Subgroups — #26220808

Order 12032.

1390 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×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20244341	C₂	1	4	yes	maximal, complemented
4	#20244337	C₂	1	4	yes	maximal, complemented
5	#20243107	C₂	1	4	yes	maximal
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689147	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
15	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
16	#15688969	—	2	1	no	—
17	C₄²×Dic₄₇	—	2	1	no	—
18	#15688885	C₂²	1	16	yes	—
19	C₂³×Q₈×C₄₇	—	2	1	no	—
20	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
21	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
23	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
24	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	—	2	8	no	—
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	#13093983	—	2	4	no	—
33	#13093983	Q₈	1	8	yes	complemented
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	2	no	—
38	C₂×C₄×Dic₄₇	—	2	2	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
40	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
41	C₄₇⋊#95	—	2	2	no	—
42	C₄₇⋊#95	—	2	2	no	—
43	C₂²×Q₈×C₄₇	—	2	8	no	—
44	C₂²×Q₈×C₄₇	—	2	2	no	—
45	C₂²×Q₈×C₄₇	—	2	2	no	—
46	C₂²×Q₈×C₄₇	—	4	8	no	—
47	C₂³×C₄×C₄₇	—	2	2	no	—
48	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	—	2	8	no	—
52	C₄⋊Q₈×C₄₇	—	2	8	no	—
53	C₄⋊Q₈×C₄₇	—	2	4	no	—
54	C₄⋊Q₈×C₄₇	—	2	4	no	—
55	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
56	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
63	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
64	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
65	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	D₄	1	4	yes	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₄₇	—	2	8	no	—
72	C₂×Q₈×C₄₇	—	2	8	no	—
73	C₂×Q₈×C₄₇	—	4	8	no	—
74	C₂×Q₈×C₄₇	—	4	32	no	—
75	C₂×Q₈×C₄₇	—	4	8	no	—
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	2	8	no	—
79	C₂²×C₄×C₄₇	—	2	4	no	—
80	C₂²×C₄×C₄₇	—	4	8	no	—
81	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
91	C₂²×C₄×C₄₇	#41	1	4	yes	—
92	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²×Dic₄₇	—	2	4	no	—
94	C₄⋊C₄×C₄₇	—	2	8	no	—
95	C₄⋊C₄×C₄₇	—	2	8	no	—
96	C₄⋊C₄×C₄₇	—	2	8	no	—
97	C₄⋊C₄×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
99	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
101	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₇	C₂×D₄	1	2	yes	—
104	C₄²×C₄₇	C₂×D₄	1	2	yes	—
105	C₂²⋊Dic₄₇	—	2	8	no	—
106	C₄⋊Dic₄₇	—	2	4	no	—
107	C₄⋊Dic₄₇	C₂×Q₈	1	8	yes	complemented
108	C₄×Dic₄₇	—	2	8	no	—
109	C₄×Dic₄₇	—	4	4	no	—
110	C₄×Dic₄₇	—	4	4	no	—
111	#10625329	—	4	8	no	—
112	C₂×C₄₇⋊C₈	—	2	4	no	—
113	C₂×C₄₇⋊C₈	—	2	8	no	—
114	C₂³×C₄₇	—	2	2	no	—
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
118	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₄₇	—	4	16	no	—
120	Q₈×C₄₇	—	4	16	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	8	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	2	8	no	—
126	C₂×C₄×C₄₇	—	4	16	no	—
127	C₂×C₄×C₄₇	—	4	8	no	—
128	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
134	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
135	C₂×C₄×C₄₇	#124	1	4	yes	—
136	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
137	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
138	C₂×C₄×C₄₇	#137	1	4	yes	—
139	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
143	C₂×Dic₄₇	—	2	8	no	—
144	C₂×Dic₄₇	—	4	4	no	—
145	C₂×Dic₄₇	—	4	4	no	—
146	C₄₇⋊C₈	—	4	4	no	—
147	#21313	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	8	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	#391	1	4	yes	—
152	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
154	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
155	C₂²×C₄₇	C₂×#137	1	2	yes	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	4	8	no	—
159	C₄×C₄₇	#391	1	4	yes	—
160	C₄×C₄₇	#418	1	4	yes	—
161	C₄×C₄₇	#419	1	4	yes	—
162	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
163	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
164	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	Dic₄₇	—	4	8	no	—
166	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
167	C₂×#423	—	47	1	no	—
168	C₄×#117	—	47	1	no	—
169	#1988	—	47	4	no	—
170	#1987	—	47	4	no	—
171	#1736	—	47	4	no	—
172	C₂×C₄₇	—	2	4	no	—
173	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
174	C₂×C₄₇	C₂×#418	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#419	1	1	yes	characteristic
176	C₂×C₄₇	#3079	1	4	yes	—
177	C₂³×Q₈	—	2	1	no	—
178	C₂²×M₁₆	—	47	2	no	—
179	C₂×C₄⋊Q₈	—	2	1	no	—
180	C₂×C₄⋊Q₈	—	2	1	no	—
181	C₂×C₄⋊Q₈	—	2	4	no	—
182	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
183	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
184	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
