Subgroups — C₄×#15688853 #26106878

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688853	C₂	1	2	yes	maximal
4	#20241227	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₄	1	2	yes	—
7	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₄₇⋊C₄	—	2	2	no	—
11	C₄×Dic₄₇⋊C₄	—	2	1	no	—
12	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
13	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₄²×Dic₄₇	—	2	1	no	—
15	#15688853	C₄	1	8	yes	direct product factor
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	—	2	1	no	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄×Dic₉₄	—	2	4	no	—
23	C₄×Dic₉₄	—	2	2	no	—
24	C₄×Dic₉₄	C₈	1	8	yes	complemented
25	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×Dic₄₇⋊C₄	—	2	2	no	—
29	C₂×Dic₄₇⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	1	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₄×C₄₇⋊C₈	—	2	8	no	—
35	C₄×C₄₇⋊C₈	—	2	2	no	—
36	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₄₇⋊#95	—	2	1	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₄₇	—	2	2	no	—
46	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₄₇	—	4	2	no	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	2	1	no	—
54	C₂²×Dic₄₇	—	2	1	no	—
55	C₂²×Dic₄₇	—	2	2	no	—
56	C₂×Dic₉₄	—	2	1	no	—
57	C₂×Dic₉₄	—	2	2	no	—
58	C₂×Dic₉₄	C₄²	1	2	yes	—
59	C₂×Dic₉₄	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₄₇	—	2	4	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₂×C₈×C₄₇	—	4	2	no	—
63	C₄²×C₄₇	—	2	4	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	C₂×C₈	1	4	yes	—
66	C₄²×C₄₇	M₁₆	1	4	yes	—
67	C₄²×C₄₇	SD₁₆	1	2	yes	—
68	C₄²×C₄₇	Q₁₆	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	#41	1	2	yes	—
71	C₄⋊Dic₄₇	—	2	1	no	—
72	C₄⋊Dic₄₇	—	2	2	no	—
73	C₄⋊Dic₄₇	C₄²	1	2	yes	—
74	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
75	Dic₄₇⋊C₄	—	2	8	no	—
76	Dic₄₇⋊C₄	—	2	4	no	—
77	C₄×Dic₄₇	—	2	8	no	—
78	C₄×Dic₄₇	—	2	4	no	—
79	C₄×Dic₄₇	—	4	4	no	—
80	C₄×Dic₄₇	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
88	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
89	C₂×C₄×C₄₇	#103	1	4	yes	—
90	C₂×C₄×C₄₇	#104	1	4	yes	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#135	1	2	yes	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	4	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	—	4	2	no	—
107	Dic₉₄	—	2	2	no	—
108	Dic₉₄	—	2	4	no	—
109	Dic₉₄	C₄×C₈	1	4	yes	complemented
110	C₈×C₄₇	—	4	4	no	—
111	C₄₇⋊C₈	—	2	8	no	—
112	C₄×Q₈⋊C₈	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
115	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
116	C₂²×C₄₇	#416	1	1	yes	characteristic
117	C₂²×C₄₇	C₂×#104	1	2	yes	—
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	Q₈⋊C₈	1	8	yes	complemented
121	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
122	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
123	C₄×C₄₇	#431	1	2	yes	—
124	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
125	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
126	Dic₄₇	—	2	4	no	—
127	Dic₄₇	—	4	4	no	—
128	Dic₄₇	—	4	2	no	—
129	C₄²×Q₈	—	47	1	no	—
130	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
131	C₄²×C₈	—	47	1	no	—
132	C₂×Q₈⋊C₈	—	47	2	no	—
133	#1279	—	47	2	no	—
134	C₂×C₄₇	C₂×Q₈⋊C₈	1	2	yes	—
135	C₂×C₄₇	#1476	1	2	yes	—
136	C₂×C₄₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
138	C₂×C₄₇	C₄×#103	1	1	yes	characteristic
139	C₂×C₄×Q₈	—	47	2	no	—
140	C₂×C₄×Q₈	—	47	1	no	—
141	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
142	C₂×C₄×C₈	—	2	1	no	—
143	C₂×C₄×C₈	—	47	2	no	—
144	C₂×C₄×C₈	—	94	2	no	—
145	C₄×C₄⋊C₄	—	47	2	no	—
146	C₄×C₄⋊C₄	—	47	1	no	—
147	C₄×C₄⋊C₄	—	94	2	no	—
148	C₄×C₄⋊C₄	—	94	1	no	—
149	C₄³	—	94	1	no	—
150	C₄³	C₂×D₄₇	1	1	yes	characteristic
151	#336	—	2	1	no	—
