Subgroups — #27108459

Order 12032.

2168 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20257166	C₂	1	1	yes	maximal, characteristic
2	#20256365	C₂	1	2	yes	maximal, complemented
3	#20256059	C₂	1	2	yes	maximal, complemented
4	#20255206	C₂	1	6	yes	maximal, complemented
5	D₄×#10625336	C₂	1	6	yes	maximal, complemented
6	#20254700	C₂	1	3	yes	maximal, complemented
7	#20254326	C₂	1	3	yes	maximal, complemented
8	#20253419	C₂	1	3	yes	maximal, complemented
9	C₂×#15689323	C₂	1	3	yes	maximal, complemented
10	#20252499	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094070	—	2	1	no	—
13	C₂²×#10625336	C₂²	1	6	yes	complemented
14	#15689795	—	2	1	no	—
15	#15689795	C₂²	1	2	yes	—
16	D₄×D₁₈₈	—	2	3	no	—
17	D₄×D₁₈₈	C₂²	1	6	yes	—
18	#15689692	—	2	3	no	—
19	#15689673	C₂²	1	3	yes	—
20	C₂×#13094021	C₂²	1	3	yes	—
21	#15689497	C₂²	1	6	yes	complemented
22	#15689471	C₂²	1	6	yes	complemented
23	#15689441	C₂²	1	6	yes	complemented
24	#15689410	C₂²	1	6	yes	complemented
25	#15689408	C₂²	1	1	yes	characteristic, complemented
26	#15689408	C₂²	1	3	yes	complemented
27	#15689354	C₂²	1	12	yes	complemented
28	#15689325	C₂²	1	6	yes	complemented
29	#15689323	C₂²	1	6	yes	complemented
30	#15689323	C₂²	1	6	yes	complemented
31	#15689323	C₂²	1	3	yes	complemented
32	#15689323	C₂²	1	6	yes	complemented
33	#15689323	C₂²	1	3	yes	complemented
34	#15689273	C₂²	1	12	yes	complemented
35	#15689262	C₂²	1	6	yes	complemented
36	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	complemented
37	C₄×#10625336	C₂²	1	6	yes	complemented
38	C₄×#10625336	C₂²	1	6	yes	complemented
39	D₄×C₄₇⋊C₈	C₂²	1	6	yes	complemented
40	#15689176	C₂²	1	3	yes	complemented
41	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	complemented
42	C₂×#13093944	C₂²	1	6	yes	complemented
43	Q₈²×C₄₇	C₂²	1	1	yes	characteristic, complemented
44	C₄₇×#557	—	2	1	no	—
45	C₄₇×#557	C₂²	1	1	yes	characteristic, complemented
46	C₄₇×#554	—	2	3	no	—
47	C₄₇×#554	C₂²	1	3	yes	complemented
48	D₄×Q₈×C₄₇	C₂²	1	3	yes	complemented
49	C₄₇×#533	—	2	3	no	—
50	C₄₇×#533	C₂²	1	3	yes	complemented
51	C₄×#41×C₄₇	—	2	1	no	—
52	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
53	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	complemented
54	#13094070	—	4	8	no	—
55	C₂×#10625354	—	2	2	no	—
56	C₂×#10625350	—	2	3	no	—
57	C₂²×D₁₈₈	—	2	3	no	—
58	C₂²×D₁₈₈	—	2	3	no	—
59	C₂²×D₁₈₈	C₂³	1	6	yes	—
60	C₂×D₄×D₄₇	—	2	6	no	—
61	C₂×#10625336	—	4	12	no	—
62	C₂×#10625336	C₂³	1	6	yes	—
63	C₂×#10625336	C₂³	1	6	yes	—
64	C₂×#10625336	C₂³	1	12	yes	—
