Subgroups — C₄×#15689158 #26214325

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689158	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093982	C₂	1	2	yes	maximal
4	C₄×#13093957	C₂	1	2	yes	maximal, complemented
5	#20243280	C₂	1	2	yes	maximal, complemented
6	#106×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₄×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243103	C₂	1	2	yes	maximal
9	C₄×#106×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
12	C₂×#13093982	C₂²	1	4	yes	—
13	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
14	#15689158	C₄	1	16	yes	direct product factor
15	C₂×#13093957	C₂²	1	4	yes	—
16	C₂×#13093956	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×D₄₇	C₄	1	8	yes	complemented
21	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	C₄×(C₂×D₄₇)⋊C₄	—	2	2	no	—
24	#15688970	—	2	2	no	—
25	#15688968	—	2	2	no	—
26	C₄²×Dic₄₇	—	2	2	no	—
27	#15688886	C₂²	1	2	yes	—
28	#15688885	C₂²	1	4	yes	—
29	C₂×#106×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
32	C₄₇×#340	C₂²	1	2	yes	—
33	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂×C₈×D₄₇	—	2	4	no	—
39	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
42	#13093982	—	2	8	no	—
43	#13093982	C₂×C₄	1	16	yes	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
45	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
46	C₂×C₄×Dic₄₇	—	2	4	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	—	2	4	no	—
49	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₄²×D₄₇	C₂×C₄	1	4	yes	—
53	C₄²×D₄₇	Q₈	1	4	yes	—
54	#13093957	C₂×C₄	1	16	yes	complemented
55	#13093956	C₂×C₄	1	8	yes	complemented
56	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
57	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
58	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
59	C₄₇⋊#95	—	2	2	no	—
60	C₄₇⋊#95	—	2	4	no	—
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
64	C₂×C₄²×C₄₇	—	2	2	no	—
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	#106×C₄₇	C₂×C₄	1	8	yes	complemented
67	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
68	#95×C₄₇	—	2	2	no	—
69	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×D₄₇	—	2	4	no	—
79	C₂×C₄×D₄₇	—	2	4	no	—
80	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
81	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
82	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
84	C₂×C₄×D₄₇	C₂×Q₈	1	2	yes	—
85	C₂×C₄×D₄₇	#41	1	4	yes	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
92	C₄²×C₄₇	—	4	4	no	—
93	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
95	(C₂×D₄₇)⋊C₄	—	2	16	no	—
96	C₄⋊Dic₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
98	C₄×Dic₄₇	—	2	16	no	—
99	C₄×Dic₄₇	—	4	4	no	—
100	C₄×Dic₄₇	—	4	8	no	—
101	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
102	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
103	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
104	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
109	C₈×D₄₇	—	2	8	no	—
110	C₈×D₄₇	C₄²	1	16	yes	complemented
111	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂²×D₄₇	—	2	2	no	—
113	C₂²×D₄₇	—	2	2	no	—
114	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂²×D₄₇	C₄×D₄	1	4	yes	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	8	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₇	#135	1	4	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
