Subgroups — H(3,ℤ/4)×Dic₄₇ #26121441

Order 12032.

954 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#117×Dic₄₇	C₂	1	2	yes	maximal
3	C₂×#15688880	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688879	C₂	1	2	yes	maximal
5	C₂×C₄₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	M₁₆×Dic₄₇	—	2	4	no	—
10	C₂×#13093934	C₄	1	2	yes	complemented
11	C₂×C₈×Dic₄₇	C₄	1	2	yes	complemented
12	C₄⋊C₄×Dic₄₇	—	2	4	no	—
13	C₂²⋊C₄×Dic₄₇	—	2	4	no	—
14	#15688969	C₄	1	4	yes	complemented
15	C₄²×Dic₄₇	C₄	1	4	yes	complemented
16	#15688880	C₄	1	4	yes	complemented
17	#15688879	C₄	1	8	yes	complemented
18	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₇	C₂²	1	2	yes	—
20	C₄₇×H(3,ℤ/4)	C₄	1	4	yes	complemented
21	C₂³×Dic₄₇	D₄	1	2	yes	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	1	no	—
24	C₂×#10625329	—	2	1	no	—
25	C₂×#10625329	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	C₂×C₄	1	4	yes	—
30	#13093983	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	8	no	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
40	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
41	#13093934	—	2	4	no	—
42	C₈×Dic₄₇	—	2	4	no	—
43	C₄₇⋊#95	—	2	8	no	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	2	2	no	—
52	#117×C₄₇	C₂×C₄	1	4	yes	—
53	#117×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	4	4	no	—
71	C₂²×Dic₄₇	C₂²⋊C₄	1	4	yes	—
72	C₂²×Dic₄₇	C₂²⋊C₄	1	4	yes	—
73	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	C₄²	1	4	yes	complemented
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₂²⋊C₄×C₄₇	—	2	4	no	—
83	C₂²⋊C₄×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	C₄²	1	8	yes	complemented
86	C₂²⋊Dic₄₇	—	2	4	no	—
87	C₂²⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄×Dic₄₇	—	2	4	no	—
91	C₄×Dic₄₇	—	4	16	no	—
92	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
93	C₄×Dic₄₇	C₄²	1	2	yes	—
94	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
95	C₄×Dic₄₇	C₄²	1	8	yes	complemented
96	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
97	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
98	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
99	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
100	#10625329	—	2	4	no	—
101	#10625329	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	C₄²	1	4	yes	complemented
105	C₂³×C₄₇	—	4	2	no	—
106	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₄₇	C₄×D₄	1	2	yes	—
109	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
110	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	—	4	8	no	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
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×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#117	1	2	yes	—
122	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#117	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
131	C₂×Dic₄₇	—	2	8	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	2	8	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	8	no	—
137	C₂×Dic₄₇	#95	1	2	yes	—
138	C₂×Dic₄₇	#95	1	4	yes	—
139	C₂×Dic₄₇	#95	1	2	yes	—
140	C₈×C₄₇	—	2	4	no	—
141	C₄₇⋊C₈	—	2	4	no	—
142	C₄×H(3,ℤ/4)	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	—	4	1	no	—
147	C₂²×C₄₇	—	4	1	no	—
148	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
149	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
150	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
151	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₄₇	—	4	8	no	—
154	C₄×C₄₇	C₄³	1	1	yes	characteristic
155	C₄×C₄₇	#376	1	1	yes	characteristic
156	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
159	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
160	Dic₄₇	—	2	4	no	—
161	Dic₄₇	—	2	2	no	—
162	Dic₄₇	—	4	8	no	—
163	Dic₄₇	H(3,ℤ/4)	1	4	yes	direct product factor
164	C₂×C₄×M₁₆	—	47	1	no	—
165	C₄×#117	—	47	2	no	—
166	C₂×H(3,ℤ/4)	—	47	1	no	—
167	C₂×H(3,ℤ/4)	—	47	2	no	—
168	C₂×H(3,ℤ/4)	D₄₇	1	1	yes	characteristic
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	—	2	2	no	—
171	C₂×C₄₇	—	2	1	no	—
172	C₂×C₄₇	—	2	1	no	—
173	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
174	C₂×C₄₇	C₂×H(3,ℤ/4)	1	2	yes	—
175	C₂²×M₁₆	—	47	1	no	—
176	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
177	C₂×#117	—	47	2	no	—
178	C₂×#117	C₂×D₄₇	1	2	yes	—
179	C₂²×C₄²	—	47	1	no	—
180	C₄×M₁₆	—	94	4	no	—
181	C₂×#97	—	47	2	no	—
182	C₂×C₄×C₈	—	47	2	no	—
183	C₄×C₄⋊C₄	—	94	4	no	—
184	C₄×C₂²⋊C₄	—	94	4	no	—
185	#376	—	47	4	no	—
186	C₄³	—	47	4	no	—
187	H(3,ℤ/4)	—	47	4	no	—
188	H(3,ℤ/4)	—	47	8	no	—
189	H(3,ℤ/4)	Dic₄₇	1	4	yes	direct product factor
190	C₄₇	C₄×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	47	2	no	—
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	47	4	no	—
196	C₂×M₁₆	—	94	2	no	—
197	C₂×M₁₆	—	94	1	no	—
198	C₂×M₁₆	—	94	1	no	—