185	C₂×#117	—	47	2	no	—
186	C₂²×C₄⋊C₄	—	2	1	no	—
187	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
188	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
189	#423	—	94	4	no	—
190	C₂×C₄⋊C₈	—	47	2	no	—
191	C₂×C₄⋊C₈	—	47	2	no	—
192	C₄×C₄⋊C₄	—	47	2	no	—
193	C₄×C₄⋊C₄	—	47	2	no	—
194	C₄×C₂²⋊C₄	—	94	1	no	—
195	C₄×C₂²⋊C₄	—	94	1	no	—
196	#376	—	94	1	no	—
197	C₄³	—	94	1	no	—
198	#340	—	47	16	no	—
199	C₄₇	#21313	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	2	8	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂²×Q₈	—	4	8	no	—
204	C₂³×C₄	—	2	2	no	—
205	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×M₁₆	—	94	8	no	—
209	C₂×M₁₆	—	94	4	no	—
210	C₂²×C₈	—	47	4	no	—
211	C₄⋊Q₈	—	2	8	no	—
212	C₄⋊Q₈	—	2	8	no	—
213	C₄⋊Q₈	—	2	4	no	—
214	C₄⋊Q₈	—	2	4	no	—
215	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
216	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
217	#117	—	47	8	no	—
218	#117	—	94	4	no	—
219	C₂×C₄⋊C₄	—	2	8	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	4	no	—
224	C₂×C₄⋊C₄	—	47	4	no	—
225	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
226	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
227	C₂×C₂²⋊C₄	—	94	2	no	—
228	C₂×C₄²	—	94	2	no	—
229	C₂×C₄²	—	94	2	no	—
230	C₂×C₄²	—	94	2	no	—
231	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
232	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
235	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
237	C₄⋊C₈	—	94	4	no	—
238	C₄⋊C₈	—	94	4	no	—
239	#95	—	94	2	no	—
240	#95	—	94	2	no	—
241	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	C₂×Q₈	—	2	8	no	—
243	C₂×Q₈	—	2	8	no	—
244	C₂×Q₈	—	4	8	no	—
245	C₂×Q₈	—	4	32	no	—
246	C₂×Q₈	—	4	8	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	8	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	4	8	no	—
252	C₂²×C₄	—	94	4	no	—
253	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
254	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
255	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
256	C₂²×C₄	#10625354	1	4	yes	—
257	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
258	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
264	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
265	M₁₆	—	188	8	no	—
266	C₂×C₈	—	94	4	no	—
267	C₂×C₈	—	94	8	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	47	8	no	—
273	C₄⋊C₄	—	94	4	no	—
274	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
275	C₂²⋊C₄	—	94	8	no	—
276	C₄²	—	94	8	no	—
277	C₄²	—	188	4	no	—
278	C₄²	—	188	4	no	—
279	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
280	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
281	C₄²	C₂²×Dic₄₇	1	2	yes	—
282	C₄²	C₂²×Dic₄₇	1	2	yes	—
283	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
284	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
288	C₂³	#13094000	1	4	yes	—
289	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
290	Q₈	—	4	16	no	—
291	Q₈	—	4	16	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	16	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	94	8	no	—
300	C₂×C₄	—	188	4	no	—
301	C₂×C₄	—	188	4	no	—
302	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
305	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
306	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
307	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
308	C₂×C₄	#13094000	1	4	yes	—
309	C₂×C₄	#13094013	1	4	yes	—
310	C₂×C₄	#13094022	1	4	yes	—
311	C₂×C₄	#13094050	1	4	yes	—
312	C₂×C₄	#13094055	1	4	yes	—
313	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
316	C₂×C₄	C₂×#10625354	1	2	yes	—
317	C₈	—	188	4	no	—
318	C₂²	—	2	8	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	#15689030	1	4	yes	—
322	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
323	C₂²	C₂×#13094000	1	2	yes	—
324	C₂²	C₂×#13094050	1	2	yes	—
325	C₂²	C₂×#13094055	1	2	yes	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	8	no	—
329	C₄	—	188	8	no	—
330	C₄	#15689030	1	4	yes	—
331	C₄	#15689128	1	4	yes	—
332	C₄	#15689132	1	4	yes	—
333	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
334	C₄	C₂×#13094013	1	1	yes	characteristic
335	C₄	C₂×#13094022	1	1	yes	characteristic
336	C₂	—	2	4	no	—
337	C₂	C₂×#15689030	1	1	yes	characteristic
338	C₂	C₂×#15689128	1	1	yes	characteristic
339	C₂	C₂×#15689132	1	1	yes	characteristic
340	C₂	#20253197	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#21313
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #20244341	C₂	semidirect
4. #20244337	C₂	semidirect
23. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
33. #13093983	Q₈	semidirect
107. C₄⋊Dic₄₇	C₂×Q₈	semidirect
182. C₂×C₄⋊Q₈	Dic₄₇	semidirect
199. C₄₇	#21313	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₄×#13093983	2
#20244341	2
#20244337	2
#20243107	2
C₂²×C₄⋊Q₈×C₄₇	2
#21313	47