65	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
66	#13094024	C₂³	1	1	yes	characteristic
67	#13094021	—	2	3	no	—
68	#13094021	—	2	3	no	—
69	#13094021	C₂³	1	6	yes	—
70	#13094021	C₂³	1	3	yes	—
71	#13094012	—	2	3	no	—
72	#13093999	—	2	6	no	—
73	#13093999	—	2	6	no	—
74	#13093998	—	2	3	no	—
75	#13093998	C₂³	1	6	yes	—
76	#13093994	—	2	12	no	—
77	C₄×D₁₈₈	—	2	3	no	—
78	C₄×D₁₈₈	C₂³	1	6	yes	—
79	#13093955	C₂³	1	6	yes	—
80	#13093955	C₂³	1	2	yes	—
81	C₄⋊(C₄₇⋊C₈)	C₂³	1	6	yes	—
82	C₄⋊(C₄₇⋊C₈)	C₂³	1	3	yes	—
83	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
84	#13093948	C₂³	1	2	yes	—
85	#13093948	C₂³	1	6	yes	—
86	#13093944	C₂³	1	6	yes	—
87	#13093944	C₂³	1	12	yes	—
88	#13093944	C₂³	1	6	yes	—
89	C₂²⋊(C₄₇⋊C₈)	C₂³	1	6	yes	—
90	C₄×C₄₇⋊C₈	C₂³	1	6	yes	—
91	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
92	C₂²×Q₈×C₄₇	C₂³	1	3	yes	—
93	C₄⋊Q₈×C₄₇	—	2	3	no	—
94	C₄⋊Q₈×C₄₇	—	2	3	no	—
95	C₄⋊Q₈×C₄₇	C₂³	1	3	yes	—
96	C₄⋊Q₈×C₄₇	C₂³	1	3	yes	—
97	#125×C₄₇	—	2	3	no	—
98	#125×C₄₇	—	2	3	no	—
99	#125×C₄₇	—	2	3	no	—
100	C₂²⋊Q₈×C₄₇	—	2	3	no	—
101	C₂²⋊Q₈×C₄₇	—	2	6	no	—
102	C₂²⋊Q₈×C₄₇	—	2	6	no	—
103	C₂²⋊Q₈×C₄₇	C₂³	1	6	yes	—
104	C₄×Q₈×C₄₇	—	2	1	no	—
105	C₄×Q₈×C₄₇	—	2	1	no	—
106	C₄×Q₈×C₄₇	—	2	3	no	—
107	C₄×Q₈×C₄₇	—	2	3	no	—
108	C₄×Q₈×C₄₇	—	2	3	no	—
109	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
110	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
111	C₄×Q₈×C₄₇	C₂³	1	3	yes	—
112	C₄×Q₈×C₄₇	C₂³	1	3	yes	—
113	C₄×Q₈×C₄₇	C₂³	1	3	yes	—
114	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
115	C₄×D₄×C₄₇	—	2	3	no	—
116	C₄×D₄×C₄₇	—	2	3	no	—
117	C₄×D₄×C₄₇	D₄	1	3	yes	—
118	C₄×D₄×C₄₇	C₂³	1	3	yes	—
119	#117×C₄₇	—	2	3	no	—
120	#117×C₄₇	D₄	1	3	yes	—
121	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
122	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
123	C₂×C₄²×C₄₇	—	2	3	no	—
124	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
125	C₂³×D₄₇	—	2	6	no	—
126	#41×C₄₇	SD₁₆	1	8	yes	complemented
127	C₂×Q₈×C₄₇	—	2	3	no	—
128	C₂×Q₈×C₄₇	—	2	3	no	—
129	C₂×Q₈×C₄₇	—	2	6	no	—
130	C₂×Q₈×C₄₇	—	4	3	no	—
131	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
132	C₂×Q₈×C₄₇	C₂⁴	1	3	yes	—
133	C₂×Q₈×C₄₇	C₂⁴	1	1	yes	characteristic
134	C₂×D₄×C₄₇	C₂×D₄	1	3	yes	—
135	C₂²×C₄×C₄₇	—	2	3	no	—
136	C₂²×C₄×C₄₇	—	2	3	no	—
137	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
138	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
139	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
140	C₂×C₂²⋊D₄₇	—	2	6	no	—
141	#10625354	—	4	8	no	—
142	D₄×D₄₇	—	4	24	no	—
143	#10625350	—	4	12	no	—
