Subgroups — C₄×#15688882 #26121450

Order 12032.

1214 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688882	C₂	1	4	yes	maximal
4	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₄₇	—	2	2	no	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	C₄×#10625329	—	2	2	no	—
9	C₄×#10625329	C₄	1	4	yes	complemented
10	C₂×#13093935	—	2	1	no	—
11	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
12	C₄²×Dic₄₇	—	2	2	no	—
13	C₄²×Dic₄₇	C₄	1	4	yes	complemented
14	#15688882	C₄	1	16	yes	direct product factor
15	C₂×#117×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	complemented
18	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
19	C₄₇×#376	—	2	1	no	—
20	C₄³×C₄₇	—	2	1	no	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	C₂×C₄	1	4	yes	—
25	C₂×#10625329	C₂×C₄	1	8	yes	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	16	no	—
28	C₂×C₄×Dic₄₇	—	2	8	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	4	no	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
33	#13093935	—	2	4	no	—
34	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₂³×C₄×C₄₇	D₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#117×C₄₇	—	2	4	no	—
38	#117×C₄₇	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	—	2	4	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	#95×C₄₇	—	2	4	no	—
51	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	4	yes	—
66	C₂²×Dic₄₇	—	2	16	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	8	no	—
69	C₄⋊C₄×C₄₇	—	2	4	no	—
70	C₄⋊C₄×C₄₇	C₄²	1	8	yes	complemented
71	C₂²⋊C₄×C₄₇	—	2	8	no	—
72	C₄²×C₄₇	—	2	8	no	—
73	C₄²×C₄₇	—	4	8	no	—
74	C₄²×C₄₇	C₄²	1	4	yes	—
75	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
80	C₄×Dic₄₇	—	2	32	no	—
81	C₄×Dic₄₇	—	2	4	no	—
82	C₄×Dic₄₇	—	2	8	no	—
83	C₄×Dic₄₇	C₄²	1	8	yes	complemented
84	#10625329	—	2	4	no	—
85	#10625329	—	2	8	no	—
86	#10625329	C₄²	1	8	yes	complemented
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	8	no	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₇	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	4	8	no	—
99	C₂×C₄×C₄₇	#95	1	4	yes	—
100	C₂×C₄×C₄₇	#95	1	4	yes	—
101	C₂×C₄×C₄₇	#104	1	8	yes	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	2	yes	—
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	#117	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
115	C₂×Dic₄₇	—	2	16	no	—
116	C₂×Dic₄₇	—	2	32	no	—
117	C₂×Dic₄₇	—	2	8	no	—
118	C₄₇⋊C₈	—	2	8	no	—
119	C₄×#339	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	8	no	—
123	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
124	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
125	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
126	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
127	C₂²×C₄₇	C₂×#104	1	4	yes	—
128	C₄×C₄₇	—	2	4	no	—
129	C₄×C₄₇	—	4	8	no	—
130	C₄×C₄₇	#339	1	8	yes	complemented
131	C₄×C₄₇	C₄³	1	1	yes	characteristic
132	C₄×C₄₇	#376	1	1	yes	characteristic
133	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
134	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
135	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
136	Dic₄₇	—	2	16	no	—
137	C₂×C₄×M₁₆	—	47	1	no	—
138	C₄×#117	D₄₇	1	1	yes	characteristic
139	C₂×C₄³	—	47	1	no	—
140	C₂×#339	—	47	4	no	—
141	C₂×C₄₇	—	2	2	no	—
142	C₂×C₄₇	—	2	2	no	—
143	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#339	1	2	yes	—
145	C₂×C₄₇	C₄×#104	1	4	yes	—
146	C₂²×M₁₆	—	47	2	no	—
147	C₂×#117	C₂×D₄₇	1	2	yes	—
148	C₂²×C₄²	—	47	2	no	—
149	C₂²×C₄²	—	94	2	no	—
150	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
151	C₄×M₁₆	—	47	4	no	—
152	C₄×M₁₆	—	94	2	no	—
153	C₂×#97	—	94	1	no	—
154	C₂×C₄×C₈	—	94	1	no	—
155	C₄×C₄⋊C₄	Dic₄₇	1	4	yes	complemented
156	C₄×C₂²⋊C₄	—	2	2	no	—
157	#376	—	2	1	no	—
158	C₄³	—	2	1	no	—
159	C₄³	—	47	4	no	—
160	C₄³	—	94	2	no	—
161	#339	—	47	16	no	—
162	C₄₇	C₄×#339	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	—	94	2	no	—
164	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
