Subgroups — C₁₃×#19809 #15706148

Order 3328.

632 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3362	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#3278	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1714	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1637	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#1631	C₂	1	1	yes	maximal, characteristic
13	C₁₃×#1623	C₂	1	1	yes	maximal, characteristic, complemented
14	C₁₃×#1583	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₁₃×#137	—	2	1	no	—
17	Q₈²×C₁₃	C₂²	1	1	yes	characteristic
18	Q₈²×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#550	C₂²	1	1	yes	characteristic
20	D₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#500	—	2	1	no	—
22	C₁₃×#500	C₂²	1	1	yes	characteristic
23	C₁₃×#498	—	2	1	no	—
24	C₁₃×#498	C₂²	1	1	yes	characteristic
25	C₁₃×#492	C₂²	1	1	yes	characteristic
26	C₁₃×#492	C₂²	1	1	yes	characteristic
27	C₁₃×#487	C₂²	1	1	yes	characteristic
28	C₁₃×#486	—	2	1	no	—
29	C₁₃×#486	C₂²	1	1	yes	characteristic
30	C₁₃×#477	—	2	1	no	—
31	C₁₃×#477	—	2	1	no	—
32	C₁₃×#477	C₂²	1	1	yes	characteristic
33	C₁₃×#477	C₂²	1	1	yes	characteristic
34	C₁₃×#476	—	2	1	no	—
35	C₁₃×#476	—	2	1	no	—
36	C₁₃×#476	C₂²	1	1	yes	characteristic
37	C₁₃×#476	C₂²	1	1	yes	characteristic
38	C₁₃×#475	—	2	1	no	—
39	C₁₃×#475	C₂²	1	1	yes	characteristic
40	C₁₃×#468	—	2	1	no	—
41	C₁₃×#462	C₂²	1	1	yes	characteristic
42	C₁₃×#460	—	2	1	no	—
43	C₁₃×#460	C₂²	1	1	yes	characteristic
44	C₁₃×#460	C₂²	1	1	yes	characteristic
45	C₁₃×#452	—	2	1	no	—
46	C₁₃×#448	—	2	1	no	—
47	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
49	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
50	C₄×C₁₃×Q₁₆	—	2	1	no	—
51	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
52	C₄×C₁₃×SD₁₆	—	2	1	no	—
53	C₄×C₁₃×SD₁₆	—	2	1	no	—
54	C₄×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
55	C₁₃×#436	C₂²	1	1	yes	characteristic
56	C₁₃×#334	C₂²	1	1	yes	characteristic
57	C₁₃×#334	C₂²	1	1	yes	characteristic
58	C₁₃×#333	C₂²	1	1	yes	characteristic
59	C₁₃×#333	C₂²	1	1	yes	characteristic
60	C₁₃×#332	C₂²	1	1	yes	characteristic
61	C₁₃×#331	C₂²	1	1	yes	characteristic
62	C₁₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₁₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₁₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₁₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₁₃×#322	C₂²	1	1	yes	characteristic
67	C₂×C₁₃×#41	—	2	1	no	—
