Subgroups — C₄×#63393 #11783744

Order 1152.

618 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63393	C₂	1	2	yes	maximal
3	C₄×#3676	C₂	1	2	yes	maximal, complemented
4	#10588203	C₂	1	2	yes	maximal, complemented
5	C₄×#834	—	3	1	no	maximal
6	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₄	1	1	yes	characteristic, complemented
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₄×D₄×C₉	—	2	2	no	—
10	C₄×D₄×C₉	—	2	2	no	—
11	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
12	C₂×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic, complemented
13	C₂×C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉	C₄	1	1	yes	characteristic, complemented
15	C₂×#3676	C₂²	1	4	yes	—
16	C₂×C₄×Dic₉	—	2	2	no	—
17	#63393	C₄	1	8	yes	direct product factor
18	C₉⋊#95	—	2	2	no	—
19	C₂×C₃×C₄×D₄	D₃	1	1	yes	characteristic
20	C₂×#834	—	3	2	no	—
21	C₄×#217	—	3	2	no	—
22	#4526	—	3	2	no	—
23	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂⁴×C₉	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₉	—	2	2	no	—
26	C₂×D₄×C₉	—	2	2	no	—
27	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	—	4	2	no	—
31	C₂²×C₄×C₉	—	4	2	no	—
32	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	D₄	1	2	yes	—
38	C₂²×C₄×C₉	D₄	1	2	yes	—
39	C₂²×Dic₉	—	2	4	no	—
40	C₉×C₄⋊C₄	—	2	2	no	—
41	C₉×C₂²⋊C₄	—	2	2	no	—
42	C₉×C₂²⋊C₄	—	2	2	no	—
43	C₄²×C₉	—	2	1	no	—
44	C₄²×C₉	—	2	1	no	—
45	#3676	—	2	4	no	—
46	#3676	C₂×C₄	1	8	yes	complemented
47	C₄×Dic₉	—	4	4	no	—
48	C₄×#99	—	9	1	no	Sylow
49	C₂²×C₃×D₄	C₂×D₃	1	1	yes	characteristic
50	C₂³×C₃×C₄	Dic₃	1	1	yes	characteristic
51	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
52	C₃×C₄×D₄	—	2	2	no	—
53	C₃×C₄×D₄	—	2	2	no	—
54	C₂×C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
55	C₂×C₃×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
56	C₂×C₃×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
58	C₂×#217	—	3	4	no	—
59	C₂×C₄×Dic₃	—	6	2	no	—
60	#834	—	3	8	no	—
61	C₃⋊#95	—	6	2	no	—
62	C₂³×C₉	—	2	1	no	—
63	C₂³×C₉	—	2	1	no	—
64	C₂³×C₉	—	4	1	no	—
65	C₂³×C₉	—	4	1	no	—
66	C₂³×C₉	—	4	1	no	—
67	C₂³×C₉	—	4	1	no	—
68	C₂³×C₉	C₄²	1	2	yes	complemented
69	C₂³×C₉	C₂²×C₄	1	2	yes	—
70	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₉	C₂×D₄	1	2	yes	—
72	C₂³×C₉	#41	1	2	yes	—
73	D₄×C₉	—	2	8	no	—
74	C₂×C₄×C₉	—	2	1	no	—
75	C₂×C₄×C₉	—	2	1	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	—	4	1	no	—
81	C₂×C₄×C₉	—	4	2	no	—
82	C₂×C₄×C₉	—	4	2	no	—
83	C₂×C₄×C₉	—	4	2	no	—
84	C₂×C₄×C₉	—	4	1	no	—
85	C₂×C₄×C₉	—	4	1	no	—
86	C₂×C₄×C₉	C₄²	1	2	yes	complemented
87	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₉	—	2	8	no	—
91	C₂×Dic₉	—	4	4	no	—
92	C₂×Dic₉	—	4	4	no	—
93	C₂×C₄×D₄	D₉	1	1	yes	characteristic
94	C₂×#99	—	9	2	no	—
95	C₄×C₂²⋊C₄	—	9	2	no	—
96	#342	—	9	2	no	—
97	C₂⁴×C₃	C₂×Dic₃	1	1	yes	characteristic
98	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₃×D₄	—	2	2	no	—