144	C₂×D₁₈₈	—	2	3	no	—
145	C₂×D₁₈₈	—	2	6	no	—
146	C₂×D₁₈₈	—	2	3	no	—
147	C₂×D₁₈₈	—	2	6	no	—
148	C₂×D₁₈₈	—	2	6	no	—
149	C₂×D₁₈₈	—	4	6	no	—
150	C₂×D₁₈₈	—	4	6	no	—
151	C₂×D₁₈₈	—	4	12	no	—
152	C₂×D₁₈₈	—	8	6	no	—
153	C₂×D₁₈₈	C₂⁴	1	6	yes	—
154	C₂×C₄×D₄₇	—	2	6	no	—
155	C₂×Dic₉₄	—	2	1	no	—
156	C₄⋊C₄×C₄₇	—	2	1	no	—
157	C₄⋊C₄×C₄₇	—	2	3	no	—
158	C₄⋊C₄×C₄₇	—	2	3	no	—
159	C₄⋊C₄×C₄₇	—	2	3	no	—
160	C₄⋊C₄×C₄₇	—	2	3	no	—
161	C₄⋊C₄×C₄₇	—	2	3	no	—
162	C₄⋊C₄×C₄₇	—	2	3	no	—
163	C₄⋊C₄×C₄₇	—	2	3	no	—
164	C₄⋊C₄×C₄₇	—	2	6	no	—
165	C₄⋊C₄×C₄₇	C₂×D₄	1	3	yes	—
166	C₄⋊C₄×C₄₇	C₂×D₄	1	3	yes	—
167	C₄⋊C₄×C₄₇	C₂⁴	1	1	yes	characteristic
168	C₄⋊C₄×C₄₇	C₂⁴	1	3	yes	—
169	C₂²⋊C₄×C₄₇	—	2	6	no	—
170	C₂²⋊C₄×C₄₇	C₂×D₄	1	6	yes	—
171	C₄²×C₄₇	—	2	1	no	—
172	C₄²×C₄₇	—	2	3	no	—
173	C₄²×C₄₇	—	2	3	no	—
174	C₄²×C₄₇	—	2	3	no	—
175	C₄²×C₄₇	C₂×D₄	1	3	yes	—
176	C₄²×C₄₇	C₂×D₄	1	3	yes	—
177	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
178	C₄²×C₄₇	C₂⁴	1	3	yes	—
179	#10625336	—	2	12	no	—
180	#10625336	—	4	12	no	—
181	#10625336	—	4	6	no	—
182	(C₂×D₄₇)⋊C₄	—	2	6	no	—
183	(C₂×D₄₇)⋊C₄	—	2	6	no	—
184	C₄⋊Dic₄₇	—	2	1	no	—
185	C₄⋊Dic₄₇	C₂⁴	1	2	yes	—
186	C₂×C₄₇⋊C₈	—	4	6	no	—
187	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
188	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
189	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
190	C₂²×D₄₇	—	2	6	no	—
191	C₂²×D₄₇	—	4	12	no	—
192	C₂²×D₄₇	—	8	6	no	—
193	C₂²×D₄₇	—	8	6	no	—
194	Q₈×C₄₇	—	2	1	no	—
195	Q₈×C₄₇	—	2	3	no	—
196	Q₈×C₄₇	—	2	3	no	—
197	Q₈×C₄₇	—	4	6	no	—
198	Q₈×C₄₇	—	8	3	no	—
199	Q₈×C₄₇	C₂×SD₁₆	1	4	yes	complemented
200	Q₈×C₄₇	H(5,𝔽₂)	1	2	yes	complemented
201	D₄×C₄₇	C₂×SD₁₆	1	12	yes	complemented
202	C₂×C₄×C₄₇	—	2	1	no	—
203	C₂×C₄×C₄₇	—	2	3	no	—
204	C₂×C₄×C₄₇	—	2	3	no	—
205	C₂×C₄×C₄₇	—	2	1	no	—
206	C₂×C₄×C₄₇	—	2	3	no	—
207	C₂×C₄×C₄₇	—	2	3	no	—
208	C₂×C₄×C₄₇	—	4	3	no	—
209	C₂×C₄×C₄₇	—	4	6	no	—
210	C₂×C₄×C₄₇	—	8	3	no	—
211	C₂×C₄×C₄₇	C₂×SD₁₆	1	12	yes	complemented
212	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
214	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
215	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
217	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
218	C₂²⋊D₄₇	—	4	12	no	—
219	C₂×Dic₄₇	—	2	2	no	—
220	D₁₈₈	—	2	6	no	—
221	D₁₈₈	—	2	6	no	—
222	D₁₈₈	—	4	6	no	—
223	D₁₈₈	—	4	6	no	—
224	D₁₈₈	—	4	12	no	—
225	D₁₈₈	—	8	6	no	—
226	D₁₈₈	—	8	6	no	—
227	C₄×D₄₇	—	4	12	no	—
228	Dic₉₄	—	4	2	no	—
229	C₄₇⋊C₈	—	2	6	no	—
230	C₄₇⋊C₈	—	2	2	no	—
231	#61951	—	47	1	no	maximal, Sylow
232	C₂²×C₄₇	—	2	3	no	—
233	C₂²×C₄₇	—	2	3	no	—