68	C₂²×Q₈×C₁₃	—	2	1	no	—
69	C₁₃×#137	—	4	4	no	—
70	C₂×C₁₃×Q₁₆	—	2	1	no	—
71	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic
72	C₂×C₁₃×SD₁₆	—	2	1	no	—
73	C₂×C₁₃×SD₁₆	—	2	1	no	—
74	C₂×C₁₃×SD₁₆	—	2	1	no	—
75	C₂×C₁₃×SD₁₆	—	2	1	no	—
76	C₂×C₁₃×SD₁₆	—	4	1	no	—
77	C₂×C₁₃×SD₁₆	D₄	1	1	yes	characteristic
78	C₂×C₁₃×M₁₆	—	2	1	no	—
79	C₁₃×C₄⋊Q₈	—	2	1	no	—
80	C₁₃×C₄⋊Q₈	—	2	1	no	—
81	C₁₃×C₄⋊Q₈	—	2	1	no	—
82	C₁₃×C₄⋊Q₈	—	2	1	no	—
83	C₁₃×C₄⋊Q₈	—	2	1	no	—
84	C₁₃×C₄⋊Q₈	—	4	1	no	—
85	C₁₃×C₄⋊Q₈	—	4	1	no	—
86	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₁₃×#127	—	2	1	no	—
90	C₁₃×#127	C₂³	1	1	yes	characteristic
91	C₁₃×C₂²⋊Q₈	—	2	1	no	—
92	C₁₃×C₂²⋊Q₈	—	4	1	no	—
93	C₁₃×#121	—	2	1	no	—
94	C₁₃×#121	—	4	1	no	—
95	C₄×Q₈×C₁₃	—	2	1	no	—
96	C₄×Q₈×C₁₃	—	2	1	no	—
97	C₄×Q₈×C₁₃	—	2	1	no	—
98	C₄×Q₈×C₁₃	—	2	1	no	—
99	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
101	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
102	C₄×D₄×C₁₃	—	2	1	no	—
103	C₄×D₄×C₁₃	—	2	1	no	—
104	C₄×D₄×C₁₃	C₂³	1	1	yes	characteristic
105	C₁₃×#107	—	2	1	no	—
106	C₁₃×#107	—	2	1	no	—
107	C₁₃×#107	—	2	1	no	—
108	C₁₃×#107	—	2	1	no	—
109	C₁₃×#107	D₄	1	1	yes	characteristic
110	C₁₃×#106	—	2	1	no	—
111	C₁₃×#106	—	2	1	no	—
112	C₁₃×#106	—	2	1	no	—
113	C₁₃×#106	—	2	1	no	—
114	C₁₃×#106	—	2	1	no	—
115	C₁₃×#106	D₄	1	1	yes	characteristic
116	C₁₃×C₄⋊C₈	—	2	1	no	—
117	C₁₃×C₄⋊C₈	—	2	1	no	—
118	C₁₃×C₄⋊C₈	—	2	1	no	—
119	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₁₃×#103	—	2	1	no	—
124	C₁₃×#103	—	2	1	no	—
125	C₁₃×#103	—	2	1	no	—
126	C₁₃×#103	—	2	1	no	—
127	C₁₃×#103	—	4	1	no	—
128	C₁₃×#103	D₄	1	1	yes	characteristic
129	C₁₃×#103	D₄	1	1	yes	characteristic
130	C₁₃×#103	D₄	1	1	yes	characteristic
131	C₁₃×#102	—	2	1	no	—
132	C₁₃×#102	—	2	1	no	—
133	C₁₃×#102	—	2	1	no	—
134	C₁₃×#102	—	2	1	no	—
135	C₁₃×#102	—	2	1	no	—
136	C₁₃×#102	—	2	1	no	—
137	C₁₃×#102	D₄	1	1	yes	characteristic
138	C₁₃×C₂²⋊C₈	—	2	1	no	—
139	C₄×C₈×C₁₃	—	2	1	no	—
140	C₄×C₈×C₁₃	—	2	1	no	—
141	C₄×C₈×C₁₃	—	2	1	no	—
142	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
145	#19809	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃×#41	—	4	2	no	—
147	C₁₃×#41	—	8	1	no	—
148	C₂×Q₈×C₁₃	—	2	1	no	—
149	C₂×Q₈×C₁₃	—	2	1	no	—
150	C₂×Q₈×C₁₃	—	2	1	no	—
151	C₂×Q₈×C₁₃	—	4	1	no	—
152	C₂×Q₈×C₁₃	—	4	1	no	—
153	C₂×Q₈×C₁₃	—	4	1	no	—
154	C₂×Q₈×C₁₃	—	4	2	no	—
155	C₂×Q₈×C₁₃	—	8	1	no	—
156	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