100	C₂×C₃×D₄	—	2	2	no	—
101	C₂×C₃×D₄	C₄×D₃	1	4	yes	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	—	4	2	no	—
105	C₂²×C₃×C₄	—	4	2	no	—
106	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	#67	1	2	yes	—
112	C₂²×C₃×C₄	#67	1	2	yes	—
113	C₂²×Dic₃	—	6	4	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₂²⋊C₄	—	2	2	no	—
116	C₃×C₂²⋊C₄	—	2	2	no	—
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	—	2	1	no	—
119	#217	—	3	8	no	—
120	#217	—	6	4	no	—
121	C₄×Dic₃	—	12	4	no	—
122	C₂²×C₉	—	2	2	no	—
123	C₂²×C₉	—	2	1	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	—	2	1	no	—
126	C₂²×C₉	—	2	2	no	—
127	C₂²×C₉	—	4	2	no	—
128	C₂²×C₉	—	4	2	no	—
129	C₂²×C₉	—	4	2	no	—
130	C₂²×C₉	—	4	2	no	—
131	C₂²×C₉	—	4	1	no	—
132	C₂²×C₉	—	4	1	no	—
133	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
134	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₉	#117	1	1	yes	characteristic
136	C₂²×C₉	C₄×D₄	1	4	yes	—
137	C₄×C₉	—	2	2	no	—
138	C₄×C₉	—	2	1	no	—
139	C₄×C₉	—	2	2	no	—
140	C₄×C₉	—	4	1	no	—
141	C₄×C₉	—	4	1	no	—
142	C₄×C₉	#99	1	2	yes	complemented
143	Dic₉	—	4	8	no	—
144	C₂²×D₄	C₂×D₉	1	1	yes	characteristic
145	C₂³×C₄	Dic₉	1	1	yes	characteristic, complemented
146	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
147	C₄×D₄	—	2	2	no	—
148	C₄×D₄	—	2	2	no	—
149	C₂×C₄⋊C₄	Dic₉	1	1	yes	characteristic, complemented
150	C₂×C₂²⋊C₄	—	9	4	no	—
151	C₂×C₂²⋊C₄	Dic₉	1	1	yes	characteristic, complemented
152	C₂×C₂²⋊C₄	C₂×D₉	1	1	yes	characteristic
153	C₂×C₄²	—	18	2	no	—
154	C₂×C₄²	Dic₉	1	1	yes	characteristic, complemented
155	#99	—	9	8	no	—
156	#95	—	18	2	no	—
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	—	2	1	no	—
159	C₂³×C₃	—	4	1	no	—
160	C₂³×C₃	—	4	1	no	—
161	C₂³×C₃	—	4	1	no	—
162	C₂³×C₃	—	4	1	no	—
163	C₂³×C₃	C₄×Dic₃	1	2	yes	—
164	C₂³×C₃	C₂×C₄×D₃	1	2	yes	—
165	C₂³×C₃	#235	1	2	yes	—
166	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
167	C₂³×C₃	C₂×#67	1	2	yes	—
168	C₃×D₄	—	2	8	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	4	1	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
182	C₂×C₃×C₄	#217	1	1	yes	characteristic
183	C₂×C₃×C₄	#217	1	2	yes	—
184	C₂×C₃×C₄	#217	1	1	yes	characteristic
185	C₂×Dic₃	—	6	8	no	—
186	C₂×Dic₃	—	12	4	no	—
187	C₂×Dic₃	—	12	4	no	—
188	C₂×C₉	—	2	4	no	—
189	C₂×C₉	—	2	1	no	—
190	C₂×C₉	—	2	1	no	—
191	C₂×C₉	—	4	2	no	—
192	C₂×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₉	C₂×#99	1	1	yes	characteristic
194	C₂×C₉	#410	1	1	yes	characteristic
195	C₂⁴	C₂×Dic₉	1	1	yes	characteristic
196	C₂⁴	C₂²×D₉	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	C₄×D₉	1	4	yes	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	18	4	no	—
205	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
210	C₂²×C₄	#611	1	2	yes	—
211	C₂²×C₄	#611	1	2	yes	—
212	C₄⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	9	8	no	—