234	C₂²×C₄₇	C₂²×SD₁₆	1	6	yes	—
235	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
236	C₂×D₄₇	—	4	6	no	—
237	C₂×D₄₇	—	4	6	no	—
238	C₂×D₄₇	—	4	6	no	—
239	C₂×D₄₇	—	16	12	no	—
240	C₄×C₄₇	—	2	1	no	—
241	C₄×C₄₇	—	2	3	no	—
242	C₄×C₄₇	—	2	3	no	—
243	C₄×C₄₇	—	2	2	no	—
244	C₄×C₄₇	—	4	1	no	—
245	C₄×C₄₇	—	4	3	no	—
246	C₄×C₄₇	—	4	3	no	—
247	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
248	C₄×C₄₇	C₂²×SD₁₆	1	6	yes	—
249	C₄×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
250	C₄×C₄₇	#585	1	1	yes	characteristic
251	Dic₄₇	—	4	2	no	—
252	#3498	—	47	1	no	—
253	Q₈×#41	D₄₇	1	1	yes	characteristic, complemented
254	#3392	—	47	2	no	—
255	#3362	—	47	2	no	—
256	#3295	—	47	6	no	—
257	D₄×SD₁₆	—	47	6	no	—
258	#3246	—	47	3	no	—
259	#3208	—	47	3	no	—
260	#3103	—	47	3	no	—
261	C₂×#460	—	47	3	no	—
262	#2988	—	47	1	no	—
263	C₂×C₄₇	—	2	6	no	—
264	C₂×C₄₇	#3488	1	1	yes	characteristic
265	C₂×C₄₇	C₂³×SD₁₆	1	1	yes	characteristic
266	C₂×C₄₇	#3585	1	1	yes	characteristic
267	D₄₇	—	8	6	no	—
268	C₂×H(5,𝔽₂)	—	94	1	no	—
269	C₂²×SD₁₆	—	47	6	no	—
270	Q₈²	C₂×D₄₇	1	1	yes	characteristic, complemented
271	#557	—	2	1	no	—
272	#557	C₂×D₄₇	1	1	yes	characteristic, complemented
273	#554	—	2	3	no	—
274	#554	C₂×D₄₇	1	3	yes	complemented
275	#550	—	47	2	no	—
276	#550	—	94	1	no	—
277	D₄×Q₈	C₂×D₄₇	1	3	yes	complemented
278	D₄²	—	47	6	no	—
279	D₄²	—	94	3	no	—
280	#535	—	94	3	no	—
281	#533	—	2	3	no	—
282	#533	C₂×D₄₇	1	3	yes	complemented
283	#532	—	47	3	no	—
284	C₂×#127	—	47	3	no	—
285	C₄×#41	—	2	1	no	—
286	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
287	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	complemented
288	#492	—	47	6	no	—
289	#486	—	47	6	no	—
290	#481	—	47	6	no	—
291	#476	—	47	6	no	—
292	#475	—	47	1	no	—
293	#475	—	47	3	no	—
294	#465	—	47	12	no	—
295	#461	—	47	6	no	—
296	#460	—	47	3	no	—
297	#460	—	47	3	no	—
298	#460	—	47	6	no	—
299	#460	—	47	6	no	—
300	#460	—	47	6	no	—
301	#450	—	47	12	no	—
302	#448	—	47	6	no	—
303	C₈×Q₈	—	47	2	no	—
304	C₄×SD₁₆	—	47	6	no	—
305	C₄×SD₁₆	—	47	6	no	—
306	C₈×D₄	—	47	6	no	—
307	#431	—	47	3	no	—
308	C₂×C₄⋊C₈	—	47	3	no	—
309	C₂×#102	—	47	6	no	—
310	C₄₇	#61951	1	1	yes	characteristic, complemented, Sylow
311	H(5,𝔽₂)	—	188	8	no	—
312	C₂×#41	—	94	2	no	—
313	C₂×#41	—	94	3	no	—
314	C₂×#41	C₂²⋊D₄₇	1	1	yes	characteristic
315	C₂²×Q₈	C₂²×D₄₇	1	3	yes	—
316	C₂²×D₄	—	47	6	no	—
317	C₂²×D₄	—	94	6	no	—
318	C₂²×D₄	—	94	3	no	—
319	C₂²×D₄	—	94	3	no	—
320	C₂×SD₁₆	—	47	6	no	—
321	C₂×SD₁₆	—	47	12	no	—
322	C₂×SD₁₆	—	47	6	no	—
323	C₂×SD₁₆	—	188	12	no	—
324	C₂²×C₈	—	47	6	no	—