157	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
158	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
159	C₂×D₄×C₁₃	—	2	1	no	—
160	C₂×D₄×C₁₃	—	2	1	no	—
161	C₂×D₄×C₁₃	—	2	1	no	—
162	C₂×D₄×C₁₃	—	4	1	no	—
163	C₂×D₄×C₁₃	—	4	1	no	—
164	C₂×D₄×C₁₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₁₃	—	2	1	no	—
166	C₂²×C₄×C₁₃	—	4	1	no	—
167	C₂²×C₄×C₁₃	—	4	1	no	—
168	C₁₃×Q₁₆	—	4	2	no	—
169	C₁₃×Q₁₆	SD₁₆	1	4	yes	complemented
170	C₁₃×SD₁₆	—	2	2	no	—
171	C₁₃×SD₁₆	—	4	2	no	—
172	C₁₃×SD₁₆	—	4	2	no	—
173	C₁₃×SD₁₆	—	4	2	no	—
174	C₁₃×SD₁₆	—	8	1	no	—
175	C₁₃×SD₁₆	—	8	2	no	—
176	C₁₃×M₁₆	—	4	2	no	—
177	C₂×C₈×C₁₃	—	2	1	no	—
178	C₂×C₈×C₁₃	—	2	1	no	—
179	C₂×C₈×C₁₃	—	2	1	no	—
180	C₂×C₈×C₁₃	—	2	1	no	—
181	C₂×C₈×C₁₃	—	2	1	no	—
182	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
186	C₁₃×C₄⋊C₄	—	2	1	no	—
187	C₁₃×C₄⋊C₄	—	2	1	no	—
188	C₁₃×C₄⋊C₄	—	2	1	no	—
189	C₁₃×C₄⋊C₄	—	2	1	no	—
190	C₁₃×C₄⋊C₄	—	2	1	no	—
191	C₁₃×C₄⋊C₄	—	2	1	no	—
192	C₁₃×C₄⋊C₄	—	2	1	no	—
193	C₁₃×C₄⋊C₄	—	2	1	no	—
194	C₁₃×C₄⋊C₄	—	2	1	no	—
195	C₁₃×C₄⋊C₄	—	4	1	no	—
196	C₁₃×C₄⋊C₄	—	4	1	no	—
197	C₁₃×C₄⋊C₄	—	4	1	no	—
198	C₁₃×C₄⋊C₄	—	4	1	no	—
199	C₁₃×C₄⋊C₄	—	4	1	no	—
200	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
201	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
202	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
203	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
204	C₁₃×C₂²⋊C₄	—	2	1	no	—
205	C₁₃×C₂²⋊C₄	—	4	1	no	—
206	C₁₃×C₂²⋊C₄	—	4	1	no	—
207	C₄²×C₁₃	—	2	1	no	—
208	C₄²×C₁₃	—	2	1	no	—
209	C₄²×C₁₃	—	2	1	no	—
210	C₄²×C₁₃	—	4	1	no	—
211	C₄²×C₁₃	—	4	1	no	—
212	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
213	Q₈×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
214	Q₈×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
215	#3362	C₂×C₁₃	1	1	yes	characteristic
216	#3278	C₂×C₁₃	1	1	yes	characteristic
217	#1714	C₂×C₁₃	1	1	yes	characteristic, complemented
218	#1695	C₂×C₁₃	1	1	yes	characteristic, complemented
219	#1678	C₂×C₁₃	1	1	yes	characteristic, complemented
220	#1672	C₂×C₁₃	1	1	yes	characteristic, complemented
221	#1666	C₂×C₁₃	1	1	yes	characteristic, complemented
222	#1645	C₂×C₁₃	1	1	yes	characteristic, complemented
223	#1637	C₂×C₁₃	1	1	yes	characteristic, complemented
224	#1631	C₂×C₁₃	1	1	yes	characteristic
225	#1623	C₂×C₁₃	1	1	yes	characteristic, complemented
226	#1583	C₂×C₁₃	1	1	yes	characteristic, complemented
227	#1579	C₂×C₁₃	1	1	yes	characteristic, complemented