152	Q₈⋊C₈	—	47	8	no	—
153	C₄₇	C₄×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	47	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	94	2	no	—
157	C₄×Q₈	—	47	8	no	—
158	C₄×Q₈	—	47	4	no	—
159	C₄×Q₈	—	94	4	no	—
160	C₄×Q₈	—	94	2	no	—
161	C₂×C₄⋊C₄	—	47	1	no	—
162	C₂×C₄⋊C₄	—	47	2	no	—
163	C₂×C₄⋊C₄	—	94	2	no	—
164	C₂×C₄⋊C₄	—	94	1	no	—
165	C₂×C₄²	—	94	2	no	—
166	C₂×C₄²	—	94	1	no	—
167	C₂×C₄²	—	94	2	no	—
168	C₂×C₄²	—	94	1	no	—
169	C₂×C₄²	C₄×D₄₇	1	2	yes	—
170	C₂×C₄²	D₁₈₈	1	2	yes	—
171	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
176	C₄×C₈	—	4	2	no	—
177	C₄×C₈	—	47	4	no	—
178	C₄×C₈	—	94	8	no	—
179	C₄×C₈	—	94	2	no	—
180	#95	—	94	2	no	—
181	#95	—	94	1	no	—
182	C₂×Q₈	—	47	2	no	—
183	C₂×Q₈	—	47	4	no	—
184	C₂×Q₈	—	94	1	no	—
185	C₂×Q₈	—	94	2	no	—
186	C₂²×C₄	—	94	1	no	—
187	C₂²×C₄	—	94	1	no	—
188	C₂²×C₄	—	94	2	no	—
189	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
190	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	94	8	no	—
198	C₂×C₈	—	94	4	no	—
199	C₄⋊C₄	—	47	2	no	—
200	C₄⋊C₄	—	47	4	no	—
201	C₄⋊C₄	—	94	8	no	—
202	C₄⋊C₄	—	94	1	no	—
203	C₄⋊C₄	—	94	4	no	—
204	C₄⋊C₄	—	94	2	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	94	8	no	—
208	C₄²	—	94	4	no	—
209	C₄²	—	188	4	no	—
210	C₄²	—	188	2	no	—
211	C₄²	C₈×D₄₇	1	4	yes	—
212	C₄²	#10625324	1	4	yes	—
213	C₄²	#10625335	1	2	yes	—
214	C₄²	#10625337	1	2	yes	—
215	C₄²	C₂×C₄×D₄₇	1	2	yes	—
216	C₄²	#10625350	1	2	yes	—
217	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
218	Q₈	—	47	4	no	—
219	Q₈	—	94	2	no	—
220	Q₈	—	94	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	94	2	no	—
225	C₂×C₄	—	94	4	no	—
226	C₂×C₄	—	94	2	no	—
227	C₂×C₄	—	94	4	no	—
228	C₂×C₄	—	188	2	no	—
229	C₂×C₄	—	188	2	no	—
230	C₂×C₄	—	188	2	no	—
231	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
232	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
233	C₂×C₄	#13093947	1	4	yes	—
234	C₂×C₄	#13093950	1	4	yes	—
235	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
236	C₂×C₄	#13093981	1	1	yes	characteristic
237	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
238	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
239	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
240	C₂×C₄	C₂×#10625324	1	2	yes	—
241	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
242	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
243	C₂×C₄	#13094044	1	2	yes	—
244	C₈	—	4	4	no	—
245	C₈	—	94	8	no	—
246	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
247	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	2	yes	—
248	C₂²	C₂×#13093947	1	1	yes	characteristic
249	C₂²	#15689117	1	1	yes	characteristic
250	C₂²	C₂×#13093950	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	2	no	—
253	C₄	—	94	4	no	—
254	C₄	—	188	4	no	—
255	C₄	—	188	2	no	—
256	C₄	#15688853	1	8	yes	direct product factor
257	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
258	C₄	C₄×#10625324	1	1	yes	characteristic
259	C₄	#15689174	1	2	yes	—
260	C₄	C₄×#10625335	1	2	yes	—
261	C₄	C₄×#10625337	1	2	yes	—
262	C₂	C₂×#15688853	1	2	yes	—
263	C₂	#20241768	1	2	yes	—
264	C₂	C₄×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
265	C₂	C₄×#13093950	1	1	yes	characteristic
266	C₂	C₄×#13093947	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₄×Q₈⋊C₈
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
15. #15688853	C₄	direct (N × H)
24. C₄×Dic₉₄	C₈	semidirect
109. Dic₉₄	C₄×C₈	semidirect
120. C₄×C₄₇	Q₈⋊C₈	semidirect
153. C₄₇	C₄×Q₈⋊C₈	semidirect
256. C₄	#15688853	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄²×Dic₉₄	2
C₄²×C₄₇⋊C₈	2
C₂×#15688853	2
#20241227	2
C₄×C₄⋊C₈×C₄₇	2
C₄×Q₈⋊C₈	47

Subgroups — C4×#15688853 #26106878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688853 #26106878