Subgroups — C₂×#26018230 #36464226

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
3	#26018230	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
6	C₂²×#13083549	—	2	1	no	—
7	C₈×Dic₁₇₈	—	2	1	no	—
8	C₈×Dic₁₇₈	—	2	1	no	—
9	C₈×Dic₁₇₈	C₄	1	2	yes	—
10	C₈×Dic₁₇₈	C₂²	1	2	yes	—
11	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
12	C₂×#15667941	—	2	1	no	—
13	C₂×C₈×Dic₈₉	—	2	1	no	—
14	C₄×#13083549	—	2	1	no	—
15	C₄×#13083549	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₈₉	—	2	1	no	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₈₉	—	2	1	no	—
19	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂²×Dic₁₇₈	C₈	1	1	yes	characteristic
21	C₂²×#10623494	—	2	1	no	—
22	C₄×Dic₁₇₈	—	2	1	no	—
23	C₄×Dic₁₇₈	—	2	1	no	—
24	C₄×Dic₁₇₈	C₈	1	2	yes	—
25	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
27	C₂×#13083553	—	2	1	no	—
28	C₂×C₄×Dic₈₉	—	2	1	no	—
29	C₂×#13083549	—	2	4	no	—
30	C₂×#13083549	—	2	2	no	—
31	C₄⋊#10623494	—	2	1	no	—
32	C₄⋊#10623494	C₂×C₄	1	2	yes	—
33	#15667941	—	2	2	no	—
34	#15667941	—	2	2	no	—
35	C₈×Dic₈₉	—	2	2	no	—
36	C₈×Dic₈₉	—	2	2	no	—
37	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₈₉	—	2	2	no	—
41	C₂×C₁₆×C₈₉	—	4	2	no	—
42	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₇₈	—	2	1	no	—
47	C₂×Dic₁₇₈	—	2	2	no	—
48	C₂×Dic₁₇₈	C₁₆	1	4	yes	complemented
49	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
50	C₂²×Dic₈₉	—	2	1	no	—
51	C₂²×Dic₈₉	—	2	1	no	—
52	C₂×#10623494	—	2	2	no	—
53	C₂×#10623494	—	4	2	no	—
54	C₄⋊Dic₈₉	—	2	1	no	—
55	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
56	#13083553	—	2	2	no	—
57	#13083553	—	2	2	no	—
58	C₄×Dic₈₉	—	2	2	no	—
59	C₄×Dic₈₉	—	2	2	no	—
60	#13083549	—	2	4	no	—
61	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
64	C₂×C₈×C₈₉	—	2	2	no	—
65	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
68	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
69	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
74	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₁₆×C₈₉	—	4	2	no	—
76	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#103	1	2	yes	—
82	C₂×C₄×C₈₉	#104	1	2	yes	—
83	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
84	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
88	C₂×Dic₈₉	—	2	4	no	—
89	C₂×Dic₈₉	—	2	2	no	—
90	C₂×Dic₈₉	—	2	2	no	—
91	C₂×Dic₈₉	—	4	2	no	—
92	Dic₁₇₈	—	2	4	no	—
93	Dic₁₇₈	—	2	2	no	—
94	Dic₁₇₈	C₂×C₁₆	1	4	yes	complemented
95	C₈×C₈₉	—	2	1	no	—
96	C₈×C₈₉	—	2	1	no	—
97	C₈×C₈₉	#103	1	2	yes	—
98	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
100	#10623494	—	4	2	no	—
101	C₂²×C₈₉	Q₈⋊C₈	1	2	yes	—
102	C₂²×C₈₉	C₂²⋊C₁₆	1	2	yes	—
103	C₂²×C₈₉	#350	1	2	yes	—
104	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₈₉	—	2	1	no	—
106	C₄×C₈₉	—	2	1	no	—
107	C₄×C₈₉	Q₈⋊C₈	1	2	yes	—
108	C₄×C₈₉	C₂²⋊C₁₆	1	2	yes	—
109	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
110	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
111	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
112	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
113	Dic₈₉	—	2	4	no	—
114	Dic₈₉	—	4	2	no	—
115	C₂×Q₈⋊C₁₆	—	89	1	no	maximal, Sylow
116	C₂×C₈₉	Q₈⋊C₁₆	1	4	yes	complemented
117	C₂×C₈₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
118	C₂×C₈₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₈₉	C₂×#350	1	1	yes	characteristic
120	C₂×C₈×Q₈	—	89	1	no	—
121	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
122	C₂×C₄×C₁₆	—	89	1	no	—
123	Q₈⋊C₁₆	—	89	4	no	—
124	C₈₉	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄×Q₈	—	89	1	no	—
126	C₂²×C₁₆	—	2	1	no	—
127	C₂²×C₁₆	—	178	1	no	—
128	C₈×Q₈	—	89	2	no	—
129	C₈×Q₈	—	89	2	no	—
130	C₈×Q₈	—	178	1	no	—
131	C₈×Q₈	—	178	1	no	—
132	C₂×C₄⋊C₈	—	89	1	no	—