325	C₄⋊Q₈	—	2	3	no	—
326	C₄⋊Q₈	—	2	3	no	—
327	C₄⋊Q₈	—	47	1	no	—
328	C₄⋊Q₈	C₂²×D₄₇	1	3	yes	—
329	C₄⋊Q₈	C₂²×D₄₇	1	3	yes	—
330	#127	—	47	3	no	—
331	#127	—	47	6	no	—
332	#127	—	94	3	no	—
333	#127	—	94	3	no	—
334	#125	—	2	3	no	—
335	#125	—	2	3	no	—
336	#125	—	2	3	no	—
337	#124	—	94	3	no	—
338	C₂²⋊Q₈	—	2	3	no	—
339	C₂²⋊Q₈	—	2	6	no	—
340	C₂²⋊Q₈	—	2	6	no	—
341	C₂²⋊Q₈	C₂²×D₄₇	1	6	yes	—
342	#121	—	47	6	no	—
343	#121	—	94	3	no	—
344	#121	—	94	6	no	—
345	#121	—	94	6	no	—
346	#120	—	94	12	no	—
347	C₄×Q₈	—	2	1	no	—
348	C₄×Q₈	—	2	1	no	—
349	C₄×Q₈	—	2	3	no	—
350	C₄×Q₈	—	2	3	no	—
351	C₄×Q₈	—	2	3	no	—
352	C₄×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
353	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
354	C₄×Q₈	C₂²×D₄₇	1	3	yes	—
355	C₄×Q₈	C₂²×D₄₇	1	3	yes	—
356	C₄×Q₈	C₂²×D₄₇	1	3	yes	—
357	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
358	C₄×D₄	—	2	3	no	—
359	C₄×D₄	—	2	3	no	—
360	C₄×D₄	—	47	6	no	—
361	C₄×D₄	—	94	3	no	—
362	C₄×D₄	C₂²⋊D₄₇	1	3	yes	—
363	C₄×D₄	C₂²×D₄₇	1	3	yes	—
364	#117	—	2	3	no	—
365	#117	C₂²⋊D₄₇	1	3	yes	—
366	C₂×C₄⋊C₄	—	2	3	no	—
367	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
368	C₂×C₄²	—	2	3	no	—
369	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
370	#106	—	47	2	no	—
371	#106	—	47	6	no	—
372	C₄⋊C₈	—	47	3	no	—
373	C₄⋊C₈	—	47	6	no	—
374	C₄⋊C₈	—	47	1	no	—
375	#103	—	47	2	no	—
376	#103	—	47	6	no	—
377	#102	—	47	6	no	—
378	#102	—	47	12	no	—
379	#102	—	47	6	no	—
380	C₂²⋊C₈	—	47	6	no	—
381	C₄×C₈	—	47	6	no	—
382	C₂⁴	—	94	6	no	—
383	#41	—	188	8	no	—
384	#41	—	188	12	no	—
385	#41	#10625336	1	8	yes	complemented
386	C₂×Q₈	—	2	3	no	—
387	C₂×Q₈	—	2	3	no	—
388	C₂×Q₈	—	2	6	no	—
389	C₂×Q₈	—	4	3	no	—
390	C₂×Q₈	—	94	1	no	—
391	C₂×Q₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
392	C₂×Q₈	C₂³×D₄₇	1	3	yes	—
393	C₂×Q₈	C₂³×D₄₇	1	1	yes	characteristic
394	C₂×D₄	—	47	6	no	—
395	C₂×D₄	—	94	6	no	—
396	C₂×D₄	—	94	6	no	—
397	C₂×D₄	—	94	6	no	—
398	C₂×D₄	—	94	3	no	—
399	C₂×D₄	—	94	6	no	—
400	C₂×D₄	—	94	3	no	—
401	C₂×D₄	—	376	6	no	—
402	C₂×D₄	—	188	6	no	—
403	C₂×D₄	—	188	6	no	—
404	C₂×D₄	—	188	24	no	—
405	C₂×D₄	—	188	12	no	—
406	C₂×D₄	C₂×C₂²⋊D₄₇	1	3	yes	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	94	6	no	—
410	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
411	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
412	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
413	SD₁₆	—	94	12	no	—
414	SD₁₆	—	188	6	no	—
415	SD₁₆	—	188	12	no	—
416	C₂×C₈	—	47	2	no	—
417	C₂×C₈	—	47	6	no	—
