Subgroups — C₄×#15689147 #26214180

Order 12032.

1382 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	2	yes	maximal
2	C₂×C₄×#10625329	C₂	1	2	yes	maximal
3	#20243241	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
5	#20243082	C₂	1	2	yes	maximal
6	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	4	yes	—
8	#15689147	C₄	1	16	yes	direct product factor
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
10	C₄×#10625329	—	2	8	no	—
11	C₂×#13093935	C₂²	1	4	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
13	#15688878	C₂²	1	8	yes	—
14	C₂²×C₄²×C₄₇	C₄	1	4	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄³×C₄₇	C₄	1	2	yes	—
19	C₄³×C₄₇	C₄	1	2	yes	—
20	C₄³×C₄₇	C₂²	1	2	yes	—
21	C₄³×C₄₇	C₂²	1	2	yes	—
22	C₂×#10625329	—	2	8	no	—
23	C₂×#10625329	—	2	8	no	—
24	C₂×#10625329	C₂×C₄	1	16	yes	—
25	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
26	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	32	yes	complemented
27	#13093935	—	2	8	no	—
28	C₄×C₄₇⋊C₈	—	2	8	no	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	—	2	4	no	—
34	C₂×C₄²×C₄₇	—	2	16	no	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	D₄	1	4	yes	—
45	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	—	2	8	no	—
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₄²×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	—	2	8	no	—
70	C₄²×C₄₇	—	2	8	no	—
71	C₄²×C₄₇	—	2	8	no	—
72	C₄²×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	C₄²	1	4	yes	—
74	C₄²×C₄₇	C₄²	1	4	yes	—
75	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
76	C₄²×C₄₇	M₁₆	1	16	yes	complemented
77	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
78	C₄²×C₄₇	C₂×D₄	1	2	yes	—
79	C₄²×C₄₇	C₂×D₄	1	2	yes	—
80	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
81	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
82	#10625329	—	2	32	no	—
83	C₂×C₄₇⋊C₈	—	2	8	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	#117	1	2	yes	—
90	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	16	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	16	no	—
97	C₂×C₄×C₄₇	—	2	16	no	—
98	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	#117	1	2	yes	—
105	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
106	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
107	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
108	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
109	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
110	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
114	C₄₇⋊C₈	—	2	16	no	—
115	C₄×#423	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
120	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
121	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
123	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	—	2	4	no	—
126	C₄×C₄₇	—	2	8	no	—
127	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
128	C₄×C₄₇	C₄×M₁₆	1	8	yes	complemented
129	C₄×C₄₇	#423	1	8	yes	complemented
130	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
131	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
132	C₂×#423	—	47	2	no	—
133	C₂×C₄×M₁₆	—	47	2	no	—
134	C₂×C₄³	D₄₇	1	1	yes	characteristic
135	#1767	—	47	4	no	—
136	C₄×C₄⋊C₈	—	47	4	no	—
137	#1728	—	47	2	no	—
138	C₂×C₄₇	—	2	4	no	—
139	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
141	C₂×C₄₇	C₂×#423	1	2	yes	—
142	C₂×C₄₇	#2920	1	2	yes	—
143	C₂²×M₁₆	—	47	4	no	—
144	C₂²×C₄²	Dic₄₇	1	4	yes	—
145	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
146	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
147	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
148	#423	—	47	16	no	—
149	C₂×C₄⋊C₈	—	47	8	no	—
150	C₄×M₁₆	—	94	8	no	—
151	C₂×#97	—	47	4	no	—
152	C₂×C₄×C₈	—	47	4	no	—
153	C₄³	Dic₄₇	1	2	yes	—
154	C₄³	Dic₄₇	1	2	yes	—
155	C₄³	C₂×D₄₇	1	2	yes	—
156	C₄³	C₂×D₄₇	1	2	yes	—
157	#336	—	47	8	no	—
158	C₄₇	C₄×#423	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
160	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
