Subgroups — C₄×#15689231 #26214783

Order 12032.

1086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689231	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
3	C₄²×Dic₉₄	C₂	1	2	yes	maximal
4	#20243285	C₂	1	2	yes	maximal
5	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
6	#20243219	C₂	1	4	yes	maximal
7	C₄×#13093949	C₂	1	2	yes	maximal
8	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
11	C₂×C₄×Dic₉₄	C₂²	1	4	yes	—
12	#15689231	C₄	1	16	yes	direct product factor
13	C₄×Dic₁₈₈	C₄	1	8	yes	complemented
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093956	C₂²	1	2	yes	—
16	C₂×#13093949	C₂²	1	4	yes	—
17	C₂×#13093949	C₂²	1	2	yes	—
18	C₄×Dic₄₇⋊C₄	—	2	2	no	—
19	C₄×Dic₄₇⋊C₄	—	2	4	no	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
22	C₄²×Dic₄₇	—	2	2	no	—
23	#15688886	C₂²	1	4	yes	—
24	#15688886	C₂²	1	2	yes	—
25	C₂×#97×C₄₇	C₂²	1	2	yes	—
26	C₂×#97×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄³×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×Dic₉₄	C₂³	1	2	yes	—
31	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
32	C₂×Dic₁₈₈	C₂×C₄	1	8	yes	—
33	C₄×Dic₉₄	—	2	4	no	—
34	C₄×Dic₉₄	—	2	8	no	—
35	C₄×Dic₉₄	C₂×C₄	1	8	yes	—
36	C₄×Dic₉₄	C₂×C₄	1	16	yes	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	C₂×Dic₄₇⋊C₄	—	2	2	no	—
40	C₂×Dic₄₇⋊C₄	—	2	4	no	—
41	C₂×C₄×Dic₄₇	—	2	4	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	C₂×C₄×Dic₄₇	—	2	4	no	—
44	C₂×C₄×Dic₄₇	—	2	2	no	—
45	#13093958	C₂×C₄	1	4	yes	—
46	#13093956	C₂×C₄	1	8	yes	—
47	#13093949	C₂×C₄	1	8	yes	—
48	#13093949	C₂×C₄	1	16	yes	—
49	C₄₇⋊#95	—	2	2	no	—
50	C₄₇⋊#95	—	2	4	no	—
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	D₄	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
58	#97×C₄₇	—	2	4	no	—
59	#97×C₄₇	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₄₇	—	2	2	no	—
61	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂×Dic₉₄	—	2	2	no	—
70	C₂×Dic₉₄	—	2	4	no	—
71	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
72	C₂×Dic₉₄	C₂²×C₄	1	8	yes	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
84	C₄²×C₄₇	#41	1	2	yes	—
85	C₄²×C₄₇	#41	1	4	yes	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
89	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
90	Dic₄₇⋊C₄	—	2	16	no	—
91	Dic₄₇⋊C₄	—	2	8	no	—
92	C₄×Dic₄₇	—	2	16	no	—
93	C₄×Dic₄₇	—	2	8	no	—
94	C₄×Dic₄₇	—	4	8	no	—
95	C₄×Dic₄₇	—	4	4	no	—
96	Dic₁₈₈	C₄²	1	16	yes	complemented
97	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	#137	1	4	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	8	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	Dic₉₄	—	2	4	no	—
117	Dic₉₄	—	2	8	no	—
118	Dic₉₄	C₂×C₄²	1	8	yes	—
119	C₈×C₄₇	—	2	4	no	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	C₂×C₄²	1	4	yes	—
122	C₄×#441	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
125	C₂²×C₄₇	C₂×#137	1	2	yes	—
126	C₂²×C₄₇	#575	1	2	yes	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	8	no	—
129	C₄×C₄₇	#441	1	4	yes	complemented
130	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
131	C₄×C₄₇	C₄×#41	1	2	yes	—
132	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
133	Dic₄₇	—	2	8	no	—
134	Dic₄₇	—	4	4	no	—
135	Dic₄₇	—	4	8	no	—
136	C₂×#441	—	47	2	no	—
137	C₂×C₄×Q₁₆	—	47	1	no	—
138	C₄²×Q₈	—	47	2	no	—
139	#1776	—	47	2	no	—
140	C₄×#106	—	47	1	no	—
141	#1763	—	47	4	no	—
142	C₄×#103	—	47	2	no	—
143	C₄×#97	D₄₇	1	1	yes	characteristic
144	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
146	C₂×C₄₇	#2943	1	1	yes	characteristic
147	C₂×C₄₇	C₄×#137	1	4	yes	—
148	C₂²×Q₁₆	—	47	1	no	—
149	C₂×C₄×Q₈	—	47	2	no	—
150	C₂×C₄×Q₈	—	47	4	no	—
151	#441	—	47	16	no	—
152	C₄×Q₁₆	—	47	8	no	—
153	C₂×#107	—	47	1	no	—
154	C₂×#106	—	47	2	no	—
155	C₂×#103	—	47	2	no	—
156	C₂×#103	—	47	4	no	—
157	C₂×#97	C₂×D₄₇	1	2	yes	—
158	C₂×#97	C₂×D₄₇	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
