Subgroups — C₄×#15688892 #26121484

Order 12032.

994 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×C₄×#10625329	C₂	1	2	yes	maximal
2	C₂×#15688892	C₂	1	4	yes	maximal
3	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	C₄×#10625329	—	2	4	no	—
6	C₄×#10625329	C₄	1	8	yes	complemented
7	C₂×#13093935	—	2	2	no	—
8	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
9	#15688892	C₄	1	16	yes	direct product factor
10	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
15	C₂×#95×C₄₇	C₄	1	2	yes	—
16	C₂×#10625329	—	2	4	no	—
17	C₂×#10625329	—	2	8	no	—
18	C₂×#10625329	C₂×C₄	1	8	yes	—
19	C₂×#10625329	C₂×C₄	1	16	yes	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	#13093935	—	2	8	no	—
22	C₄×C₄₇⋊C₈	—	2	8	no	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	—	4	4	no	—
33	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	2	yes	—
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
38	#95×C₄₇	—	2	4	no	—
39	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	—	4	4	no	—
46	C₂²×C₄×C₄₇	—	4	4	no	—
47	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
48	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
49	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	4	yes	—
61	C₄⋊C₄×C₄₇	—	2	16	no	—
62	C₄²×C₄₇	—	4	4	no	—
63	C₄²×C₄₇	—	4	4	no	—
64	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
65	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
66	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
68	#10625329	—	2	8	no	—
69	#10625329	—	2	16	no	—
70	#10625329	C₄²	1	16	yes	complemented
71	C₂×C₄₇⋊C₈	—	2	8	no	—
72	C₂×C₄₇⋊C₈	—	2	16	no	—
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	—	2	1	no	—
75	C₂³×C₄₇	—	2	2	no	—
76	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
77	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
78	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂³×C₄₇	#117	1	2	yes	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	4	2	no	—
88	C₂×C₄×C₄₇	—	4	2	no	—
89	C₂×C₄×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	—	4	4	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	—	4	8	no	—
93	C₂×C₄×C₄₇	#95	1	2	yes	—
94	C₂×C₄×C₄₇	#95	1	4	yes	—
95	C₂×C₄×C₄₇	#95	1	2	yes	—
96	C₂×C₄×C₄₇	#100	1	2	yes	—
97	C₂×C₄×C₄₇	#101	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	#117	1	4	yes	—
102	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
105	C₄₇⋊C₈	—	2	16	no	—
106	C₄×#343	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	—	2	1	no	—
110	C₂²×C₄₇	—	2	1	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	C₄³	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
117	C₂²×C₄₇	#376	1	1	yes	characteristic
118	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
119	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
120	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
122	C₂²×C₄₇	#413	1	2	yes	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	—	4	8	no	—
126	C₄×C₄₇	#343	1	4	yes	complemented
127	C₄×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
128	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
129	C₂×C₄×M₁₆	—	47	2	no	—
130	C₂×C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic
131	C₂×#343	—	47	4	no	—
132	C₂×C₄₇	—	2	1	no	—
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	2	2	no	—
135	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
136	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
137	C₂×C₄₇	#1742	1	1	yes	characteristic
138	C₂×C₄₇	C₄×#100	1	2	yes	—
139	C₂×C₄₇	C₄×#101	1	2	yes	—
140	C₂²×M₁₆	—	47	4	no	—
141	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
142	C₂²×C₄²	Dic₄₇	1	2	yes	—
143	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
144	C₄×M₁₆	—	47	8	no	—
145	C₄×M₁₆	—	94	4	no	—
146	C₂×#97	—	94	2	no	—
147	C₂×C₄×C₈	—	94	2	no	—
148	C₄×C₄⋊C₄	—	2	2	no	—
149	C₄×C₄⋊C₄	—	2	2	no	—
150	C₂×#95	Dic₄₇	1	2	yes	—
151	#343	—	47	16	no	—
152	C₄₇	C₄×#343	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
154	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
155	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
156	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
157	C₂×M₁₆	—	47	16	no	—
158	C₂×M₁₆	—	47	8	no	—
159	C₂×M₁₆	—	94	4	no	—
160	C₂×M₁₆	—	94	8	no	—
161	C₂²×C₈	—	94	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	C₄×D₄₇	1	8	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	4	4	no	—
168	C₂×C₄²	Dic₉₄	1	2	yes	—
169	C₂×C₄²	C₄×D₄₇	1	4	yes	—
170	C₂×C₄²	D₁₈₈	1	2	yes	—
171	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
173	#97	—	94	8	no	—
174	C₄×C₈	—	94	8	no	—
175	#95	—	2	4	no	—
176	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
185	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
186	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
189	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
195	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
196	C₂²×C₄	#10625350	1	4	yes	—
197	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
198	M₁₆	—	47	16	no	—
199	M₁₆	—	94	8	no	—
200	M₁₆	—	94	16	no	—
201	C₂×C₈	—	94	8	no	—
202	C₂×C₈	—	94	16	no	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄²	—	4	4	no	—
205	C₄²	—	4	4	no	—
206	C₄²	Dic₄₇⋊C₄	1	4	yes	—
207	C₄²	Dic₄₇⋊C₄	1	4	yes	—
208	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
209	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
214	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
215	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
216	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
217	C₂³	#13093983	1	2	yes	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	C₄₇⋊#95	1	2	yes	—
231	C₂×C₄	C₄₇⋊#95	1	4	yes	—
232	C₂×C₄	C₄₇⋊#95	1	2	yes	—
233	C₂×C₄	#13093941	1	2	yes	—
234	C₂×C₄	#13093943	1	2	yes	—
235	C₂×C₄	C₄²×D₄₇	1	4	yes	—
236	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
237	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
238	C₂×C₄	#13093981	1	4	yes	—
239	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
240	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
241	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
242	C₈	—	94	16	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
252	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
253	C₂²	#15688969	1	1	yes	characteristic
254	C₂²	C₄×C₂²⋊Dic₄₇	1	2	yes	—
255	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
256	C₂²	C₂×#13093941	1	1	yes	characteristic
257	C₂²	C₂×#13093943	1	1	yes	characteristic
258	C₂²	#15689107	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	#15688892	1	4	yes	direct product factor
263	C₄	C₄×(C₂×D₄₇)⋊C₄	1	4	yes	—
264	C₄	C₄×Dic₄₇⋊C₄	1	4	yes	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	2	2	no	—
268	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
269	C₂	C₂×#15688892	1	1	yes	characteristic
270	C₂	#20243127	1	1	yes	characteristic
271	C₂	C₄×#13093941	1	2	yes	—
272	C₂	C₄×#13093943	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₄×#343
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₄×#10625329	C₄	semidirect
9. #15688892	C₄	direct (N × H)
70. #10625329	C₄²	semidirect
126. C₄×C₄₇	#343	semidirect
152. C₄₇	C₄×#343	semidirect
262. C₄	#15688892	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×#15688892	2
C₂×C₄×C₄⋊C₄×C₄₇	2
C₄×#343	47

Subgroups — C4×#15688892 #26121484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688892 #26121484