Subgroups — #27085812

Order 12032.

3250 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689934	C₂	1	4	yes	maximal, complemented
2	#20257173	C₂	1	1	yes	maximal, characteristic, complemented
3	#20253860	C₂	1	12	yes	maximal, complemented
4	#20253848	C₂	1	3	yes	maximal, complemented
5	#20253838	C₂	1	3	yes	maximal, complemented
6	#20253814	C₂	1	3	yes	maximal, complemented
7	C₂×#15689495	C₂	1	3	yes	maximal, complemented
8	C₄×#13094031	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3498	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094071	C₂²	1	2	yes	—
11	#15689934	—	2	32	no	—
12	C₂×#13094054	C₂²	1	12	yes	complemented
13	C₂×#13094050	C₂²	1	12	yes	complemented
14	C₂×#13094044	C₂²	1	12	yes	complemented
15	C₂²×#10625335	C₂²	1	12	yes	complemented
16	C₂×#13094031	C₂²	1	2	yes	—
17	Q₈×D₁₈₈	—	2	2	no	—
18	#15689777	—	2	6	no	—
19	#15689699	C₂²	1	6	yes	complemented
20	#15689673	C₂²	1	3	yes	complemented
21	C₂×#13094024	C₂²	1	3	yes	complemented
22	#15689518	C₂²	1	12	yes	complemented
23	#15689510	C₂²	1	12	yes	complemented
24	#15689505	C₂²	1	12	yes	complemented
25	#15689502	C₂²	1	3	yes	complemented
26	#15689499	C₂²	1	3	yes	complemented
27	#15689497	C₂²	1	1	yes	characteristic, complemented
28	#15689495	C₂²	1	3	yes	complemented
29	#15689495	C₂²	1	6	yes	complemented
30	#15689495	C₂²	1	12	yes	complemented
31	#15689079	C₂²	1	6	yes	complemented
32	C₄×#10625329	C₂²	1	3	yes	complemented
33	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	complemented
34	C₂×H(5,𝔽₂)×C₄₇	C₂²	1	2	yes	—
35	C₄₇×#550	—	2	2	no	—
36	D₄²×C₄₇	—	2	6	no	—
37	C₄₇×#535	C₂²	1	6	yes	—
38	C₄₇×#532	C₂²	1	3	yes	—
39	C₂×#127×C₄₇	C₂²	1	3	yes	—
40	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
41	#13094071	—	2	16	no	—
42	C₂×#10625350	C₂³	1	6	yes	—
43	C₂×#10625352	—	2	6	no	—
44	C₂²×Dic₉₄	C₂³	1	6	yes	—
45	C₂×Q₈×D₄₇	—	2	2	no	—
46	#13094054	—	2	48	no	—
47	#13094050	—	2	48	no	—
48	#13094044	—	2	48	no	—
49	C₂×#10625337	C₂³	1	12	yes	—
50	C₂×#10625335	—	2	48	no	—
51	C₂×#10625335	C₂³	1	12	yes	—
52	C₂×#10625335	C₂³	1	24	yes	—
53	C₂×#10625336	C₂³	1	4	yes	—
54	C₂×#10625334	C₂³	1	12	yes	—
55	#13094031	D₄	1	16	yes	complemented
56	C₂×#10625329	C₂³	1	6	yes	—
57	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
58	#13094024	—	2	6	no	—
59	#13094024	C₂³	1	3	yes	—
60	#13094024	C₂³	1	6	yes	—
61	#13094021	C₂³	1	1	yes	characteristic
62	#13094012	C₂³	1	6	yes	—
63	#13094008	—	2	12	no	—
64	#13094004	—	2	12	no	—
65	#13093998	—	2	6	no	—
66	C₄×D₁₈₈	—	2	6	no	—
67	C₂×C₄⋊Dic₄₇	—	2	6	no	—
68	#13093935	C₂³	1	6	yes	—
69	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₄×C₄₇⋊C₈	C₂³	1	3	yes	—