418	C₂×C₈	—	188	6	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	3	no	—
421	C₄⋊C₄	—	2	3	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	—	2	3	no	—
424	C₄⋊C₄	—	2	3	no	—
425	C₄⋊C₄	—	2	3	no	—
426	C₄⋊C₄	—	2	3	no	—
427	C₄⋊C₄	—	2	6	no	—
428	C₄⋊C₄	—	47	2	no	—
429	C₄⋊C₄	—	94	1	no	—
430	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
431	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
432	C₄⋊C₄	C₂³×D₄₇	1	1	yes	characteristic
433	C₄⋊C₄	C₂³×D₄₇	1	3	yes	—
434	C₂²⋊C₄	—	2	6	no	—
435	C₂²⋊C₄	—	94	6	no	—
436	C₂²⋊C₄	—	94	6	no	—
437	C₂²⋊C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	3	no	—
440	C₄²	—	2	3	no	—
441	C₄²	—	2	3	no	—
442	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
443	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
444	C₄²	C₂³×D₄₇	1	3	yes	—
445	C₄²	C₂³×D₄₇	1	1	yes	characteristic
446	C₂³	—	94	6	no	—
447	C₂³	—	376	6	no	—
448	C₂³	—	376	6	no	—
449	C₂³	—	188	12	no	—
450	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
451	Q₈	—	2	1	no	—
452	Q₈	—	2	3	no	—
453	Q₈	—	2	3	no	—
454	Q₈	—	4	6	no	—
455	Q₈	—	8	3	no	—
456	Q₈	—	188	2	no	—
457	Q₈	C₂×#10625336	1	4	yes	complemented
458	Q₈	#13094070	1	2	yes	complemented
459	D₄	—	94	6	no	—
460	D₄	—	94	6	no	—
461	D₄	—	376	6	no	—
462	D₄	—	376	6	no	—
463	D₄	—	188	6	no	—
464	D₄	—	188	12	no	—
465	D₄	—	188	12	no	—
466	D₄	—	188	6	no	—
467	D₄	C₂×#10625336	1	12	yes	complemented
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	3	no	—
470	C₂×C₄	—	2	3	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	3	no	—
473	C₂×C₄	—	2	3	no	—
474	C₂×C₄	—	4	3	no	—
475	C₂×C₄	—	4	6	no	—
476	C₂×C₄	—	8	3	no	—
477	C₂×C₄	—	94	2	no	—
478	C₂×C₄	—	188	12	no	—
479	C₂×C₄	C₂×#10625336	1	12	yes	complemented
480	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
481	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
482	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
483	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
484	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
485	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
486	C₈	—	94	6	no	—
487	C₈	—	94	2	no	—
488	C₂²	—	2	3	no	—
489	C₂²	—	2	3	no	—
490	C₂²	—	188	6	no	—
491	C₂²	—	188	6	no	—
492	C₂²	—	188	6	no	—
493	C₂²	—	752	12	no	—
494	C₂²	C₂²×#10625336	1	6	yes	—
495	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
496	C₄	—	2	3	no	—
497	C₄	—	2	2	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	3	no	—
500	C₄	—	4	3	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	3	no	—
503	C₄	—	188	2	no	—
504	C₄	C₂²×#10625336	1	6	yes	—
505	C₄	C₂²×#10625336	1	2	yes	—
