Subgroups — C₂×#26019619 #36536759

Order 22784.

702 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151880	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	#26019619	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151344	C₂	1	2	yes	maximal
6	C₂×#494×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
8	#19151880	—	2	4	no	—
9	#19151880	C₂²	1	8	yes	—
10	C₈×Dic₁₇₈	—	2	2	no	—
11	C₈×Dic₁₇₈	C₂²	1	4	yes	—
12	C₂×#15667943	—	2	2	no	—
13	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
15	C₂×#15667941	—	2	1	no	—
16	C₂×#15667935	—	2	2	no	—
17	C₂×C₈×Dic₈₉	—	2	1	no	—
18	#19151351	C₂²	1	4	yes	—
19	#19151344	C₂²	1	8	yes	—
20	C₂²×Q₁₆×C₈₉	—	2	1	no	—
21	C₂²×Q₁₆×C₈₉	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
23	#494×C₈₉	—	2	2	no	—
24	#494×C₈₉	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
27	C₂²×#10623494	—	2	1	no	—
28	C₂²×#10623494	—	2	2	no	—
29	C₄×Dic₁₇₈	—	2	1	no	—
30	C₄×Dic₁₇₈	—	2	1	no	—
31	C₄×Dic₁₇₈	D₄	1	2	yes	—
32	C₄×Dic₁₇₈	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
34	C₂×#13083553	—	2	1	no	—
35	C₂×C₄×Dic₈₉	—	2	1	no	—
36	#15667943	—	2	8	no	—
37	C₄⋊#10623494	—	2	2	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	C₂³	1	4	yes	—
40	C₄⋊#10623494	C₂³	1	2	yes	—
41	#15667941	—	2	4	no	—
42	#15667935	—	2	8	no	—
43	C₈×Dic₈₉	—	2	4	no	—
44	C₂²×Q₈×C₈₉	—	2	2	no	—
45	C₂×Q₁₆×C₈₉	—	2	4	no	—
46	C₂×Q₁₆×C₈₉	—	2	4	no	—
47	C₂×Q₁₆×C₈₉	—	4	4	no	—
48	C₂×Q₁₆×C₈₉	—	4	4	no	—
49	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₈₉	—	2	2	no	—
52	C₄⋊Q₈×C₈₉	—	2	2	no	—
53	C₄⋊Q₈×C₈₉	—	2	2	no	—
54	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
55	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₈₉	D₄	1	2	yes	—
59	C₄×C₈×C₈₉	C₂³	1	2	yes	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	—	2	2	no	—
62	C₂×Dic₁₇₈	Q₁₆	1	4	yes	complemented
63	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	—	2	4	no	—
68	C₂×#10623494	—	4	4	no	—
69	C₂×#10623494	—	4	2	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
72	#13083553	—	2	2	no	—
73	#13083553	—	2	2	no	—
74	C₄×Dic₈₉	—	2	2	no	—
75	C₄×Dic₈₉	—	2	2	no	—
76	C₂×Q₈×C₈₉	—	2	4	no	—
77	C₂×Q₈×C₈₉	—	4	2	no	—
78	C₂×Q₈×C₈₉	—	4	4	no	—
79	C₂×Q₈×C₈₉	—	4	4	no	—
80	C₂²×C₄×C₈₉	—	4	2	no	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
84	Q₁₆×C₈₉	—	4	8	no	—
85	Q₁₆×C₈₉	—	4	8	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
88	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₈₉	—	2	4	no	—
91	C₄⋊C₄×C₈₉	—	2	4	no	—
92	C₄⋊C₄×C₈₉	—	4	2	no	—
93	C₄⋊C₄×C₈₉	—	4	2	no	—
94	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₈₉	#41	1	1	yes	characteristic
97	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₂³×C₈₉	#121	1	1	yes	characteristic
99	Q₈×C₈₉	—	4	4	no	—
100	Q₈×C₈₉	—	4	4	no	—
101	Q₈×C₈₉	—	8	4	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	4	4	no	—
104	C₂×C₄×C₈₉	—	8	4	no	—
105	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₈₉	#135	1	2	yes	—
111	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₈₉	—	2	2	no	—
116	C₂×Dic₈₉	—	2	4	no	—
117	C₂×Dic₈₉	—	2	2	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	Dic₁₇₈	—	2	2	no	—
120	Dic₁₇₈	—	2	4	no	—
121	Dic₁₇₈	C₂×Q₁₆	1	4	yes	complemented
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
124	#10623494	—	4	4	no	—
125	#10623494	—	4	2	no	—
126	C₂²×C₈₉	#460	1	2	yes	—
127	C₂²×C₈₉	#467	1	2	yes	—
128	C₂²×C₈₉	#473	1	2	yes	—
129	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	8	4	no	—
133	C₄×C₈₉	#460	1	2	yes	—
134	C₄×C₈₉	#467	1	2	yes	—
135	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
136	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
137	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
138	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
139	Dic₈₉	—	2	4	no	—
140	Dic₈₉	—	4	2	no	—
141	C₂×#1670	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	#1670	1	4	yes	complemented
143	C₂×C₈₉	C₂×#460	1	1	yes	characteristic
144	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
145	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
146	C₂×#494	D₈₉	1	1	yes	characteristic
147	C₂×#475	—	89	2	no	—
148	C₂×C₈×Q₈	—	89	1	no	—
149	#1670	—	89	8	no	—
150	C₂×#335	—	89	1	no	—
151	C₂×#332	—	89	2	no	—
152	C₈₉	C₂×#1670	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₁₆	—	2	1	no	—