161	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×M₁₆	—	47	16	no	—
164	C₂×M₁₆	—	94	8	no	—
165	C₂×M₁₆	—	94	8	no	—
166	C₂²×C₈	—	47	8	no	—
167	C₂×C₄²	—	2	4	no	—
168	C₂×C₄²	—	2	16	no	—
169	C₂×C₄²	Dic₉₄	1	4	yes	—
170	C₂×C₄²	C₄×D₄₇	1	8	yes	—
171	C₂×C₄²	D₁₈₈	1	4	yes	—
172	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
173	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
174	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
175	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
176	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
177	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
180	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
181	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	C₄⋊C₈	—	47	32	no	—
185	#97	—	94	8	no	—
186	C₄×C₈	—	94	8	no	—
187	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
193	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
194	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
196	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
197	C₂²×C₄	#10625350	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
201	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
204	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
205	M₁₆	—	94	32	no	—
206	C₂×C₈	—	47	16	no	—
207	C₂×C₈	—	94	8	no	—
208	C₂×C₈	—	94	8	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	2	8	no	—
212	C₄²	—	2	8	no	—
213	C₄²	—	2	4	no	—
214	C₄²	#10625329	1	16	yes	complemented
215	C₄²	C₄×Dic₄₇	1	4	yes	—
216	C₄²	C₄×Dic₄₇	1	4	yes	—
217	C₄²	C₄⋊Dic₄₇	1	8	yes	—
218	C₄²	C₂×Dic₉₄	1	2	yes	—
219	C₄²	C₂×Dic₉₄	1	2	yes	—
220	C₄²	C₂×C₄×D₄₇	1	8	yes	—
221	C₄²	C₂×D₁₈₈	1	2	yes	—
222	C₄²	C₂×D₁₈₈	1	2	yes	—
223	C₂³	—	2	4	no	—
224	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
225	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
226	C₂³	#13093983	1	2	yes	—
227	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
236	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
237	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
241	C₂×C₄	#13093983	1	2	yes	—
242	C₂×C₄	C₄×D₁₈₈	1	8	yes	—
243	C₂×C₄	C₄×Dic₉₄	1	8	yes	—
244	C₂×C₄	C₂×#10625329	1	8	yes	—
245	C₂×C₄	C₂×#10625329	1	8	yes	—
246	C₂×C₄	C₂×#10625329	1	8	yes	—
247	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
248	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
249	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
250	C₂×C₄	C₂×#10625350	1	2	yes	—
251	C₈	—	94	16	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₄×C₄⋊Dic₄₇	1	4	yes	—
256	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
257	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
258	C₂²	C₂×#13093983	1	1	yes	characteristic
259	C₂²	C₂²×#10625329	1	4	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	2	8	no	—
262	C₄	—	2	4	no	—
263	C₄	C₄×C₄⋊Dic₄₇	1	4	yes	—
264	C₄	C₄×#10625329	1	8	yes	complemented
265	C₄	#15689147	1	8	yes	direct product factor
266	C₄	C₂×C₄×D₁₈₈	1	2	yes	—
267	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
268	C₂	—	2	4	no	—
269	C₂	C₂×C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
270	C₂	C₂×C₄×#10625329	1	2	yes	—
271	C₂	C₂×#15689147	1	2	yes	—
272	C₂	#20252021	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	C₄×#423
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. #20243241	C₂	semidirect
4. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
8. #15689147	C₄	direct (N × H)
26. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
76. C₄²×C₄₇	M₁₆	semidirect
128. C₄×C₄₇	C₄×M₁₆	semidirect
129. C₄×C₄₇	#423	semidirect
158. C₄₇	C₄×#423	semidirect
214. C₄²	#10625329	semidirect
264. C₄	C₄×#10625329	semidirect
265. C₄	#15689147	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×#15689147	2
C₂×C₄×#10625329	2
#20243241	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243082	2
C₂×C₄³×C₄₇	2
C₄×#423	47

Subgroups — C4×#15689147 #26214180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689147 #26214180