71	C₄×C₄₇⋊C₈	C₂³	1	6	yes	—
72	H(5,𝔽₂)×C₄₇	—	2	16	no	—
73	C₂×#41×C₄₇	—	2	2	no	—
74	C₂×#41×C₄₇	C₂³	1	6	yes	—
75	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₄₇	—	2	6	no	—
77	C₂²×D₄×C₄₇	—	2	6	no	—
78	C₂²×D₄×C₄₇	C₂³	1	6	yes	—
79	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
80	#127×C₄₇	—	2	6	no	—
81	#127×C₄₇	C₂³	1	3	yes	—
82	#127×C₄₇	C₂³	1	6	yes	—
83	#124×C₄₇	C₂³	1	6	yes	—
84	#121×C₄₇	—	2	6	no	—
85	#121×C₄₇	—	2	12	no	—
86	#120×C₄₇	—	2	12	no	—
87	C₄×Q₈×C₄₇	D₄	1	2	yes	—
88	C₄×D₄×C₄₇	—	2	6	no	—
89	C₄×D₄×C₄₇	D₄	1	6	yes	—
90	#117×C₄₇	C₂³	1	3	yes	—
91	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
92	C₂⁴×C₄₇	—	2	6	no	—
93	#41×C₄₇	—	2	24	no	—
94	#41×C₄₇	—	4	8	no	—
95	#41×C₄₇	C₂×D₄	1	8	yes	—
96	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×Q₈×C₄₇	C₂⁴	1	2	yes	—
98	C₂×D₄×C₄₇	—	2	6	no	—
99	C₂×D₄×C₄₇	—	2	24	no	—
100	C₂×D₄×C₄₇	—	2	6	no	—
101	C₂×D₄×C₄₇	—	2	6	no	—
102	C₂×D₄×C₄₇	—	2	12	no	—
103	C₂×D₄×C₄₇	—	4	24	no	—
104	C₂×D₄×C₄₇	—	8	12	no	—
105	C₂×D₄×C₄₇	C₂×D₄	1	3	yes	—
106	C₂×D₄×C₄₇	C₂⁴	1	6	yes	—
107	C₂²×C₄×C₄₇	—	2	6	no	—
108	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
109	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
110	C₂×C₂²⋊D₄₇	—	2	6	no	—
111	C₂²×Dic₄₇	—	2	6	no	—
112	Q₈×D₄₇	—	4	8	no	—
113	#10625352	—	4	24	no	—
114	#10625350	—	2	24	no	—
115	C₂×D₁₈₈	—	2	2	no	—
116	C₂×D₁₈₈	C₂⁴	1	2	yes	—
117	C₂×C₄×D₄₇	—	2	6	no	—
118	C₂×Dic₉₄	—	2	24	no	—
119	C₂×Dic₉₄	—	2	6	no	—
120	C₂×Dic₉₄	C₂⁴	1	6	yes	—
121	C₄⋊C₄×C₄₇	—	2	2	no	—
122	C₄⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
123	C₂²⋊C₄×C₄₇	—	2	12	no	—
124	C₂²⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
125	C₄²×C₄₇	C₂×D₄	1	6	yes	—
126	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
127	C₄²×C₄₇	C₂⁴	1	3	yes	—
128	#10625337	—	2	24	no	—
129	#10625336	—	2	8	no	—
130	#10625335	—	2	24	no	—
131	#10625335	—	2	48	no	—
132	#10625334	—	2	24	no	—
133	(C₂×D₄₇)⋊C₄	—	2	12	no	—
134	C₄⋊Dic₄₇	—	2	6	no	—
135	C₄⋊Dic₄₇	—	2	12	no	—
136	#10625329	C₂×D₄	1	24	yes	complemented
137	C₂×C₄₇⋊C₈	C₂×D₄	1	24	yes	complemented
138	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
139	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
140	C₂³×C₄₇	—	2	6	no	—
141	C₂³×C₄₇	—	4	12	no	—
142	C₂³×C₄₇	—	8	12	no	—
143	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
144	C₂²×D₄₇	—	2	2	no	—
145	Q₈×C₄₇	—	2	4	no	—
146	Q₈×C₄₇	#127	1	4	yes	—
147	D₄×C₄₇	—	2	12	no	—
148	D₄×C₄₇	—	4	12	no	—
149	D₄×C₄₇	—	4	12	no	—
150	D₄×C₄₇	—	4	12	no	—
151	D₄×C₄₇	—	8	12	no	—
152	D₄×C₄₇	#127	1	12	yes	—
153	C₂×C₄×C₄₇	—	2	12	no	—