165	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×M₁₆	—	47	8	no	—
167	C₂×M₁₆	—	47	4	no	—
168	C₂×M₁₆	—	94	2	no	—
169	C₂×M₁₆	—	94	4	no	—
170	C₂²×C₈	—	94	2	no	—
171	#117	—	2	4	no	—
172	#117	C₄×D₄₇	1	8	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	—	47	8	no	—
178	C₂×C₄²	—	47	4	no	—
179	C₂×C₄²	—	94	16	no	—
180	C₂×C₄²	—	94	8	no	—
181	C₂×C₄²	—	94	2	no	—
182	C₂×C₄²	—	94	4	no	—
183	C₂×C₄²	Dic₉₄	1	2	yes	—
184	C₂×C₄²	C₄×D₄₇	1	2	yes	—
185	C₂×C₄²	C₄×D₄₇	1	2	yes	—
186	C₂×C₄²	D₁₈₈	1	2	yes	—
187	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
188	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
190	#97	—	94	4	no	—
191	C₄×C₈	—	94	4	no	—
192	#95	—	2	4	no	—
193	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	94	16	no	—
197	C₂²×C₄	—	94	4	no	—
198	C₂²×C₄	—	94	8	no	—
199	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
200	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
201	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
202	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
203	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
207	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
208	C₂²×C₄	#10625350	1	4	yes	—
209	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
211	M₁₆	—	47	8	no	—
212	M₁₆	—	94	4	no	—
213	M₁₆	—	94	8	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	—	94	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	complemented
218	C₂²⋊C₄	—	2	8	no	—
219	C₄²	—	2	8	no	—
220	C₄²	—	4	8	no	—
221	C₄²	—	47	8	no	—
222	C₄²	—	94	32	no	—
223	C₄²	—	94	4	no	—
224	C₄²	—	94	8	no	—
225	C₄²	C₄×Dic₄₇	1	4	yes	—
226	C₄²	Dic₄₇⋊C₄	1	2	yes	—
227	C₄²	Dic₄₇⋊C₄	1	2	yes	—
228	C₄²	C₄⋊Dic₄₇	1	4	yes	—
229	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
230	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
233	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
234	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
235	C₂³	C₄×C₂²⋊D₄₇	1	4	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	94	16	no	—
242	C₂×C₄	—	94	32	no	—
243	C₂×C₄	—	94	8	no	—
244	C₂×C₄	C₄₇⋊#95	1	4	yes	—
245	C₂×C₄	C₄₇⋊#95	1	4	yes	—
246	C₂×C₄	#13093951	1	8	yes	—
247	C₂×C₄	C₄²×D₄₇	1	2	yes	—
248	C₂×C₄	C₄²×D₄₇	1	2	yes	—
249	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
251	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
254	C₂×C₄	#13093981	1	2	yes	—
255	C₂×C₄	#13093981	1	2	yes	—
256	C₂×C₄	#13093983	1	2	yes	—
257	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
258	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
259	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
260	C₈	—	94	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
265	C₂²	C₄×C₂²⋊Dic₄₇	1	2	yes	—
266	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
267	C₂²	C₄×Dic₄₇⋊C₄	1	2	yes	—
268	C₂²	C₂×#13093951	1	4	yes	—
269	C₄	—	2	4	no	—
270	C₄	—	4	8	no	—
271	C₄	—	94	16	no	—
272	C₄	#15688882	1	8	yes	direct product factor
273	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
274	C₄	#15688969	1	1	yes	characteristic
275	C₄	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
276	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
277	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
281	C₂	C₂×#15688882	1	2	yes	—
282	C₂	C₄×#13093951	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	C₄×#339
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
9. C₄×#10625329	C₄	semidirect
13. C₄²×Dic₄₇	C₄	semidirect
14. #15688882	C₄	direct (N × H)
17. C₄×C₄⋊C₄×C₄₇	C₄	semidirect
70. C₄⋊C₄×C₄₇	C₄²	semidirect
83. C₄×Dic₄₇	C₄²	semidirect
86. #10625329	C₄²	semidirect
130. C₄×C₄₇	#339	semidirect
155. C₄×C₄⋊C₄	Dic₄₇	semidirect
162. C₄₇	C₄×#339	semidirect
217. C₄⋊C₄	C₄×Dic₄₇	semidirect
272. C₄	#15688882	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×#10625329	2
C₂×C₄²×Dic₄₇	2
C₂×#15688882	2
C₄×#117×C₄₇	2
C₄×#339	47

Subgroups — C4×#15688882 #26121450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688882 #26121450