Subgroups — C₄×#15689171 #26214386

Order 12032.

1034 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689171	C₂	1	2	yes	maximal
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
4	#20243295	C₂	1	4	yes	maximal, complemented
5	C₄×#13093959	C₂	1	4	yes	maximal, complemented
6	#20243114	C₂	1	2	yes	maximal
7	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	#15689171	C₄	1	16	yes	direct product factor
12	C₂×#13093959	C₂²	1	8	yes	—
13	C₄×#10625329	—	2	1	no	—
14	C₄×#10625329	—	2	1	no	—
15	C₄×#10625329	C₄	1	4	yes	—
16	C₄×#10625329	C₄	1	4	yes	—
17	C₄×#10625329	C₂²	1	2	yes	—
18	C₄×#10625329	C₂²	1	2	yes	—
19	C₂×#13093935	—	2	1	no	—
20	C₂×#13093935	C₂²	1	2	yes	—
21	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
23	#15688887	C₂²	1	8	yes	—
24	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₄₇	—	2	1	no	—
27	C₄×M₁₆×C₄₇	—	2	1	no	—
28	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
29	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
30	C₂×#97×C₄₇	—	2	1	no	—
31	C₂×C₄×C₈×C₄₇	—	2	1	no	—
32	C₂×#10625329	—	2	2	no	—
33	C₂×#10625329	—	2	4	no	—
34	C₂×#10625329	—	2	4	no	—
35	C₂×#10625329	C₂×C₄	1	8	yes	—
36	C₂×#10625329	C₂×C₄	1	2	yes	—
37	C₂×#10625329	C₂×C₄	1	2	yes	—
38	C₂×#10625329	C₂×C₄	1	2	yes	—
39	C₂×#10625329	C₂×C₄	1	2	yes	—
40	C₂×#10625329	C₂×C₄	1	8	yes	—
41	C₂×#10625329	C₂³	1	4	yes	—
42	C₂²×C₄₇⋊C₈	—	2	2	no	—
43	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
45	#13093959	C₂×C₄	1	32	yes	complemented
46	#13093935	—	2	2	no	—
47	#13093935	—	2	2	no	—
48	#13093935	C₂×C₄	1	8	yes	—
49	C₄×C₄₇⋊C₈	—	2	2	no	—
50	C₄×C₄₇⋊C₈	—	2	2	no	—
51	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
52	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
53	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₄₇	—	2	2	no	—
55	C₂×M₁₆×C₄₇	—	2	4	no	—
56	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
57	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₄₇	—	2	2	no	—
59	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	#97×C₄₇	—	2	2	no	—
69	#97×C₄₇	—	2	2	no	—
70	C₄×C₈×C₄₇	—	2	2	no	—
71	C₄×C₈×C₄₇	—	2	2	no	—
72	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	—	4	4	no	—
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	#41	1	4	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
88	M₁₆×C₄₇	—	2	8	no	—
89	M₁₆×C₄₇	—	2	4	no	—
90	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
91	C₂×C₈×C₄₇	—	2	8	no	—
92	C₂×C₈×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
94	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
95	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
96	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	#10625329	—	2	4	no	—
104	#10625329	—	2	4	no	—
105	#10625329	—	2	8	no	—
106	#10625329	—	2	4	no	—
107	#10625329	C₄²	1	8	yes	—
108	#10625329	C₄²	1	8	yes	—
109	#10625329	C₂²×C₄	1	8	yes	—
110	C₂×C₄₇⋊C₈	—	2	8	no	—
111	C₂×C₄₇⋊C₈	—	2	4	no	—
112	C₂×C₄₇⋊C₈	—	2	4	no	—
113	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
114	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
115	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
116	C₂³×C₄₇	—	4	2	no	—
117	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
120	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₄₇	#117	1	2	yes	—
136	C₂×C₄×C₄₇	#117	1	2	yes	—
137	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
138	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
139	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
144	C₈×C₄₇	—	2	8	no	—
145	C₄₇⋊C₈	—	2	4	no	—
146	C₄₇⋊C₈	—	2	8	no	—
147	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
148	C₄×#430	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	4	4	no	—
153	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
154	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
155	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
161	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
162	C₄×C₄₇	#430	1	4	yes	complemented
163	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
164	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
165	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
166	C₂×#430	—	47	2	no	—
167	C₂×C₄×M₁₆	—	47	1	no	—
168	C₂×C₄×M₁₆	—	47	1	no	—
169	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
170	#1778	—	47	4	no	—
171	C₄×#108	—	47	4	no	—
172	#1738	—	47	2	no	—
173	C₂×C₄₇	—	2	2	no	—
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	—	2	2	no	—
176	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#430	1	2	yes	—
178	C₂²×M₁₆	—	47	1	no	—
179	C₂²×M₁₆	—	47	1	no	—
180	C₂²×M₁₆	—	47	2	no	—
181	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	#430	—	47	16	no	—
184	C₂×#108	—	47	8	no	—
185	C₄×M₁₆	—	2	1	no	—
186	C₄×M₁₆	—	2	1	no	—
187	C₄×M₁₆	—	47	4	no	—
188	C₄×M₁₆	—	47	4	no	—
189	C₄×M₁₆	—	47	2	no	—