154	C₂×C₄×C₄₇	—	2	2	no	—
155	C₂×C₄×C₄₇	—	4	12	no	—
156	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
157	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
158	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
159	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
160	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
161	C₂²⋊D₄₇	—	4	12	no	—
162	C₂×Dic₄₇	—	2	6	no	—
163	C₂×Dic₄₇	—	4	12	no	—
164	D₁₈₈	—	2	4	no	—
165	D₁₈₈	—	4	4	no	—
166	C₄×D₄₇	—	4	12	no	—
167	Dic₉₄	—	2	12	no	—
168	Dic₉₄	—	4	12	no	—
169	C₄₇⋊C₈	C₂²×D₄	1	4	yes	—
170	C₄₇⋊C₈	C₂²×D₄	1	12	yes	—
171	#60646	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	6	no	—
173	C₂²×C₄₇	—	4	6	no	—
174	C₂²×C₄₇	—	4	12	no	—
175	C₂²×C₄₇	—	16	12	no	—
176	C₂²×C₄₇	C₂×#127	1	6	yes	—
177	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
178	C₂×D₄₇	—	4	2	no	—
179	C₂×D₄₇	—	4	4	no	—
180	C₄×C₄₇	—	2	6	no	—
181	C₄×C₄₇	—	4	2	no	—
182	C₄×C₄₇	C₂×#127	1	6	yes	—
183	C₄×C₄₇	#577	1	4	yes	complemented
184	C₄×C₄₇	C₂³×D₄	1	2	yes	—
185	Dic₄₇	—	4	6	no	—
186	C₂×#577	—	47	4	no	—
187	#3499	—	47	1	no	—
188	#3498	D₄₇	1	1	yes	characteristic, complemented
189	#3154	—	47	12	no	—
190	#3152	—	47	3	no	—
191	#3151	—	47	3	no	—
192	#3147	—	47	3	no	—
193	C₂×#492	—	47	3	no	—
194	C₄×#131	—	47	1	no	—
195	C₂×C₄₇	—	2	6	no	—
196	C₂×C₄₇	—	8	6	no	—
197	C₂×C₄₇	C₂²×#127	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#577	1	2	yes	—
199	D₄₇	—	8	2	no	—
200	C₂×2₋¹⁺⁴	—	47	2	no	—
201	C₂×H(5,𝔽₂)	C₂×D₄₇	1	2	yes	—
202	#577	—	94	32	no	—
203	C₂×#137	—	47	12	no	—
204	C₂×AGL(1,ℤ/8)	—	47	12	no	—
205	C₂×#135	—	47	12	no	—
206	C₂²×SD₁₆	—	47	12	no	—
207	C₂×#131	—	47	2	no	—
208	#550	—	2	2	no	—
209	D₄×Q₈	—	94	2	no	—
210	#547	—	94	6	no	—
211	D₄²	—	2	6	no	—
212	#536	—	47	6	no	—
213	#535	C₂×D₄₇	1	6	yes	—
214	#532	—	47	3	no	—
215	#532	C₂×D₄₇	1	3	yes	—
216	C₂×C₄⋊Q₈	—	47	3	no	—
217	C₂×#127	C₂×D₄₇	1	3	yes	—
218	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
219	#497	—	47	12	no	—
220	#496	—	47	12	no	—
221	#495	—	47	12	no	—
222	#494	—	47	3	no	—
223	#493	—	47	3	no	—
224	#492	—	47	1	no	—
225	#492	—	47	3	no	—
226	#492	—	47	6	no	—
227	#492	—	47	12	no	—
228	#405	—	47	6	no	—
229	C₄×M₁₆	—	47	3	no	—
230	C₂×C₄×C₈	—	47	3	no	—
231	C₄₇	#60646	1	1	yes	characteristic, complemented, Sylow
232	2₋¹⁺⁴	—	94	16	no	—
233	H(5,𝔽₂)	—	2	16	no	—
234	C₂×#41	—	2	2	no	—
235	C₂×#41	—	47	6	no	—
236	C₂×#41	—	94	6	no	—
237	C₂×#41	C₂²×D₄₇	1	6	yes	—
238	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
239	C₂²×Q₈	—	47	6	no	—
240	C₂²×Q₈	—	94	2	no	—
241	C₂²×D₄	—	2	6	no	—
242	C₂²×D₄	—	2	6	no	—
243	C₂²×D₄	C₂²×D₄₇	1	6	yes	—
244	#137	—	94	48	no	—
