Subgroups — C₄×#15668042 #26029069

Order 11392.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668042	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151703	C₂	1	2	yes	maximal, complemented
4	#19151698	C₂	1	2	yes	maximal, complemented
5	C₄×#13083564	C₂	1	2	yes	maximal, complemented
6	C₄×#13083562	C₂	1	2	yes	maximal, complemented
7	#19151641	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151606	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151599	C₂	1	1	yes	maximal, characteristic
10	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	#15668042	C₄	1	8	yes	direct product factor
14	C₂×#13083564	C₂²	1	2	yes	—
15	C₂×#13083562	C₂²	1	2	yes	—
16	C₂×#13083558	C₂²	1	1	yes	characteristic
17	C₂×#13083558	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₇₈	—	2	1	no	—
19	C₄×Dic₁₇₈	—	2	1	no	—
20	C₄×Dic₁₇₈	C₂²	1	2	yes	—
21	C₄×Dic₁₇₈	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
23	C₂×#13083553	—	2	1	no	—
24	C₂×C₄×Dic₈₉	—	2	1	no	—
25	#15667945	C₂²	1	2	yes	—
26	#15667943	C₂²	1	2	yes	—
27	#15667936	C₂²	1	2	yes	—
28	#15667935	C₂²	1	2	yes	—
29	#15667935	C₂²	1	2	yes	—
30	#15667933	C₂²	1	2	yes	—
31	#15667923	C₂²	1	2	yes	—
32	C₄×#10623494	C₂²	1	2	yes	—
33	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₈₉	—	2	1	no	—
37	C₄×D₄×C₈₉	—	2	1	no	—
38	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	#117×C₈₉	—	2	1	no	—
41	#117×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	—	2	1	no	—
43	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
44	C₂×Dic₁₇₈	—	2	1	no	—
45	C₂×Dic₁₇₈	—	2	2	no	—
46	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₁₇₈	C₂³	1	2	yes	—
48	C₂²×Dic₈₉	—	2	1	no	—
49	C₂²×Dic₈₉	—	2	1	no	—
50	#13083564	C₂×C₄	1	8	yes	complemented
51	#13083562	C₂×C₄	1	8	yes	complemented
52	#13083558	—	2	2	no	—
53	#13083558	C₂×C₄	1	4	yes	complemented
54	C₂×#10623494	C₂³	1	2	yes	—
55	C₂×#10623494	C₂³	1	2	yes	—
56	C₄⋊Dic₈₉	—	2	1	no	—
57	C₄⋊Dic₈₉	C₂³	1	2	yes	—
58	#13083553	—	2	2	no	—
59	#13083553	—	2	2	no	—
60	C₄×Dic₈₉	—	2	2	no	—
61	C₄×Dic₈₉	—	2	2	no	—
62	#41×C₈₉	—	2	2	no	—
63	#41×C₈₉	C₂×C₄	1	4	yes	—
64	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₈₉	—	2	1	no	—
66	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	—	2	1	no	—
68	C₂²×C₄×C₈₉	—	2	1	no	—
69	C₂²×C₄×C₈₉	—	2	1	no	—
70	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₈₉	—	2	1	no	—
74	C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₈₉	—	2	1	no	—
78	C₂²⋊C₄×C₈₉	—	2	1	no	—
79	C₂²⋊C₄×C₈₉	—	2	1	no	—
80	C₄²×C₈₉	—	2	1	no	—
81	C₄²×C₈₉	—	2	1	no	—
82	C₄²×C₈₉	—	2	1	no	—
83	C₄²×C₈₉	D₄	1	1	yes	characteristic
84	C₄²×C₈₉	D₄	1	1	yes	characteristic
85	C₄²×C₈₉	C₂³	1	1	yes	characteristic
86	C₄²×C₈₉	C₂³	1	1	yes	characteristic
87	C₂³×C₈₉	—	2	1	no	—
88	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₈₉	—	2	1	no	—
90	Q₈×C₈₉	C₂²×C₄	1	2	yes	—
91	D₄×C₈₉	—	2	4	no	—
92	D₄×C₈₉	—	2	1	no	—
93	D₄×C₈₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	4	1	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	—	4	2	no	—
102	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#41	1	2	yes	—
109	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₂×Dic₈₉	—	2	4	no	—
111	C₂×Dic₈₉	—	2	2	no	—
112	C₂×Dic₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	Dic₁₇₈	—	2	4	no	—
115	Dic₁₇₈	—	2	2	no	—
116	Dic₁₇₈	C₂²×C₄	1	4	yes	—
117	#10623494	—	2	2	no	—
118	#10623494	C₂²×C₄	1	4	yes	—
119	C₂²×C₈₉	—	2	1	no	—
120	C₂²×C₈₉	—	2	1	no	—
121	C₂²×C₈₉	—	2	2	no	—
122	C₂²×C₈₉	—	4	1	no	—
123	C₂²×C₈₉	—	4	1	no	—
124	C₂²×C₈₉	C₄×D₄	1	2	yes	—
125	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	4	1	no	—
131	C₄×C₈₉	—	4	1	no	—
132	C₄×C₈₉	C₄×D₄	1	2	yes	—
133	C₄×C₈₉	#137	1	2	yes	complemented
134	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
136	Dic₈₉	—	2	4	no	—
137	Dic₈₉	—	4	2	no	—
138	C₂×C₈₉	—	2	2	no	—
139	C₂×C₈₉	—	4	2	no	—
140	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
142	C₂×C₈₉	#575	1	1	yes	characteristic
143	C₄×#137	—	89	1	no	maximal, Sylow
