇₈	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
33	C₂×#13083553	—	2	2	no	—
34	C₂×C₄×Dic₈₉	—	2	2	no	—
35	#15667946	C₂×C₄	1	4	yes	—
36	C₄⋊#10623494	—	2	2	no	—
37	C₄⋊#10623494	C₂³	1	4	yes	—
38	#15667941	—	2	8	no	—
39	#15667937	C₂×C₄	1	8	yes	—
40	#15667922	—	2	8	no	—
41	C₂²×C₈×C₈₉	—	2	1	no	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	D₄	1	2	yes	—
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₈₉	—	2	1	no	—
47	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₈₉	D₄	1	2	yes	—
49	C₄×C₈×C₈₉	C₂³	1	2	yes	—
50	C₄×C₈×C₈₉	C₂³	1	2	yes	—
51	C₄×C₈×C₈₉	C₂³	1	2	yes	—
52	C₂×Dic₁₇₈	—	2	4	no	—
53	C₂×Dic₁₇₈	—	2	4	no	—
54	C₂×Dic₁₇₈	—	2	2	no	—
55	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
56	C₂²×Dic₈₉	—	2	2	no	—
57	C₂²×Dic₈₉	—	2	2	no	—
58	Dic₃₅₆	—	2	8	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	4	4	no	—
61	C₄⋊Dic₈₉	—	2	2	no	—
62	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
63	#13083553	—	2	4	no	—
64	#13083553	—	2	4	no	—
65	C₄×Dic₈₉	—	2	4	no	—
66	C₄×Dic₈₉	—	2	4	no	—
67	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	—	2	4	no	—
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	—	4	2	no	—
74	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
75	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
78	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₈₉	#41	1	1	yes	characteristic
81	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
82	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₈₉	#131	1	2	yes	—
89	C₂×C₄×C₈₉	#137	1	4	yes	—
90	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
93	C₂×Dic₈₉	—	2	4	no	—
94	C₂×Dic₈₉	—	2	4	no	—
95	C₂×Dic₈₉	—	4	4	no	—
96	C₂×Dic₈₉	—	4	4	no	—
97	Dic₁₇₈	—	2	4	no	—
98	Dic₁₇₈	—	2	4	no	—
99	Dic₁₇₈	—	4	4	no	—
100	C₈×C₈₉	—	2	2	no	—
101	C₈×C₈₉	—	2	4	no	—
102	C₈×C₈₉	—	4	2	no	—
103	C₈×C₈₉	C₂×M₁₆	1	4	yes	—
104	#10623494	—	4	4	no	—
105	C₂²×C₈₉	#436	1	2	yes	—
106	C₂²×C₈₉	#441	1	2	yes	—
107	C₂²×C₈₉	#444	1	2	yes	—
108	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
109	C₄×C₈₉	—	2	1	no	—
110	C₄×C₈₉	—	2	1	no	—
111	C₄×C₈₉	#436	1	2	yes	—
112	C₄×C₈₉	#441	1	2	yes	—
113	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
114	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
115	C₄×C₈₉	C₂×#137	1	2	yes	—
116	Dic₈₉	—	4	4	no	—
117	Dic₈₉	—	4	4	no	—
118	C₂×#1593	—	89	1	no	maximal, Sylow
119	C₂×C₈₉	#1593	1	4	yes	complemented
120	C₂×C₈₉	C₂×#436	1	1	yes	characteristic
121	C₂×C₈₉	C₂×#441	1	1	yes	characteristic
122	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
123	C₂×#446	—	89	2	no	—
124	C₂×C₄×Q₁₆	—	89	1	no	—
125	#1593	—	89	8	no	—
126	C₂×#334	—	89	1	no	—
127	C₂×Q₈⋊C₈	—	89	2	no	—
128	C₂×#322	D₈₉	1	1	yes	characteristic
129	C₈₉	C₂×#1593	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	—	89	1	no	—
131	C₂×C₄×Q₈	—	89	2	no	—
132	#446	—	89	8	no	—
133	#446	—	178	4	no	—
134	C₄×Q₁₆	—	89	4	no	—
135	C₄×Q₁₆	—	89	4	no	—
136	C₂×#107	—	89	1	no	—
137	C₂×C₄⋊C₈	—	89	2	no	—
138	C₂×C₄⋊C₈	—	178	2	no	—
139	C₂×#103	—	89	2	no	—
140	C₂×#97	—	178	2	no	—
141	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
144	#334	—	89	4	no	—
145	Q₈⋊C₈	—	89	8	no	—
146	#322	C₂×D₈₉	1	4	yes	—
147	C₂²×Q₈	—	89	2	no	—
148	C₂×Q₁₆	—	89	4	no	—
149	C₂×Q₁₆	—	178	4	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	178	2	no	—