245	AGL(1,ℤ/8)	—	94	48	no	—
246	#135	—	94	48	no	—
247	C₂×Q₁₆	—	47	12	no	—
248	C₂×SD₁₆	—	47	4	no	—
249	C₂×SD₁₆	—	47	12	no	—
250	C₂×SD₁₆	—	47	24	no	—
251	C₂×SD₁₆	—	94	48	no	—
252	C₂×D₈	—	47	12	no	—
253	#131	—	47	16	no	—
254	C₂×M₁₆	—	47	6	no	—
255	C₂²×C₈	—	47	6	no	—
256	C₄⋊Q₈	—	47	3	no	—
257	C₄⋊Q₈	—	47	6	no	—
258	C₄⋊Q₈	—	94	6	no	—
259	C₄⋊Q₈	C₂²×D₄₇	1	1	yes	characteristic
260	#127	—	2	6	no	—
261	#127	—	47	1	no	—
262	#127	C₂²×D₄₇	1	3	yes	—
263	#127	C₂²×D₄₇	1	6	yes	—
264	#124	—	47	6	no	—
265	#124	C₂²×D₄₇	1	6	yes	—
266	#123	—	94	12	no	—
267	C₂²⋊Q₈	—	94	12	no	—
268	#121	—	2	6	no	—
269	#121	—	2	12	no	—
270	#121	—	94	6	no	—
271	#120	—	2	12	no	—
272	C₄×Q₈	C₂²⋊D₄₇	1	2	yes	—
273	C₄×D₄	—	2	6	no	—
274	C₄×D₄	—	94	6	no	—
275	C₄×D₄	C₂²⋊D₄₇	1	6	yes	—
276	#117	C₂²×D₄₇	1	3	yes	—
277	C₂×C₄⋊C₄	—	94	6	no	—
278	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
279	#97	—	47	6	no	—
280	C₄×C₈	—	47	1	no	—
281	C₄×C₈	—	47	3	no	—
282	C₄×C₈	—	47	6	no	—
283	C₂⁴	—	2	6	no	—
284	#41	—	2	24	no	—
285	#41	—	4	8	no	—
286	#41	—	94	24	no	—
287	#41	—	188	24	no	—
288	#41	D₄×D₄₇	1	8	yes	—
289	C₂×Q₈	—	47	6	no	—
290	C₂×Q₈	—	94	6	no	—
291	C₂×Q₈	—	94	24	no	—
292	C₂×Q₈	—	188	8	no	—
293	C₂×Q₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
294	C₂×Q₈	C₂³×D₄₇	1	2	yes	—
295	C₂×D₄	—	2	6	no	—
296	C₂×D₄	—	2	24	no	—
297	C₂×D₄	—	2	6	no	—
298	C₂×D₄	—	2	6	no	—
299	C₂×D₄	—	2	12	no	—
300	C₂×D₄	—	4	24	no	—
301	C₂×D₄	—	8	12	no	—
302	C₂×D₄	—	47	2	no	—
303	C₂×D₄	—	94	6	no	—
304	C₂×D₄	—	94	2	no	—
305	C₂×D₄	C₂×C₂²⋊D₄₇	1	3	yes	—
306	C₂×D₄	C₂³×D₄₇	1	6	yes	—
307	C₂²×C₄	—	2	6	no	—
308	C₂²×C₄	—	94	6	no	—
309	C₂²×C₄	—	94	6	no	—
310	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
311	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
312	Q₁₆	—	94	24	no	—
313	SD₁₆	—	94	24	no	—
314	SD₁₆	—	94	48	no	—
315	SD₁₆	—	94	8	no	—
316	D₈	—	94	24	no	—
317	M₁₆	—	47	24	no	—
318	C₂×C₈	—	47	24	no	—
319	C₂×C₈	—	47	2	no	—
320	C₂×C₈	—	47	6	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	94	6	no	—
323	C₄⋊C₄	—	94	12	no	—
324	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
325	C₂²⋊C₄	—	2	12	no	—
326	C₂²⋊C₄	—	94	12	no	—
327	C₂²⋊C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
328	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
329	C₄²	C₂³×D₄₇	1	1	yes	characteristic
330	C₄²	C₂³×D₄₇	1	3	yes	—
331	C₂³	—	2	6	no	—
332	C₂³	—	4	12	no	—
333	C₂³	—	8	12	no	—
334	C₂³	—	94	2	no	—
335	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
336	Q₈	—	2	4	no	—
337	Q₈	—	94	12	no	—
338	Q₈	—	188	12	no	—
339	Q₈	#13094020	1	4	yes	—
340	D₄	—	2	12	no	—
341	D₄	—	4	12	no	—