127	C₂×Dic₄₇	—	2	2	no	—
128	C₂×Dic₄₇	—	2	8	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₄₇	C₄×D₄	1	4	yes	—
135	C₂×Dic₄₇	#135	1	4	yes	—
136	C₄×D₄₇	—	2	8	no	—
137	C₄×D₄₇	—	2	8	no	—
138	C₄×D₄₇	C₂×C₄²	1	4	yes	—
139	C₄×D₄₇	#117	1	4	yes	—
140	C₄×D₄₇	C₄×Q₈	1	8	yes	—
141	C₈×C₄₇	—	2	2	no	—
142	C₈×C₄₇	C₂×C₄²	1	4	yes	—
143	C₄₇⋊C₈	—	2	2	no	—
144	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
145	C₄×#427	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
148	C₂²×C₄₇	C₂×#135	1	2	yes	—
149	C₂²×C₄₇	C₂×#135	1	2	yes	—
150	C₂×D₄₇	—	2	2	no	—
151	C₂×D₄₇	—	2	2	no	—
152	C₂×D₄₇	—	2	8	no	—
153	C₂×D₄₇	C₄×C₄⋊C₄	1	4	yes	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	4	8	no	—
156	C₄×C₄₇	#427	1	4	yes	complemented
157	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
158	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
159	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
160	Dic₄₇	—	2	2	no	—
161	Dic₄₇	—	4	8	no	—
162	Dic₄₇	C₄×C₄⋊C₄	1	4	yes	—
163	Dic₄₇	#427	1	4	yes	complemented
164	C₂×#427	—	47	2	no	—
165	C₂²×C₄×C₈	—	47	1	no	—
166	C₄×#117	—	47	2	no	—
167	C₄×#107	—	47	2	no	—
168	C₄×#107	—	47	2	no	—
169	C₄×#106	—	47	2	no	—
170	C₄×#106	—	47	1	no	—
171	C₄×#106	D₄₇	1	1	yes	characteristic, complemented
172	#1735	—	47	2	no	—
173	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
176	C₂×C₄₇	C₄×#135	1	4	yes	—
177	D₄₇	—	2	4	no	—
178	C₂³×C₈	—	47	1	no	—
179	C₂³×C₈	—	47	1	no	—
180	C₂×#117	—	47	4	no	—
181	C₂²×C₄²	—	47	1	no	—
182	#427	—	47	16	no	—
183	C₂×#107	—	47	4	no	—
184	C₂×#106	—	47	2	no	—
185	C₂×#106	C₂×D₄₇	1	2	yes	—
186	C₂×C₄×C₈	—	47	1	no	—
187	C₂×C₄×C₈	—	47	1	no	—
188	C₂×C₄×C₈	—	47	1	no	—
189	C₂×C₄×C₈	—	47	8	no	—
190	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	47	4	no	—
192	C₄×C₄⋊C₄	—	47	2	no	—
193	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
194	C₄×C₂²⋊C₄	—	94	2	no	—
195	C₄×C₂²⋊C₄	—	94	2	no	—
196	#376	—	94	2	no	—
197	C₄³	—	94	2	no	—
198	#340	—	47	2	no	—
199	#340	—	47	4	no	—
200	#340	C₂×D₄₇	1	2	yes	—
201	C₄₇	C₄×#427	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₄	—	47	1	no	—
203	C₂³×C₄	—	47	1	no	—
204	C₂³×C₄	—	47	1	no	—
205	C₂²×C₈	—	47	1	no	—
206	C₂²×C₈	—	47	4	no	—
207	C₂²×C₈	—	47	8	no	—
208	C₂²×C₈	—	47	1	no	—
209	C₂²×C₈	—	47	4	no	—
210	C₂²×C₈	—	94	4	no	—
211	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	#117	—	47	16	no	—
214	#117	—	94	8	no	—
215	C₂×C₄⋊C₄	—	47	4	no	—
216	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
217	C₂×C₂²⋊C₄	—	94	4	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	47	1	no	—
220	C₂×C₄²	—	47	1	no	—
221	C₂×C₄²	—	47	1	no	—
222	C₂×C₄²	—	47	4	no	—
223	C₂×C₄²	—	47	4	no	—
224	C₂×C₄²	—	94	4	no	—
225	C₂×C₄²	—	94	4	no	—
226	C₂×C₄²	—	94	2	no	—
227	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
228	#107	—	47	16	no	—
229	#106	—	47	8	no	—
230	#106	C₄×D₄₇	1	8	yes	complemented
231	C₄×C₈	—	47	4	no	—
232	C₄×C₈	—	47	4	no	—
233	C₄×C₈	—	47	4	no	—
234	C₄×C₈	C₄×D₄₇	1	4	yes	—
235	#95	—	2	2	no	—
236	#95	—	94	4	no	—
237	#95	—	94	2	no	—
238	C₂⁴	—	47	1	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	47	1	no	—
241	C₂²×C₄	—	47	1	no	—
242	C₂²×C₄	—	47	2	no	—
243	C₂²×C₄	—	47	4	no	—
244	C₂²×C₄	—	47	2	no	—
245	C₂²×C₄	—	47	4	no	—