161	C₄×C₄⋊C₄	—	47	2	no	—
162	C₄×C₄⋊C₄	—	47	4	no	—
163	C₄×C₄⋊C₄	—	94	4	no	—
164	C₄×C₄⋊C₄	—	94	2	no	—
165	C₄³	—	94	2	no	—
166	C₄³	C₂×D₄₇	1	1	yes	characteristic
167	#340	—	47	4	no	—
168	#340	—	47	2	no	—
169	C₄₇	C₄×#441	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	—	47	2	no	—
171	C₂×Q₁₆	—	47	4	no	—
172	C₂×Q₁₆	—	47	8	no	—
173	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
174	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
176	C₄×Q₈	—	47	8	no	—
177	C₄×Q₈	—	47	16	no	—
178	C₄×Q₈	—	94	4	no	—
179	C₄×Q₈	—	94	8	no	—
180	C₂×C₄⋊C₄	—	47	4	no	—
181	C₂×C₄⋊C₄	—	47	2	no	—
182	C₂×C₄⋊C₄	—	94	2	no	—
183	C₂×C₄⋊C₄	—	94	4	no	—
184	C₂×C₄²	—	94	4	no	—
185	C₂×C₄²	—	94	2	no	—
186	C₂×C₄²	—	94	4	no	—
187	C₂×C₄²	—	94	2	no	—
188	C₂×C₄²	D₁₈₈	1	2	yes	—
189	C₂×C₄²	D₁₈₈	1	2	yes	—
190	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	#107	—	47	4	no	—
193	#106	—	47	8	no	—
194	#103	—	47	8	no	—
195	#103	—	47	16	no	—
196	#97	—	2	4	no	—
197	#97	C₄×D₄₇	1	8	yes	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₄×D₄₇	1	4	yes	—
200	#95	—	94	2	no	—
201	#95	—	94	4	no	—
202	C₂×Q₈	—	47	4	no	—
203	C₂×Q₈	—	47	8	no	—
204	C₂×Q₈	—	94	2	no	—
205	C₂×Q₈	—	94	4	no	—
206	C₂²×C₄	—	94	2	no	—
207	C₂²×C₄	—	94	4	no	—
208	C₂²×C₄	—	94	2	no	—
209	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
211	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
212	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
213	Q₁₆	—	47	16	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
219	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
221	C₄⋊C₄	—	47	8	no	—
222	C₄⋊C₄	—	47	4	no	—
223	C₄⋊C₄	—	94	16	no	—
224	C₄⋊C₄	—	94	8	no	—
225	C₄⋊C₄	—	94	4	no	—
226	C₄⋊C₄	—	94	2	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	4	2	no	—
229	C₄²	—	94	16	no	—
230	C₄²	—	94	8	no	—
231	C₄²	—	188	8	no	—
232	C₄²	—	188	4	no	—
233	C₄²	C₂×C₄×D₄₇	1	2	yes	—
234	C₄²	C₂×C₄×D₄₇	1	4	yes	—
235	C₄²	#10625350	1	2	yes	—
236	C₄²	#10625350	1	4	yes	—
237	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
238	Q₈	—	47	8	no	—
239	Q₈	—	94	4	no	—
240	Q₈	—	94	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	94	4	no	—
245	C₂×C₄	—	94	8	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	94	8	no	—
248	C₂×C₄	—	188	4	no	—
249	C₂×C₄	—	188	4	no	—
250	C₂×C₄	—	188	4	no	—
251	C₂×C₄	C₄×D₁₈₈	1	8	yes	—
252	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
253	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
254	C₂×C₄	#13094057	1	4	yes	—
255	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
257	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
258	C₂×C₄	C₂×#10625350	1	2	yes	—
259	C₈	—	2	4	no	—
260	C₈	—	2	2	no	—
261	C₈	C₄²×D₄₇	1	4	yes	—
262	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
263	C₂²	C₂×C₄×D₁₈₈	1	2	yes	—
264	C₂²	C₂×#13094057	1	2	yes	—
265	C₂²	#15689923	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	8	no	—
268	C₄	—	94	8	no	—
269	C₄	—	188	4	no	—
270	C₄	—	188	8	no	—
271	C₄	#15689231	1	4	yes	direct product factor
272	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
273	C₄	C₄×#10625350	1	2	yes	—
274	C₄	C₄×#10625350	1	1	yes	characteristic
275	C₂	C₄²×D₁₈₈	1	1	yes	characteristic
276	C₂	C₂×#15689231	1	1	yes	characteristic
277	C₂	#20252190	1	1	yes	characteristic
278	C₂	C₄×#13094057	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	C₄×#441
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
12. #15689231	C₄	direct (N × H)
13. C₄×Dic₁₈₈	C₄	semidirect
96. Dic₁₈₈	C₄²	semidirect
129. C₄×C₄₇	#441	semidirect
169. C₄₇	C₄×#441	semidirect
271. C₄	#15689231	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×#15689231	2
C₂×C₄×Dic₁₈₈	2
C₄²×Dic₉₄	2
#20243285	2
C₄×#13093956	2
#20243219	2
C₄×#13093949	2
C₄×#97×C₄₇	2
C₄×#441	47

Subgroups — C4×#15689231 #26214783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689231 #26214783