342	D₄	—	4	12	no	—
343	D₄	—	4	12	no	—
344	D₄	—	8	12	no	—
345	D₄	—	94	4	no	—
346	D₄	—	188	12	no	—
347	D₄	—	188	4	no	—
348	D₄	#13094020	1	12	yes	—
349	C₂×C₄	—	2	12	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	12	no	—
352	C₂×C₄	—	94	6	no	—
353	C₂×C₄	—	188	12	no	—
354	C₂×C₄	—	188	12	no	—
355	C₂×C₄	C₂×D₄×D₄₇	1	12	yes	—
356	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
357	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
358	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
359	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
360	C₈	—	47	4	no	—
361	C₈	—	47	12	no	—
362	C₂²	—	2	6	no	—
363	C₂²	—	4	12	no	—
364	C₂²	—	4	6	no	—
365	C₂²	—	16	12	no	—
366	C₂²	—	188	4	no	—
367	C₂²	—	188	2	no	—
368	C₂²	C₂×#13094020	1	6	yes	—
369	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
370	C₄	—	2	6	no	—
371	C₄	—	4	2	no	—
372	C₄	—	188	6	no	—
373	C₄	C₂×#13094020	1	6	yes	—
374	C₄	#15689934	1	4	yes	complemented
375	C₄	C₂²×D₄×D₄₇	1	2	yes	—
376	C₂	—	2	6	no	—
377	C₂	—	8	6	no	—
378	C₂	—	376	2	no	—
379	C₂	C₂²×#13094020	1	1	yes	characteristic
380	C₂	C₂×#15689934	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3498	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3498	D₄₇
hypocenter	C₄₇	#60646
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #3498

Complements

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

N	H ≅ G/N	type
1. C₂×#15689934	C₂	semidirect
2. #20257173	C₂	semidirect
3. #20253860	C₂	semidirect
4. #20253848	C₂	semidirect
5. #20253838	C₂	semidirect
6. #20253814	C₂	semidirect
7. C₂×#15689495	C₂	semidirect
8. C₄×#13094031	C₂	semidirect
9. C₄₇×#3498	C₂	semidirect
12. C₂×#13094054	C₂²	semidirect
13. C₂×#13094050	C₂²	semidirect
14. C₂×#13094044	C₂²	semidirect
15. C₂²×#10625335	C₂²	semidirect
19. #15689699	C₂²	semidirect
20. #15689673	C₂²	semidirect
21. C₂×#13094024	C₂²	semidirect
22. #15689518	C₂²	semidirect
23. #15689510	C₂²	semidirect
24. #15689505	C₂²	semidirect
25. #15689502	C₂²	semidirect
26. #15689499	C₂²	semidirect
27. #15689497	C₂²	semidirect
28. #15689495	C₂²	semidirect
29. #15689495	C₂²	semidirect
30. #15689495	C₂²	semidirect
31. #15689079	C₂²	semidirect
32. C₄×#10625329	C₂²	semidirect
33. C₂×C₄×C₄₇⋊C₈	C₂²	semidirect
55. #13094031	D₄	semidirect
136. #10625329	C₂×D₄	semidirect
137. C₂×C₄₇⋊C₈	C₂×D₄	semidirect
183. C₄×C₄₇	#577	semidirect
188. #3498	D₄₇	semidirect
231. C₄₇	#60646	semidirect
374. C₄	#15689934	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689934	2
#20257173	2
#20253860	2
#20253848	2
#20253838	2
#20253814	2
C₂×#15689495	2
C₄×#13094031	2
C₄₇×#3498	2
#60646	47