216	C₂²⋊C₄	—	18	4	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	36	4	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	4	2	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	—	4	2	no	—
229	C₂²×C₃	—	4	1	no	—
230	C₂²×C₃	—	4	1	no	—
231	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
232	C₂²×C₃	#926	1	1	yes	characteristic
233	C₂²×C₃	C₄×#67	1	4	yes	—
234	C₂²×C₃	C₂×#217	1	1	yes	characteristic
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	4	1	no	—
239	C₃×C₄	—	4	1	no	—
240	C₃×C₄	#834	1	2	yes	—
241	Dic₃	—	12	8	no	—
242	C₉	C₄×#99	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	4	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	1	no	—
249	C₂³	C₄×Dic₉	1	2	yes	complemented
250	C₂³	C₂×C₄×D₉	1	2	yes	—
251	C₂³	#3697	1	2	yes	—
252	C₂³	C₂²×Dic₉	1	1	yes	characteristic
253	C₂³	C₂×#611	1	2	yes	—
254	D₄	—	2	8	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	18	8	no	—
268	C₂×C₄	—	36	4	no	—
269	C₂×C₄	—	36	4	no	—
270	C₂×C₄	C₄×Dic₉	1	2	yes	complemented
271	C₂×C₄	#3676	1	1	yes	characteristic
272	C₂×C₄	#3676	1	2	yes	—
273	C₂×C₄	#3676	1	1	yes	characteristic
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	4	2	no	—
278	C₂×C₃	C₄×#217	1	1	yes	characteristic
279	C₂×C₃	C₂×#834	1	1	yes	characteristic
280	C₂×C₃	#5240	1	1	yes	characteristic
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	C₂×C₄×Dic₉	1	1	yes	characteristic
293	C₂²	#63488	1	1	yes	characteristic
294	C₂²	C₄×#611	1	4	yes	—
295	C₂²	C₂×#3676	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	36	8	no	—
302	C₄	#63393	1	2	yes	direct product factor
303	C₃	C₄×#834	1	1	yes	characteristic
304	C₂	—	2	4	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	4	2	no	—
308	C₂	C₄×#3676	1	1	yes	characteristic
309	C₂	C₂×#63393	1	1	yes	characteristic
310	C₂	#10588920	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3676
commutator	C₂²×C₉	C₂×C₄²
Frattini	C₂⁴×C₃	C₂²×D₃
Fitting	C₂×C₄×D₄×C₉	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₂²×C₃×D₄	C₂×D₃
cosocle	C₂⁴×C₉	C₂³
hypercenter	C₂×C₄×D₄	D₉
hypocenter	C₉	C₄×#99
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
3. C₄×#3676	C₂	semidirect
4. #10588203	C₂	semidirect
7. C₂³×C₄×C₉	C₄	semidirect
11. C₂×C₉×C₄⋊C₄	C₄	semidirect
12. C₂×C₉×C₂²⋊C₄	C₄	semidirect
14. C₂×C₄²×C₉	C₄	semidirect
17. #63393	C₄	direct (N × H)
46. #3676	C₂×C₄	semidirect
68. C₂³×C₉	C₄²	semidirect
86. C₂×C₄×C₉	C₄²	semidirect
142. C₄×C₉	#99	semidirect
145. C₂³×C₄	Dic₉	semidirect
149. C₂×C₄⋊C₄	Dic₉	semidirect
151. C₂×C₂²⋊C₄	Dic₉	semidirect
154. C₂×C₄²	Dic₉	semidirect
242. C₉	C₄×#99	semidirect
249. C₂³	C₄×Dic₉	semidirect
270. C₂×C₄	C₄×Dic₉	semidirect
302. C₄	#63393	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×D₄×C₉	2
C₂×#63393	2
C₄×#3676	2
#10588203	2
C₄×#834	3

Subgroups — C4×#63393 #11783744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#63393 #11783744