199	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
200	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	C₄×D₄₇	1	2	yes	—
203	#117	—	2	2	no	—
204	#117	—	2	2	no	—
205	#117	—	47	4	no	—
206	#117	—	47	4	no	—
207	#117	—	94	2	no	—
208	#117	—	94	2	no	—
209	#117	C₄×D₄₇	1	4	yes	—
210	#117	C₂×Dic₄₇	1	4	yes	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	—	94	2	no	—
215	C₂×C₄²	—	47	2	no	—
216	C₂×C₄²	—	47	2	no	—
217	C₂×C₄²	—	47	1	no	—
218	C₂×C₄²	—	47	1	no	—
219	C₂×C₄²	—	47	4	no	—
220	C₂×C₄²	—	47	2	no	—
221	C₂×C₄²	—	47	4	no	—
222	C₂×C₄²	—	94	8	no	—
223	C₂×C₄²	C₄×D₄₇	1	4	yes	—
224	#97	—	94	4	no	—
225	C₄×C₈	—	94	4	no	—
226	#95	—	94	8	no	—
227	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	47	4	no	—
230	C₂²×C₄	—	47	2	no	—
231	C₂²×C₄	—	47	4	no	—
232	C₂²×C₄	—	47	2	no	—
233	C₂²×C₄	—	94	4	no	—
234	C₂²×C₄	—	188	4	no	—
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
239	C₂²×C₄	D₄×D₄₇	1	2	yes	—
240	C₂²×C₄	#10625352	1	1	yes	characteristic
241	C₂²×C₄	#10625352	1	2	yes	—
242	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	#10625354	1	1	yes	characteristic
244	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
245	M₁₆	—	2	4	no	—
246	M₁₆	—	2	4	no	—
247	M₁₆	—	94	4	no	—
248	M₁₆	—	94	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	47	4	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	—	94	2	no	—
254	C₂×C₈	C₄×Dic₄₇	1	4	yes	complemented
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₄⋊C₄	—	94	4	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	94	4	no	—
262	C₂²⋊C₄	—	94	4	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	47	1	no	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	1	no	—
267	C₄²	—	47	2	no	—
268	C₄²	—	47	2	no	—
269	C₄²	—	47	4	no	—
270	C₄²	—	47	4	no	—
271	C₄²	—	47	8	no	—
272	C₄²	—	94	4	no	—
273	C₄²	—	188	16	no	—
274	C₄²	C₄×Dic₄₇	1	8	yes	complemented
275	C₂³	—	4	2	no	—
276	C₂³	C₂²⋊C₄×D₄₇	1	2	yes	—
277	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
278	C₂³	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
279	C₂³	#13093986	1	1	yes	characteristic
280	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	8	no	—
283	C₂×C₄	—	47	2	no	—
284	C₂×C₄	—	47	2	no	—
285	C₂×C₄	—	47	4	no	—
286	C₂×C₄	—	94	8	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	94	8	no	—
289	C₂×C₄	—	188	4	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	—	188	8	no	—
292	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
293	C₂×C₄	C₄²×D₄₇	1	2	yes	—
294	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
297	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
299	C₂×C₄	#13093980	1	1	yes	characteristic
300	C₂×C₄	#13093980	1	2	yes	—
301	C₂×C₄	#13093982	1	1	yes	characteristic
302	C₂×C₄	#13093982	1	2	yes	—
303	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
304	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
305	C₂×C₄	#13093986	1	2	yes	—
306	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
307	C₂×C₄	#13093989	1	1	yes	characteristic
308	C₂×C₄	#13093989	1	2	yes	—
309	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	94	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	#95×D₄₇	1	1	yes	characteristic
318	C₂²	#15688970	1	2	yes	—
319	C₂²	C₂²⋊C₄×Dic₄₇	1	2	yes	—
320	C₂²	#15688974	1	1	yes	characteristic
321	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
322	C₄	—	4	8	no	—
323	C₄	—	47	4	no	—
324	C₄	—	94	4	no	—
325	C₄	—	94	2	no	—
326	C₄	—	188	8	no	—
327	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
328	C₄	#15688968	1	1	yes	characteristic
329	C₄	#15688970	1	1	yes	characteristic
330	C₄	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
331	C₄	#15688974	1	2	yes	—
332	C₄	C₄⋊C₄×Dic₄₇	1	2	yes	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	#95×Dic₄₇	1	1	yes	characteristic
338	C₂	H(3,ℤ/4)×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₇
commutator	C₄×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×H(3,ℤ/4)	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₄₇
hypocenter	C₄₇	C₄×H(3,ℤ/4)
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
10. C₂×#13093934	C₄	semidirect
11. C₂×C₈×Dic₄₇	C₄	semidirect
14. #15688969	C₄	semidirect
15. C₄²×Dic₄₇	C₄	semidirect
16. #15688880	C₄	semidirect
17. #15688879	C₄	semidirect
20. C₄₇×H(3,ℤ/4)	C₄	semidirect
79. C₂×C₈×C₄₇	C₄²	semidirect
85. C₄²×C₄₇	C₄²	semidirect
95. C₄×Dic₄₇	C₄²	semidirect
104. C₂×C₄₇⋊C₈	C₄²	semidirect
163. Dic₄₇	H(3,ℤ/4)	direct (N × H)
189. H(3,ℤ/4)	Dic₄₇	direct (N × H)
190. C₄₇	C₄×H(3,ℤ/4)	semidirect
254. C₂×C₈	C₄×Dic₄₇	semidirect
274. C₄²	C₄×Dic₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×M₁₆×Dic₄₇	2
#117×Dic₄₇	2
C₂×#15688880	2
C₂×#15688879	2
C₂×C₄₇×H(3,ℤ/4)	2
C₄×H(3,ℤ/4)	47

Subgroups — H(3,ℤ/4)×Dic47 #26121441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,ℤ/4)×Dic₄₇ #26121441