133	C₂×C₄⋊C₈	—	178	1	no	—
134	C₂×C₄×C₈	—	178	1	no	—
135	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
136	C₄⋊C₁₆	—	2	1	no	—
137	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
138	C₄×C₁₆	—	89	2	no	—
139	C₄×C₁₆	—	178	1	no	—
140	C₂²×Q₈	—	89	1	no	—
141	C₂²×C₈	—	178	1	no	—
142	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
144	C₄×Q₈	—	89	2	no	—
145	C₄×Q₈	—	89	2	no	—
146	C₄×Q₈	—	178	1	no	—
147	C₄×Q₈	—	178	1	no	—
148	C₂×C₄⋊C₄	—	89	1	no	—
149	C₂×C₄⋊C₄	—	178	1	no	—
150	C₂×C₄²	—	178	1	no	—
151	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	4	2	no	—
154	C₂×C₁₆	—	178	4	no	—
155	C₂×C₁₆	—	178	2	no	—
156	C₄⋊C₈	—	89	2	no	—
157	C₄⋊C₈	—	178	2	no	—
158	C₄⋊C₈	—	178	2	no	—
159	C₄⋊C₈	—	178	1	no	—
160	C₄×C₈	—	178	2	no	—
161	C₄×C₈	—	178	2	no	—
162	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
163	C₄×C₈	D₃₅₆	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
166	C₂×Q₈	—	89	2	no	—
167	C₂×Q₈	—	89	4	no	—
168	C₂×Q₈	—	178	1	no	—
169	C₂×Q₈	—	178	2	no	—
170	C₂²×C₄	—	178	1	no	—
171	C₂²×C₄	—	178	1	no	—
172	C₂²×C₄	#13083551	1	1	yes	characteristic
173	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
174	C₂²×C₄	#13083557	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	356	2	no	—
177	C₂×C₈	—	178	2	no	—
178	C₂×C₈	#13083551	1	1	yes	characteristic
179	C₂×C₈	#13083551	1	1	yes	characteristic
180	C₂×C₈	#13083562	1	2	yes	—
181	C₂×C₈	#13083564	1	2	yes	—
182	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
183	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
184	C₄⋊C₄	—	89	2	no	—
185	C₄⋊C₄	—	178	2	no	—
186	C₄⋊C₄	—	178	1	no	—
187	C₄⋊C₄	—	178	2	no	—
188	C₄²	—	178	2	no	—
189	C₄²	—	178	2	no	—
190	C₄²	#13083551	1	1	yes	characteristic
191	C₄²	C₈×D₈₉	1	1	yes	characteristic
192	C₄²	#13083557	1	1	yes	characteristic
193	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₁₆	—	178	4	no	—
196	C₂³	#15667924	1	1	yes	characteristic
197	Q₈	—	89	4	no	—
198	Q₈	—	178	4	no	—
199	Q₈	—	178	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	356	2	no	—
202	C₂×C₄	—	178	2	no	—
203	C₂×C₄	—	178	4	no	—
204	C₂×C₄	—	178	2	no	—
205	C₂×C₄	#15667924	1	1	yes	characteristic
206	C₂×C₄	#15667924	1	1	yes	characteristic
207	C₂×C₄	#15667924	1	1	yes	characteristic
208	C₂×C₄	#15667935	1	2	yes	—
209	C₂×C₄	#15667938	1	2	yes	—
210	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
211	C₂×C₄	#15667951	1	2	yes	—
212	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
213	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
214	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	356	2	no	—
218	C₈	#15667935	1	2	yes	—
219	C₈	C₂×#13083562	1	1	yes	characteristic
220	C₈	C₂×#13083564	1	1	yes	characteristic
221	C₂²	#19151328	1	2	yes	—
222	C₂²	#19151377	1	2	yes	—
223	C₂²	#19151383	1	2	yes	—
224	C₂²	C₂×#15667924	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	356	2	no	—
228	C₄	—	178	4	no	—
229	C₄	#19151328	1	2	yes	—
230	C₄	#19151377	1	2	yes	—
231	C₄	C₂×#15667935	1	1	yes	characteristic
232	C₄	C₂×#15667938	1	1	yes	characteristic
233	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
234	C₄	C₂×#15667951	1	1	yes	characteristic
235	C₂	#26018230	1	4	yes	direct product factor
236	C₂	C₂×#19151328	1	1	yes	characteristic
237	C₂	C₂×#19151377	1	1	yes	characteristic
238	C₂	C₂×#19151383	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #26018230	C₂	direct (N × H)
48. C₂×Dic₁₇₈	C₁₆	semidirect
94. Dic₁₇₈	C₂×C₁₆	semidirect
116. C₂×C₈₉	Q₈⋊C₁₆	semidirect
124. C₈₉	C₂×Q₈⋊C₁₆	semidirect
235. C₂	#26018230	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×C₈×Dic₁₇₈	2
C₂×C₄×#13083549	2
#26018230	2
C₂×C₄⋊C₁₆×C₈₉	2
C₂×Q₈⋊C₁₆	89

Subgroups — C2×#26018230 #36464226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018230 #36464226