228	C₂³×C₁₃	—	2	1	no	—
229	C₂³×C₁₃	—	4	1	no	—
230	Q₈×C₁₃	—	2	1	no	—
231	Q₈×C₁₃	—	2	1	no	—
232	Q₈×C₁₃	—	2	1	no	—
233	Q₈×C₁₃	—	4	1	no	—
234	Q₈×C₁₃	—	4	1	no	—
235	Q₈×C₁₃	—	8	2	no	—
236	Q₈×C₁₃	—	8	1	no	—
237	Q₈×C₁₃	—	8	1	no	—
238	Q₈×C₁₃	—	16	1	no	—
239	Q₈×C₁₃	—	16	1	no	—
240	Q₈×C₁₃	C₂×SD₁₆	1	2	yes	—
241	Q₈×C₁₃	C₂×SD₁₆	1	2	yes	—
242	Q₈×C₁₃	#137	1	2	yes	complemented
243	D₄×C₁₃	—	2	1	no	—
244	D₄×C₁₃	—	2	1	no	—
245	D₄×C₁₃	—	4	1	no	—
246	D₄×C₁₃	—	4	1	no	—
247	D₄×C₁₃	—	4	1	no	—
248	D₄×C₁₃	—	4	1	no	—
249	D₄×C₁₃	—	8	2	no	—
250	D₄×C₁₃	—	8	1	no	—
251	D₄×C₁₃	—	16	1	no	—
252	C₂×C₄×C₁₃	—	2	1	no	—
253	C₂×C₄×C₁₃	—	2	1	no	—
254	C₂×C₄×C₁₃	—	2	1	no	—
255	C₂×C₄×C₁₃	—	2	1	no	—
256	C₂×C₄×C₁₃	—	2	1	no	—
257	C₂×C₄×C₁₃	—	2	1	no	—
258	C₂×C₄×C₁₃	—	2	1	no	—
259	C₂×C₄×C₁₃	—	4	1	no	—
260	C₂×C₄×C₁₃	—	4	1	no	—
261	C₂×C₄×C₁₃	—	4	1	no	—
262	C₂×C₄×C₁₃	—	4	1	no	—
263	C₂×C₄×C₁₃	—	4	1	no	—
264	C₂×C₄×C₁₃	—	8	2	no	—
265	C₂×C₄×C₁₃	—	8	2	no	—
266	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
269	C₈×C₁₃	—	2	1	no	—
270	C₈×C₁₃	—	2	1	no	—
271	C₈×C₁₃	—	2	1	no	—
272	C₈×C₁₃	—	4	1	no	—
273	C₈×C₁₃	—	4	1	no	—
274	C₈×C₁₃	—	4	1	no	—
275	C₈×C₁₃	—	4	1	no	—
276	C₈×C₁₃	—	4	1	no	—
277	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
278	C₂×#137	—	2	1	no	—
279	Q₈²	C₂²×C₁₃	1	1	yes	characteristic
280	Q₈²	C₂²×C₁₃	1	1	yes	characteristic
281	#550	C₂²×C₁₃	1	1	yes	characteristic
282	D₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
283	#500	—	2	1	no	—
284	#500	C₂²×C₁₃	1	1	yes	characteristic
285	#498	—	2	1	no	—
286	#498	C₂²×C₁₃	1	1	yes	characteristic
287	#492	C₂²×C₁₃	1	1	yes	characteristic
288	#492	C₂²×C₁₃	1	1	yes	characteristic
289	#487	C₂²×C₁₃	1	1	yes	characteristic
290	#486	—	2	1	no	—
291	#486	C₂²×C₁₃	1	1	yes	characteristic
292	#477	—	2	1	no	—
293	#477	—	2	1	no	—
294	#477	C₂²×C₁₃	1	1	yes	characteristic
295	#477	C₂²×C₁₃	1	1	yes	characteristic
296	#476	—	2	1	no	—
297	#476	—	2	1	no	—
298	#476	C₂²×C₁₃	1	1	yes	characteristic
299	#476	C₂²×C₁₃	1	1	yes	characteristic
300	#475	—	2	1	no	—
301	#475	C₂²×C₁₃	1	1	yes	characteristic
302	#468	—	2	1	no	—
303	#462	C₂²×C₁₃	1	1	yes	characteristic
304	#460	—	2	1	no	—
305	#460	C₂²×C₁₃	1	1	yes	characteristic
306	#460	C₂²×C₁₃	1	1	yes	characteristic
307	#452	—	2	1	no	—
308	#448	—	2	1	no	—
309	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
310	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
311	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
