Subgroups — C₄×#15688887 #26121464

Order 12032.

910 conjugacy classes of subgroups.

The subgroup lattice has 266 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	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688887	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₄×#10625329	—	2	2	no	—
8	C₄×#10625329	C₄	1	4	yes	—
9	C₂×#13093935	—	2	1	no	—
10	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
11	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
12	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
14	#15688887	C₄	1	16	yes	direct product factor
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₄₇	—	2	2	no	—
18	C₄×M₁₆×C₄₇	C₄	1	4	yes	—
19	C₂×#97×C₄₇	—	2	1	no	—
20	C₂×C₄×C₈×C₄₇	—	2	1	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	C₂×C₄	1	4	yes	—
24	C₂×#10625329	C₂×C₄	1	8	yes	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	4	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
31	#13093935	—	2	4	no	—
32	C₄×C₄₇⋊C₈	—	2	8	no	—
33	C₄×C₄₇⋊C₈	—	2	4	no	—
34	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	—	2	2	no	—
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	#97×C₄₇	—	2	4	no	—
49	C₄×C₈×C₄₇	—	2	4	no	—
50	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	4	yes	—
64	M₁₆×C₄₇	—	2	4	no	—
65	M₁₆×C₄₇	—	2	8	no	—
66	M₁₆×C₄₇	C₄²	1	8	yes	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	8	no	—
69	C₄²×C₄₇	C₄²	1	4	yes	—
70	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
71	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
75	#10625329	—	2	4	no	—
76	#10625329	—	2	8	no	—
77	#10625329	C₄²	1	8	yes	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	2	8	no	—
83	C₂×C₄₇⋊C₈	—	2	8	no	—
84	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
85	C₂³×C₄₇	—	2	4	no	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	#95	1	4	yes	—
94	C₂×C₄×C₄₇	#95	1	4	yes	—
95	C₂×C₄×C₄₇	#108	1	8	yes	—
96	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#117	1	2	yes	—
104	C₂×C₄×C₄₇	#117	1	2	yes	—
105	C₂×C₄×C₄₇	#117	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
107	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
108	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
109	C₈×C₄₇	—	2	8	no	—
110	C₄₇⋊C₈	—	2	8	no	—
111	C₄₇⋊C₈	—	2	8	no	—
112	C₄×#341	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	8	no	—
116	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
117	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
118	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
119	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
120	C₂²×C₄₇	C₂×#108	1	4	yes	—
121	C₄×C₄₇	—	2	4	no	—
122	C₄×C₄₇	#341	1	8	yes	complemented
123	C₄×C₄₇	C₄³	1	1	yes	characteristic
124	C₄×C₄₇	#376	1	1	yes	characteristic
125	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
126	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
127	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×M₁₆	—	47	1	no	—
129	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
130	C₂²×C₄×C₈	—	47	1	no	—
131	C₂×#341	—	47	4	no	—
132	C₂×C₄₇	—	2	2	no	—
133	C₂×C₄₇	—	2	2	no	—
134	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
135	C₂×C₄₇	C₂×#341	1	2	yes	—
136	C₂×C₄₇	C₄×#108	1	4	yes	—
137	C₂²×M₁₆	—	47	2	no	—
138	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
139	C₂³×C₈	—	47	2	no	—
140	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
141	C₄×M₁₆	—	2	2	no	—
142	C₄×M₁₆	—	47	4	no	—
143	C₄×M₁₆	—	94	2	no	—
144	C₄×M₁₆	Dic₄₇	1	4	yes	—
145	C₂×#97	—	2	1	no	—
146	C₂×#97	—	94	1	no	—
147	C₂×C₄×C₈	—	2	1	no	—
148	C₂×C₄×C₈	—	47	4	no	—
149	C₂×C₄×C₈	—	94	2	no	—
150	C₂×C₄×C₈	—	94	2	no	—
151	C₂×C₄×C₈	—	94	1	no	—
152	#341	—	47	16	no	—
153	C₄₇	C₄×#341	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	Dic₉₄	1	2	yes	—
155	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	47	8	no	—
159	C₂×M₁₆	—	47	4	no	—
160	C₂×M₁₆	—	94	2	no	—
161	C₂×M₁₆	—	94	4	no	—
162	C₂×M₁₆	C₄×D₄₇	1	8	yes	—
163	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	47	8	no	—
166	C₂²×C₈	—	47	4	no	—
167	C₂²×C₈	—	94	2	no	—
168	C₂²×C₈	—	94	4	no	—
169	C₂²×C₈	—	94	2	no	—
170	C₂²×C₈	—	94	2	no	—
171	C₂×C₄²	C₄×D₄₇	1	2	yes	—
172	C₂×C₄²	C₄×D₄₇	1	2	yes	—
173	C₂×C₄²	D₁₈₈	1	2	yes	—
174	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
175	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
176	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
178	#97	—	2	4	no	—
179	#97	—	94	4	no	—
180	C₄×C₈	—	2	4	no	—
181	C₄×C₈	—	94	8	no	—
182	C₄×C₈	—	94	4	no	—
183	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
186	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
187	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
189	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
192	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
193	C₂²×C₄	#10625350	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	8	no	—
199	M₁₆	—	47	8	no	—
200	M₁₆	—	94	4	no	—
201	M₁₆	—	94	8	no	—
202	M₁₆	C₄×Dic₄₇	1	8	yes	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	—	47	8	no	—
206	C₂×C₈	—	94	4	no	—
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	8	no	—
210	C₂×C₈	—	94	8	no	—
211	C₂×C₈	—	94	8	no	—
212	C₄²	C₄×Dic₄₇	1	4	yes	—
213	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
214	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
215	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
216	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
217	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
218	C₂³	—	2	4	no	—
219	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
220	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
221	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
222	C₂³	C₄×Dic₉₄	1	4	yes	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	C₄₇⋊#95	1	4	yes	—
227	C₂×C₄	C₄₇⋊#95	1	4	yes	—
228	C₂×C₄	#13093959	1	8	yes	—
229	C₂×C₄	C₄²×D₄₇	1	2	yes	—
230	C₂×C₄	C₄²×D₄₇	1	2	yes	—
231	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
233	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
236	C₂×C₄	#13093981	1	2	yes	—
237	C₂×C₄	#13093981	1	2	yes	—
238	C₂×C₄	#13093983	1	2	yes	—
239	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
240	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
241	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
242	C₈	—	2	8	no	—
243	C₈	—	94	8	no	—
244	C₈	—	94	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
249	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
250	C₂²	C₄×Dic₄₇⋊C₄	1	2	yes	—
251	C₂²	C₄×Dic₄₇⋊C₄	1	2	yes	—
252	C₂²	C₂×#13093959	1	4	yes	—
253	C₄	—	2	4	no	—
254	C₄	#15688887	1	8	yes	direct product factor
255	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
256	C₄	#15688969	1	1	yes	characteristic
257	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
258	C₄	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
259	C₄	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	2	2	no	—
262	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
263	C₂	C₂×#15688887	1	2	yes	—
264	C₂	C₄×#13093959	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	C₄×#341
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
14. #15688887	C₄	direct (N × H)
122. C₄×C₄₇	#341	semidirect
153. C₄₇	C₄×#341	semidirect
254. C₄	#15688887	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
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688887	2
C₂×C₄×M₁₆×C₄₇	2
C₄×#341	47

Subgroups — C4×#15688887 #26121464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688887 #26121464