Subgroups — C₄×#15688895 #26121488

Order 12032.

906 conjugacy classes of subgroups.

The subgroup lattice has 316 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	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688895	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂×#13093983	C₂²	1	2	yes	—
7	C₄×#10625329	—	2	1	no	—
8	C₄×#10625329	—	2	1	no	—
9	C₄×#10625329	—	2	1	no	—
10	C₄×#10625329	—	2	1	no	—
11	C₂×#13093935	C₄	1	2	yes	complemented
12	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	complemented
13	C₄×C₄⋊Dic₄₇	—	2	2	no	—
14	C₄×C₂²⋊Dic₄₇	C₄	1	4	yes	—
15	#15688969	—	2	1	no	—
16	C₄²×Dic₄₇	—	2	1	no	—
17	#15688895	C₄	1	16	yes	direct product factor
18	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₄₇	—	2	2	no	—
21	C₄×M₁₆×C₄₇	C₄	1	4	yes	complemented
22	C₂×#97×C₄₇	—	2	1	no	—
23	C₂×C₄×C₈×C₄₇	—	2	1	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	C₂×C₄	1	8	yes	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	#13093983	—	2	4	no	—
30	#13093983	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	#13093935	—	2	2	no	—
36	#13093935	—	2	2	no	—
37	C₄×C₄₇⋊C₈	—	2	2	no	—
38	C₄×C₄₇⋊C₈	—	2	2	no	—
39	C₄₇⋊#95	—	2	4	no	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	4	no	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
56	#97×C₄₇	—	2	4	no	—
57	C₄×C₈×C₄₇	—	2	4	no	—
58	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
61	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	2	8	no	—
77	M₁₆×C₄₇	C₄²	1	8	yes	complemented
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	8	no	—
80	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
90	C₂²⋊Dic₄₇	—	2	4	no	—
91	C₂²⋊Dic₄₇	C₄²	1	8	yes	—
92	C₄⋊Dic₄₇	—	2	8	no	—
93	C₄×Dic₄₇	—	2	8	no	—
94	C₄×Dic₄₇	—	4	8	no	—
95	#10625329	—	2	8	no	—
96	#10625329	—	2	8	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	C₄²	1	8	yes	complemented
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
103	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₇	#117	1	2	yes	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	#95	1	2	yes	—
113	C₂×C₄×C₄₇	#95	1	2	yes	—
114	C₂×C₄×C₄₇	#95	1	2	yes	—
115	C₂×C₄×C₄₇	#95	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#117	1	2	yes	—
123	C₂×C₄×C₄₇	#117	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
125	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
127	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
128	C₂×Dic₄₇	—	2	8	no	—
129	C₂×Dic₄₇	—	4	8	no	—
130	C₈×C₄₇	—	2	8	no	—
131	C₄₇⋊C₈	—	2	8	no	—
132	C₄×#344	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	4	2	no	—
137	C₂²×C₄₇	—	4	2	no	—
138	C₂²×C₄₇	C₄³	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
140	C₂²×C₄₇	#376	1	1	yes	characteristic
141	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
142	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	#344	1	4	yes	complemented
147	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
148	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
149	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
150	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
151	Dic₄₇	—	4	8	no	—
152	C₂×C₄×M₁₆	—	47	1	no	—
153	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
154	C₄×#117	—	47	1	no	—
155	C₂×#344	—	47	4	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	—	2	1	no	—
159	C₂×C₄₇	—	2	2	no	—
160	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#344	1	2	yes	—
162	C₂²×M₁₆	—	47	2	no	—
163	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
164	C₂×#117	—	47	2	no	—
165	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
166	C₄×M₁₆	—	2	2	no	—
167	C₄×M₁₆	—	94	1	no	—
168	C₄×M₁₆	—	94	1	no	—
169	C₄×M₁₆	—	94	1	no	—
170	C₄×M₁₆	—	94	1	no	—
171	C₄×M₁₆	Dic₄₇	1	4	yes	complemented
172	C₂×#97	—	2	1	no	—
173	C₂×#97	—	47	2	no	—
174	C₂×C₄×C₈	—	2	1	no	—
175	C₂×C₄×C₈	—	47	2	no	—
176	C₄×C₄⋊C₄	—	94	2	no	—
177	C₄×C₂²⋊C₄	—	47	4	no	—
178	#376	—	94	1	no	—
179	C₄³	—	94	1	no	—
180	#344	—	47	16	no	—
181	C₄₇	C₄×#344	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	C₄×D₄₇	1	2	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	47	8	no	—
187	C₂×M₁₆	—	94	2	no	—
188	C₂×M₁₆	—	94	4	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	C₄×D₄₇	1	8	yes	—