144	C₈₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
145	C₂×#137	—	89	1	no	—
146	C₄×#41	D₈₉	1	1	yes	characteristic
147	C₂×C₄×Q₈	—	89	1	no	—
148	#441	—	89	2	no	—
149	#440	—	89	2	no	—
150	C₄×Q₁₆	—	89	2	no	—
151	C₄×SD₁₆	—	89	2	no	—
152	#428	—	89	1	no	—
153	#419	—	89	1	no	—
154	#417	—	89	1	no	—
155	C₄×M₁₆	—	89	1	no	—
156	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
157	C₂²×Q₈	—	89	1	no	—
158	#137	—	89	8	no	—
159	C₂×Q₁₆	—	89	2	no	—
160	C₂×SD₁₆	—	89	2	no	—
161	C₂×M₁₆	—	89	1	no	—
162	C₂×M₁₆	—	89	1	no	—
163	C₄×Q₈	—	89	2	no	—
164	C₄×Q₈	—	89	2	no	—
165	C₄×Q₈	—	178	1	no	—
166	C₄×Q₈	—	178	1	no	—
167	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
168	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
172	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	C₂×D₈₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	89	1	no	—
176	C₂×C₄⋊C₄	—	178	1	no	—
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	178	1	no	—
179	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
180	#107	—	89	2	no	—
181	#106	—	89	2	no	—
182	#103	—	89	2	no	—
183	#103	—	89	2	no	—
184	#103	—	89	2	no	—
185	#102	—	89	2	no	—
186	#97	—	89	2	no	—
187	C₄×C₈	—	89	2	no	—
188	#41	—	2	2	no	—
189	#41	C₄×D₈₉	1	4	yes	—
190	C₂×Q₈	—	89	4	no	—
191	C₂×Q₈	—	89	2	no	—
192	C₂×Q₈	—	178	1	no	—
193	C₂×Q₈	—	178	2	no	—
194	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	178	1	no	—
201	C₂²×C₄	—	178	1	no	—
202	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
205	Q₁₆	—	89	8	no	—
206	SD₁₆	—	89	8	no	—
207	M₁₆	—	89	4	no	—
208	M₁₆	—	178	2	no	—
209	C₂×C₈	—	89	2	no	—
210	C₂×C₈	—	89	2	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	89	2	no	—
214	C₄⋊C₄	—	178	1	no	—
215	C₄⋊C₄	—	178	2	no	—
216	C₄⋊C₄	—	178	2	no	—
217	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	178	2	no	—
226	C₄²	—	178	2	no	—
227	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
229	C₄²	C₂²×D₈₉	1	1	yes	characteristic
230	C₄²	C₂²×D₈₉	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
233	Q₈	—	2	1	no	—
234	Q₈	—	89	4	no	—
235	Q₈	—	178	4	no	—
236	Q₈	—	178	2	no	—
237	Q₈	C₂×C₄×D₈₉	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂×C₄×D₈₉	1	2	yes	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	356	2	no	—
250	C₂×C₄	—	178	2	no	—
251	C₂×C₄	—	178	4	no	—
252	C₂×C₄	—	178	2	no	—
253	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
254	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₂×C₄	#13083574	1	2	yes	—
260	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
261	C₈	—	89	4	no	—
262	C₈	—	178	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₄×C₂²⋊D₈₉	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	356	2	no	—
277	C₄	—	178	4	no	—
278	C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
279	C₄	#15668042	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
281	C₄	C₂×#13083574	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
285	C₂	C₂×#15668042	1	1	yes	characteristic
286	C₂	#19152396	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₄×#41×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#41×C₈₉	C₂²
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₄×#41	D₈₉
hypocenter	C₈₉	C₄×#137
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₁₇₈	C₂	semidirect
3. #19151703	C₂	semidirect
4. #19151698	C₂	semidirect
5. C₄×#13083564	C₂	semidirect
6. C₄×#13083562	C₂	semidirect
7. #19151641	C₂	semidirect
8. #19151606	C₂	semidirect
10. C₄×#13083558	C₂	semidirect
13. #15668042	C₄	direct (N × H)
46. C₂×Dic₁₇₈	C₂×C₄	semidirect
50. #13083564	C₂×C₄	semidirect
51. #13083562	C₂×C₄	semidirect
53. #13083558	C₂×C₄	semidirect
133. C₄×C₈₉	#137	semidirect
144. C₈₉	C₄×#137	semidirect
279. C₄	#15668042	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×#15668042	2
C₂×C₄×Dic₁₇₈	2
#19151703	2
#19151698	2
C₄×#13083564	2
C₄×#13083562	2
#19151641	2
#19151606	2
#19151599	2
C₄×#13083558	2
C₄×#41×C₈₉	2
C₄×#137	89

Subgroups — C4×#15668042 #26029069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15668042 #26029069