152	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
154	C₂²×C₈	D₃₅₆	1	2	yes	—
155	C₄×Q₈	—	89	4	no	—
156	C₄×Q₈	—	89	4	no	—
157	C₄×Q₈	—	178	2	no	—
158	C₄×Q₈	—	178	2	no	—
159	C₂×C₄⋊C₄	—	89	2	no	—
160	C₂×C₄⋊C₄	—	178	2	no	—
161	C₂×C₄²	—	178	2	no	—
162	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
163	#107	—	89	4	no	—
164	C₄⋊C₈	—	89	4	no	—
165	C₄⋊C₈	—	178	8	no	—
166	C₄⋊C₈	—	178	2	no	—
167	#103	—	89	8	no	—
168	#97	—	178	8	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₄×D₈₉	1	2	yes	—
171	C₄×C₈	D₃₅₆	1	2	yes	—
172	C₄×C₈	C₂²×D₈₉	1	2	yes	—
173	C₄×C₈	C₂²×D₈₉	1	2	yes	—
174	C₄×C₈	C₂²×D₈₉	1	2	yes	—
175	C₂×Q₈	—	89	4	no	—
176	C₂×Q₈	—	178	4	no	—
177	C₂×Q₈	—	178	4	no	—
178	C₂×Q₈	—	178	2	no	—
179	C₂²×C₄	—	178	2	no	—
180	C₂²×C₄	—	178	2	no	—
181	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
183	C₂²×C₄	#13083574	1	1	yes	characteristic
184	Q₁₆	—	178	8	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	356	4	no	—
190	C₂×C₈	—	178	4	no	—
191	C₂×C₈	#13083557	1	4	yes	—
192	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
193	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
194	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
195	C₄⋊C₄	—	89	4	no	—
196	C₄⋊C₄	—	178	2	no	—
197	C₄⋊C₄	—	178	4	no	—
198	C₄⋊C₄	—	178	4	no	—
199	C₄²	—	178	4	no	—
200	C₄²	—	178	4	no	—
201	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
202	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
203	C₄²	#13083574	1	1	yes	characteristic
204	C₄²	C₂³×D₈₉	1	1	yes	characteristic
205	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
206	Q₈	—	356	4	no	—
207	Q₈	—	178	4	no	—
208	Q₈	—	178	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	356	4	no	—
211	C₂×C₄	—	356	4	no	—
212	C₂×C₄	—	178	4	no	—
213	C₂×C₄	—	178	4	no	—
214	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
215	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
216	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
217	C₂×C₄	C₂×#13083557	1	2	yes	—
218	C₂×C₄	#15668018	1	2	yes	—
219	C₂×C₄	#15668045	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
222	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
223	C₈	—	2	4	no	—
224	C₈	—	2	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	356	4	no	—
227	C₈	C₂×#13083557	1	4	yes	—
228	C₂²	#19151677	1	2	yes	—
229	C₂²	#19151706	1	2	yes	—
230	C₂²	#19151721	1	2	yes	—
231	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	356	4	no	—
235	C₄	—	356	4	no	—
236	C₄	#19151677	1	2	yes	—
237	C₄	#19151706	1	2	yes	—
238	C₄	C₂²×#13083557	1	1	yes	characteristic
239	C₄	C₂×#15668018	1	1	yes	characteristic
240	C₄	C₂×#15668045	1	2	yes	—
241	C₂	#26019162	1	4	yes	direct product factor
242	C₂	C₂×#19151677	1	1	yes	characteristic
243	C₂	C₂×#19151706	1	1	yes	characteristic
244	C₂	C₂×#19151721	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	C₂×#1593
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #26019162	C₂	direct (N × H)
119. C₂×C₈₉	#1593	semidirect
129. C₈₉	C₂×#1593	semidirect
241. C₂	#26019162	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×#19151728	2
C₂×C₄×Dic₃₅₆	2
#26019162	2
C₂×#19151349	2
C₂×#19151330	2
C₂×#322×C₈₉	2
C₂×#1593	89

Subgroups — C2×#26019162 #36524193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019162 #36524193