312	C₄×Q₁₆	—	2	1	no	—
313	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
314	C₄×SD₁₆	—	2	1	no	—
315	C₄×SD₁₆	—	2	1	no	—
316	C₄×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
317	#436	C₂²×C₁₃	1	1	yes	characteristic
318	#334	C₂²×C₁₃	1	1	yes	characteristic
319	#334	C₂²×C₁₃	1	1	yes	characteristic
320	#333	C₂²×C₁₃	1	1	yes	characteristic
321	#333	C₂²×C₁₃	1	1	yes	characteristic
322	#332	C₂²×C₁₃	1	1	yes	characteristic
323	#331	C₂²×C₁₃	1	1	yes	characteristic
324	Q₈⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
325	Q₈⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
326	Q₈⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
327	D₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
328	#322	C₂²×C₁₃	1	1	yes	characteristic
329	C₂²×C₁₃	—	4	1	no	—
330	C₂²×C₁₃	—	4	1	no	—
331	C₂²×C₁₃	—	4	1	no	—
332	C₂²×C₁₃	—	8	1	no	—
333	C₂²×C₁₃	—	8	1	no	—
334	C₂²×C₁₃	—	8	1	no	—
335	C₂²×C₁₃	D₄²	1	1	yes	characteristic
336	C₄×C₁₃	—	2	2	no	—
337	C₄×C₁₃	—	2	2	no	—
338	C₄×C₁₃	—	2	2	no	—
339	C₄×C₁₃	—	2	1	no	—
340	C₄×C₁₃	—	4	1	no	—
341	C₄×C₁₃	—	4	1	no	—
342	C₄×C₁₃	—	4	1	no	—
343	C₄×C₁₃	—	4	1	no	—
344	C₄×C₁₃	—	8	1	no	—
345	C₄×C₁₃	—	8	1	no	—
346	C₄×C₁₃	—	8	1	no	—
347	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
348	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
349	C₄×C₁₃	#577	1	1	yes	characteristic
350	C₄×C₁₃	#578	1	1	yes	characteristic
351	C₂×#41	—	2	1	no	—
352	C₂²×Q₈	—	2	1	no	—
353	#137	—	4	4	no	—
354	C₂×Q₁₆	—	2	1	no	—
355	C₂×Q₁₆	D₄×C₁₃	1	1	yes	characteristic
356	C₂×SD₁₆	—	2	1	no	—
357	C₂×SD₁₆	—	2	1	no	—
358	C₂×SD₁₆	—	2	1	no	—
359	C₂×SD₁₆	—	2	1	no	—
360	C₂×SD₁₆	—	4	1	no	—
361	C₂×SD₁₆	D₄×C₁₃	1	1	yes	characteristic
362	C₂×M₁₆	—	2	1	no	—
363	C₄⋊Q₈	—	2	1	no	—
364	C₄⋊Q₈	—	2	1	no	—
365	C₄⋊Q₈	—	2	1	no	—
366	C₄⋊Q₈	—	2	1	no	—
367	C₄⋊Q₈	—	2	1	no	—
368	C₄⋊Q₈	—	4	1	no	—
369	C₄⋊Q₈	—	4	1	no	—
370	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
372	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
373	#127	—	2	1	no	—
374	#127	C₂³×C₁₃	1	1	yes	characteristic
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	4	1	no	—
377	#121	—	2	1	no	—
378	#121	—	4	1	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	—	2	1	no	—
383	C₄×Q₈	C₂³×C₁₃	1	1	yes	characteristic
384	C₄×Q₈	C₂³×C₁₃	1	1	yes	characteristic
385	C₄×Q₈	C₂³×C₁₃	1	1	yes	characteristic
386	C₄×D₄	—	2	1	no	—
387	C₄×D₄	—	2	1	no	—
388	C₄×D₄	C₂³×C₁₃	1	1	yes	characteristic
389	#107	—	2	1	no	—