191	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	47	4	no	—
194	#117	—	47	8	no	—
195	#117	—	94	4	no	—
196	C₂×C₄⋊C₄	—	94	2	no	—
197	C₂×C₂²⋊C₄	—	47	4	no	—
198	C₂×C₄²	—	94	2	no	—
199	C₂×C₄²	—	94	4	no	—
200	C₂×C₄²	Dic₉₄	1	2	yes	—
201	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
203	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
204	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
206	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
209	#97	—	2	4	no	—
210	#97	—	94	2	no	—
211	#97	—	94	2	no	—
212	C₄×C₈	—	2	4	no	—
213	C₄×C₈	—	94	2	no	—
214	C₄×C₈	—	94	2	no	—
215	#95	—	94	4	no	—
216	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	—	94	4	no	—
219	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
220	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
221	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
222	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
223	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
224	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
225	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	#10625350	1	4	yes	—
230	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	8	no	—
235	M₁₆	—	94	8	no	—
236	M₁₆	—	94	8	no	—
237	M₁₆	C₄×Dic₄₇	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	—	47	8	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	94	4	no	—
243	C₄⋊C₄	—	94	8	no	—
244	C₂²⋊C₄	—	47	8	no	—
245	C₂²⋊C₄	—	94	4	no	—
246	C₄²	—	94	8	no	—
247	C₄²	—	188	8	no	—
248	C₄²	Dic₄₇⋊C₄	1	2	yes	—
249	C₄²	Dic₄₇⋊C₄	1	2	yes	—
250	C₄²	C₄⋊Dic₄₇	1	2	yes	—
251	C₄²	C₄⋊Dic₄₇	1	2	yes	—
252	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
253	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
254	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
255	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
257	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
258	C₂³	—	4	2	no	—
259	C₂³	C₄²×D₄₇	1	2	yes	—
260	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
261	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
262	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
263	C₂³	#13093981	1	2	yes	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	94	8	no	—
271	C₂×C₄	—	188	8	no	—
272	C₂×C₄	C₄₇⋊#95	1	2	yes	—
273	C₂×C₄	C₄₇⋊#95	1	2	yes	—
274	C₂×C₄	C₄₇⋊#95	1	2	yes	—
275	C₂×C₄	C₄₇⋊#95	1	2	yes	—
276	C₂×C₄	C₄²×D₄₇	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
278	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
280	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
282	C₂×C₄	#13093981	1	2	yes	—
283	C₂×C₄	#13093983	1	2	yes	—
284	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
285	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
286	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
287	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
288	C₈	—	2	8	no	—
289	C₈	—	94	8	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
296	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
297	C₂²	#15688969	1	1	yes	characteristic
298	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
299	C₂²	C₄×Dic₄₇⋊C₄	1	2	yes	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	188	8	no	—
304	C₄	#15688895	1	4	yes	direct product factor
305	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
306	C₄	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
307	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
308	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
314	C₂	C₂×#15688895	1	2	yes	—

	subgroup	G/H
center	C₄²	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	C₄×#344
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
11. C₂×#13093935	C₄	semidirect
12. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
17. #15688895	C₄	direct (N × H)
21. C₄×M₁₆×C₄₇	C₄	semidirect
77. M₁₆×C₄₇	C₄²	semidirect
99. C₂×C₄₇⋊C₈	C₄²	semidirect
146. C₄×C₄₇	#344	semidirect
171. C₄×M₁₆	Dic₄₇	semidirect
181. C₄₇	C₄×#344	semidirect
237. M₁₆	C₄×Dic₄₇	semidirect
304. C₄	#15688895	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
C₄×#13093983	2
C₂×#15688895	2
C₂×C₄×M₁₆×C₄₇	2
C₄×#344	47

Subgroups — C4×#15688895 #26121488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688895 #26121488