246	C₂²×C₄	—	47	1	no	—
247	C₂²×C₄	—	47	4	no	—
248	C₂²×C₄	—	47	4	no	—
249	C₂²×C₄	—	94	4	no	—
250	C₂²×C₄	—	94	4	no	—
251	C₂²×C₄	—	94	4	no	—
252	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	47	2	no	—
257	C₂×C₈	—	47	4	no	—
258	C₂×C₈	—	47	16	no	—
259	C₂×C₈	—	47	2	no	—
260	C₂×C₈	—	94	8	no	—
261	C₂×C₈	—	94	2	no	—
262	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
263	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	47	8	no	—
267	C₄⋊C₄	—	94	4	no	—
268	C₄⋊C₄	C₂×C₄×D₄₇	1	8	yes	—
269	C₂²⋊C₄	—	94	16	no	—
270	C₄²	—	4	4	no	—
271	C₄²	—	47	4	no	—
272	C₄²	—	47	4	no	—
273	C₄²	—	47	2	no	—
274	C₄²	—	47	2	no	—
275	C₄²	—	94	16	no	—
276	C₄²	—	188	8	no	—
277	C₄²	—	188	4	no	—
278	C₄²	C₂×C₄×D₄₇	1	2	yes	—
279	C₄²	Q₈×D₄₇	1	2	yes	—
280	C₂³	—	47	2	no	—
281	C₂³	—	47	4	no	—
282	C₂³	—	94	2	no	—
283	C₂³	—	94	2	no	—
284	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	47	2	no	—
290	C₂×C₄	—	47	2	no	—
291	C₂×C₄	—	47	4	no	—
292	C₂×C₄	—	47	4	no	—
293	C₂×C₄	—	47	4	no	—
294	C₂×C₄	—	47	4	no	—
295	C₂×C₄	—	47	8	no	—
296	C₂×C₄	—	47	2	no	—
297	C₂×C₄	—	94	8	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	8	no	—
300	C₂×C₄	—	94	8	no	—
301	C₂×C₄	—	188	4	no	—
302	C₂×C₄	—	188	4	no	—
303	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
304	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
305	C₂×C₄	#13094045	1	4	yes	—
306	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
308	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
309	C₂×C₄	#41×D₄₇	1	2	yes	—
310	C₈	—	2	2	no	—
311	C₈	—	47	4	no	—
312	C₈	—	94	2	no	—
313	C₈	C₄²×D₄₇	1	4	yes	—
314	C₂²	—	47	4	no	—
315	C₂²	—	94	2	no	—
316	C₂²	—	94	2	no	—
317	C₂²	—	94	8	no	—
318	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
319	C₂²	C₄×D₄×D₄₇	1	2	yes	—
320	C₂²	#135×D₄₇	1	2	yes	—
321	C₂²	C₂×#13094045	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	4	8	no	—
324	C₄	—	47	4	no	—
325	C₄	—	47	4	no	—
326	C₄	—	94	2	no	—
327	C₄	—	188	8	no	—
328	C₄	#15689158	1	4	yes	direct product factor
329	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
330	C₄	#117×D₄₇	1	1	yes	characteristic
331	C₄	C₄×Q₈×D₄₇	1	2	yes	—
332	C₂	—	94	4	no	—
333	C₂	C₄×C₄⋊C₄×D₄₇	1	1	yes	characteristic
334	C₂	#427×D₄₇	1	1	yes	characteristic
335	C₂	C₂×#15689158	1	1	yes	characteristic
336	C₂	C₄×#13094045	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#106×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#106	D₄₇
hypocenter	C₄₇	C₄×#427
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
4. C₄×#13093957	C₂	semidirect
5. #20243280	C₂	semidirect
6. #106×Dic₄₇	C₂	semidirect
7. C₄×#13093956	C₂	semidirect
9. C₄×#106×C₄₇	C₂	semidirect
14. #15689158	C₄	direct (N × H)
20. C₄×C₈×D₄₇	C₄	semidirect
54. #13093957	C₂×C₄	semidirect
55. #13093956	C₂×C₄	semidirect
66. #106×C₄₇	C₂×C₄	semidirect
110. C₈×D₄₇	C₄²	semidirect
156. C₄×C₄₇	#427	semidirect
163. Dic₄₇	#427	semidirect
171. C₄×#106	D₄₇	semidirect
201. C₄₇	C₄×#427	semidirect
230. #106	C₄×D₄₇	semidirect
328. C₄	#15689158	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×#15689158	2
C₂×C₄×C₈×D₄₇	2
C₄×#13093982	2
C₄×#13093957	2
#20243280	2
#106×Dic₄₇	2
C₄×#13093956	2
#20243103	2
C₄×#106×C₄₇	2
C₄×#427	47

Subgroups — C4×#15689158 #26214325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689158 #26214325