154	C₂²×Q₁₆	—	2	1	no	—
155	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
156	C₂×C₄×Q₈	—	89	1	no	—
157	#494	—	2	2	no	—
158	#494	C₂×D₈₉	1	4	yes	—
159	#475	—	89	8	no	—
160	#475	—	178	4	no	—
161	C₈×Q₈	—	89	4	no	—
162	C₈×Q₈	—	178	2	no	—
163	C₂×#106	—	178	2	no	—
164	C₂×C₄⋊C₈	—	89	2	no	—
165	C₂×C₄⋊C₈	—	89	1	no	—
166	C₂×C₄⋊C₈	—	178	1	no	—
167	C₂×#103	—	178	2	no	—
168	C₂×C₄×C₈	—	178	1	no	—
169	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
170	#335	—	89	4	no	—
171	#332	—	89	8	no	—
172	C₂²×Q₈	—	2	2	no	—
173	C₂²×Q₈	—	89	1	no	—
174	C₂×Q₁₆	—	2	4	no	—
175	C₂×Q₁₆	—	2	4	no	—
176	C₂×Q₁₆	—	4	4	no	—
177	C₂×Q₁₆	—	4	4	no	—
178	C₂²×C₈	—	178	2	no	—
179	C₂²×C₈	—	178	1	no	—
180	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	C₂²×D₈₉	1	4	yes	—
186	C₄×Q₈	—	89	2	no	—
187	C₄×Q₈	—	89	2	no	—
188	C₄×Q₈	—	178	1	no	—
189	C₄×Q₈	—	178	1	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	89	1	no	—
193	C₂×C₄⋊C₄	—	178	1	no	—
194	C₂×C₄²	—	178	1	no	—
195	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
196	#106	—	178	8	no	—
197	C₄⋊C₈	—	89	2	no	—
198	C₄⋊C₈	—	89	4	no	—
199	C₄⋊C₈	—	178	4	no	—
200	C₄⋊C₈	—	178	2	no	—
201	C₄⋊C₈	—	178	1	no	—
202	#103	—	178	8	no	—
203	C₄×C₈	—	178	4	no	—
204	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
205	C₄×C₈	C₂²×D₈₉	1	2	yes	—
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	4	4	no	—
209	C₂×Q₈	—	4	4	no	—
210	C₂×Q₈	—	89	4	no	—
211	C₂×Q₈	—	89	2	no	—
212	C₂×Q₈	—	178	1	no	—
213	C₂×Q₈	—	178	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	178	1	no	—
216	C₂²×C₄	—	178	1	no	—
217	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
219	C₂²×C₄	#13083573	1	1	yes	characteristic
220	Q₁₆	—	4	8	no	—
221	Q₁₆	—	4	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	356	4	no	—
224	C₂×C₈	—	356	2	no	—
225	C₂×C₈	—	178	4	no	—
226	C₂×C₈	—	178	2	no	—
227	C₂×C₈	#13083562	1	4	yes	—
228	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
229	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	89	2	no	—
235	C₄⋊C₄	—	178	1	no	—
236	C₄⋊C₄	—	178	2	no	—
237	C₄⋊C₄	—	178	2	no	—
238	C₄²	—	178	2	no	—
239	C₄²	—	178	2	no	—
240	C₄²	D₄×D₈₉	1	1	yes	characteristic
241	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
242	C₄²	#13083573	1	1	yes	characteristic
243	C₄²	C₂³×D₈₉	1	1	yes	characteristic
244	C₂³	#15667983	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	8	4	no	—
248	Q₈	—	89	4	no	—
249	Q₈	—	178	4	no	—
250	Q₈	—	178	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	—	178	2	no	—
256	C₂×C₄	—	178	2	no	—
257	C₂×C₄	—	178	4	no	—
258	C₂×C₄	#15667983	1	1	yes	characteristic
259	C₂×C₄	#15667983	1	1	yes	characteristic
260	C₂×C₄	#15667983	1	1	yes	characteristic
261	C₂×C₄	C₂×#13083562	1	2	yes	—
262	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
263	C₂×C₄	#15668035	1	2	yes	—
264	C₂×C₄	#15668038	1	2	yes	—
265	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
267	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	356	2	no	—
270	C₈	—	356	4	no	—
271	C₈	C₂×#13083562	1	4	yes	—
272	C₂²	#19151795	1	2	yes	—
273	C₂²	#19151836	1	2	yes	—
274	C₂²	#19151870	1	2	yes	—
275	C₂²	C₂×#15667983	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	4	no	—
279	C₄	—	356	2	no	—
280	C₄	—	178	4	no	—
281	C₄	#19151795	1	2	yes	—
282	C₄	#19151836	1	2	yes	—
283	C₄	C₂²×#13083562	1	1	yes	characteristic
284	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
285	C₄	C₂×#15668035	1	1	yes	characteristic
286	C₄	C₂×#15668038	1	1	yes	characteristic
287	C₂	#26019619	1	4	yes	direct product factor
288	C₂	C₂×#19151795	1	1	yes	characteristic
289	C₂	C₂×#19151836	1	1	yes	characteristic
290	C₂	C₂×#19151870	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#494×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#121
hypercenter	C₂×#494	D₈₉
hypocenter	C₈₉	C₂×#1670
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #26019619	C₂	direct (N × H)
62. C₂×Dic₁₇₈	Q₁₆	semidirect
121. Dic₁₇₈	C₂×Q₁₆	semidirect
142. C₂×C₈₉	#1670	semidirect
152. C₈₉	C₂×#1670	semidirect
287. C₂	#26019619	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×#19151880	2
C₂×C₈×Dic₁₇₈	2
#26019619	2
C₂×#19151351	2
C₂×#19151344	2
C₂×#494×C₈₉	2
C₂×#1670	89

Subgroups — C2×#26019619 #36536759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019619 #36536759