506	C₄	C₂×#13094070	1	1	yes	characteristic
507	C₄	#15689959	1	1	yes	characteristic
508	C₂	—	2	6	no	—
509	C₂	—	376	6	no	—
510	C₂	#20257057	1	1	yes	characteristic
511	C₂	C₂³×#10625336	1	1	yes	characteristic
512	C₂	#20258167	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	Q₈×#41×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#20257166	C₂
hypercenter	Q₈×#41	D₄₇
hypocenter	C₄₇	#61951
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ Q₈×#41

Complements

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

N	H ≅ G/N	type
2. #20256365	C₂	semidirect
3. #20256059	C₂	semidirect
4. #20255206	C₂	semidirect
5. D₄×#10625336	C₂	semidirect
6. #20254700	C₂	semidirect
7. #20254326	C₂	semidirect
8. #20253419	C₂	semidirect
9. C₂×#15689323	C₂	semidirect
10. #20252499	C₂	semidirect
11. Q₈×#41×C₄₇	C₂	semidirect
13. C₂²×#10625336	C₂²	semidirect
21. #15689497	C₂²	semidirect
22. #15689471	C₂²	semidirect
23. #15689441	C₂²	semidirect
24. #15689410	C₂²	semidirect
25. #15689408	C₂²	semidirect
26. #15689408	C₂²	semidirect
27. #15689354	C₂²	semidirect
28. #15689325	C₂²	semidirect
29. #15689323	C₂²	semidirect
30. #15689323	C₂²	semidirect
31. #15689323	C₂²	semidirect
32. #15689323	C₂²	semidirect
33. #15689323	C₂²	semidirect
34. #15689273	C₂²	semidirect
35. #15689262	C₂²	semidirect
36. Q₈×C₄₇⋊C₈	C₂²	semidirect
37. C₄×#10625336	C₂²	semidirect
38. C₄×#10625336	C₂²	semidirect
39. D₄×C₄₇⋊C₈	C₂²	semidirect
40. #15689176	C₂²	semidirect
41. C₂×C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
42. C₂×#13093944	C₂²	semidirect
43. Q₈²×C₄₇	C₂²	semidirect
45. C₄₇×#557	C₂²	semidirect
47. C₄₇×#554	C₂²	semidirect
48. D₄×Q₈×C₄₇	C₂²	semidirect
50. C₄₇×#533	C₂²	semidirect
53. C₂×C₄×Q₈×C₄₇	C₂²	semidirect
126. #41×C₄₇	SD₁₆	semidirect
199. Q₈×C₄₇	C₂×SD₁₆	semidirect
200. Q₈×C₄₇	H(5,𝔽₂)	semidirect
201. D₄×C₄₇	C₂×SD₁₆	semidirect
211. C₂×C₄×C₄₇	C₂×SD₁₆	semidirect
253. Q₈×#41	D₄₇	semidirect
270. Q₈²	C₂×D₄₇	semidirect
272. #557	C₂×D₄₇	semidirect
274. #554	C₂×D₄₇	semidirect
277. D₄×Q₈	C₂×D₄₇	semidirect
282. #533	C₂×D₄₇	semidirect
287. C₂×C₄×Q₈	C₂×D₄₇	semidirect
310. C₄₇	#61951	semidirect
385. #41	#10625336	semidirect
457. Q₈	C₂×#10625336	semidirect
458. Q₈	#13094070	semidirect
467. D₄	C₂×#10625336	semidirect
479. C₂×C₄	C₂×#10625336	semidirect

Maximal subgroups

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

subgroup	index
#20257166	2
#20256365	2
#20256059	2
#20255206	2
D₄×#10625336	2
#20254700	2
#20254326	2
#20253419	2
C₂×#15689323	2
#20252499	2
Q₈×#41×C₄₇	2
#61951	47