190	C₄×M₁₆	—	47	2	no	—
191	C₄×M₁₆	—	94	1	no	—
192	C₄×M₁₆	—	94	1	no	—
193	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
194	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
195	C₂×#97	—	2	1	no	—
196	C₂×#97	—	47	2	no	—
197	C₂×#97	—	94	1	no	—
198	C₂×C₄×C₈	—	2	1	no	—
199	C₂×C₄×C₈	—	47	2	no	—
200	C₂×C₄×C₈	—	94	1	no	—
201	#341	—	47	8	no	—
202	C₄₇	C₄×#430	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	Dic₉₄	1	2	yes	—
204	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
205	C₂×M₁₆	—	2	4	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	47	2	no	—
208	C₂×M₁₆	—	47	2	no	—
209	C₂×M₁₆	—	47	8	no	—
210	C₂×M₁₆	—	47	2	no	—
211	C₂×M₁₆	—	47	2	no	—
212	C₂×M₁₆	—	47	8	no	—
213	C₂×M₁₆	—	47	4	no	—
214	C₂×M₁₆	—	94	4	no	—
215	C₂×M₁₆	—	94	4	no	—
216	C₂×M₁₆	—	94	2	no	—
217	C₂×M₁₆	C₄×D₄₇	1	8	yes	—
218	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
219	C₂²×C₈	—	2	2	no	—
220	C₂²×C₈	—	47	2	no	—
221	C₂²×C₈	—	47	2	no	—
222	C₂²×C₈	—	94	2	no	—
223	C₂×C₄²	Dic₉₄	1	2	yes	—
224	C₂×C₄²	C₄×D₄₇	1	2	yes	—
225	C₂×C₄²	C₄×D₄₇	1	2	yes	—
226	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
227	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	#108	—	47	32	no	—
233	#97	—	2	2	no	—
234	#97	—	2	2	no	—
235	#97	—	47	8	no	—
236	#97	—	94	2	no	—
237	#97	—	94	2	no	—
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	—	2	2	no	—
240	C₄×C₈	—	47	8	no	—
241	C₄×C₈	—	94	2	no	—
242	C₄×C₈	—	94	2	no	—
243	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
246	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
247	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
248	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
249	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
250	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
251	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
255	C₂²×C₄	#10625350	1	4	yes	—
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
259	M₁₆	—	2	8	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	47	8	no	—
262	M₁₆	—	47	8	no	—
263	M₁₆	—	47	8	no	—
264	M₁₆	—	94	4	no	—
265	M₁₆	—	94	4	no	—
266	M₁₆	—	94	8	no	—
267	M₁₆	—	94	4	no	—
268	M₁₆	C₂×C₄×D₄₇	1	8	yes	—
269	C₂×C₈	—	2	8	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	47	4	no	—
272	C₂×C₈	—	47	8	no	—
273	C₂×C₈	—	47	4	no	—
274	C₂×C₈	—	94	8	no	—
275	C₂×C₈	—	94	4	no	—
276	C₂×C₈	—	94	4	no	—
277	C₄²	Dic₄₇⋊C₄	1	2	yes	—
278	C₄²	Dic₄₇⋊C₄	1	2	yes	—
279	C₄²	Dic₄₇⋊C₄	1	2	yes	—
280	C₄²	Dic₄₇⋊C₄	1	2	yes	—
281	C₄²	C₂×C₄×D₄₇	1	2	yes	—
282	C₄²	C₂×C₄×D₄₇	1	2	yes	—
283	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
287	C₂³	—	4	2	no	—
288	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
289	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
290	C₂³	C₄×Dic₉₄	1	4	yes	—
291	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	C₄²×D₄₇	1	2	yes	—
299	C₂×C₄	C₄²×D₄₇	1	2	yes	—
300	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
301	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
302	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
303	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
306	C₂×C₄	#13093981	1	2	yes	—
307	C₂×C₄	#13093981	1	2	yes	—
308	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
309	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
310	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
311	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
313	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂×#10625350	1	2	yes	—
315	C₈	—	2	8	no	—
316	C₈	—	47	8	no	—
317	C₈	—	94	4	no	—
318	C₈	—	94	8	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	4	no	—
323	C₂²	C₄×Dic₄₇⋊C₄	1	2	yes	—
324	C₂²	C₄×Dic₄₇⋊C₄	1	2	yes	—
325	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
326	C₂²	C₂×C₄×Dic₉₄	1	2	yes	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
331	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
332	C₄	#15689171	1	4	yes	direct product factor
333	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
334	C₄	C₂×#13093981	1	1	yes	characteristic
335	C₄	C₂×C₄×C₂²⋊D₄₇	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	C₂×C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
340	C₂	C₂×#15689171	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	C₄×#430
LCS	G ▷ C₄×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
4. #20243295	C₂	semidirect
5. C₄×#13093959	C₂	semidirect
11. #15689171	C₄	direct (N × H)
45. #13093959	C₂×C₄	semidirect
162. C₄×C₄₇	#430	semidirect
202. C₄₇	C₄×#430	semidirect
332. C₄	#15689171	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×#15689171	2
C₂×C₄×#10625329	2
#20243295	2
C₄×#13093959	2
#20243114	2
C₂×C₄×M₁₆×C₄₇	2
C₄×#430	47

Subgroups — C4×#15689171 #26214386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689171 #26214386