390	#107	—	2	1	no	—
391	#107	—	2	1	no	—
392	#107	—	2	1	no	—
393	#107	D₄×C₁₃	1	1	yes	characteristic
394	#106	—	2	1	no	—
395	#106	—	2	1	no	—
396	#106	—	2	1	no	—
397	#106	—	2	1	no	—
398	#106	—	2	1	no	—
399	#106	D₄×C₁₃	1	1	yes	characteristic
400	C₄⋊C₈	—	2	1	no	—
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	—	2	1	no	—
403	C₄⋊C₈	C₂³×C₁₃	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₁₃	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₁₃	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₁₃	1	1	yes	characteristic
407	#103	—	2	1	no	—
408	#103	—	2	1	no	—
409	#103	—	2	1	no	—
410	#103	—	2	1	no	—
411	#103	—	4	1	no	—
412	#103	D₄×C₁₃	1	1	yes	characteristic
413	#103	D₄×C₁₃	1	1	yes	characteristic
414	#103	D₄×C₁₃	1	1	yes	characteristic
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	—	2	1	no	—
418	#102	—	2	1	no	—
419	#102	—	2	1	no	—
420	#102	—	2	1	no	—
421	#102	D₄×C₁₃	1	1	yes	characteristic
422	C₂²⋊C₈	—	2	1	no	—
423	C₄×C₈	—	2	1	no	—
424	C₄×C₈	—	2	1	no	—
425	C₄×C₈	—	2	1	no	—
426	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
427	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
428	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
429	C₂×C₁₃	—	8	1	no	—
430	C₂×C₁₃	—	16	1	no	—
431	C₂×C₁₃	#3270	1	1	yes	characteristic
432	C₂×C₁₃	D₄×SD₁₆	1	1	yes	characteristic
433	C₂×C₁₃	#3288	1	1	yes	characteristic
434	#41	—	4	2	no	—
435	#41	—	8	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	4	1	no	—
440	C₂×Q₈	—	4	1	no	—
441	C₂×Q₈	—	4	1	no	—
442	C₂×Q₈	—	4	2	no	—
443	C₂×Q₈	—	8	1	no	—
444	C₂×Q₈	C₂×D₄×C₁₃	1	1	yes	characteristic
445	C₂×Q₈	C₂×D₄×C₁₃	1	1	yes	characteristic
446	C₂×Q₈	C₂×D₄×C₁₃	1	1	yes	characteristic
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	4	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	C₂×D₄×C₁₃	1	1	yes	characteristic
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	4	1	no	—
455	C₂²×C₄	—	4	1	no	—
456	Q₁₆	—	4	2	no	—
457	Q₁₆	C₁₃×SD₁₆	1	4	yes	complemented
458	SD₁₆	—	2	2	no	—
459	SD₁₆	—	4	2	no	—
460	SD₁₆	—	4	2	no	—
461	SD₁₆	—	4	2	no	—
462	SD₁₆	—	8	1	no	—
463	SD₁₆	—	8	2	no	—
464	M₁₆	—	4	2	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
471	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
472	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
473	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	4	1	no	—
484	C₄⋊C₄	—	4	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	—	4	1	no	—
487	C₄⋊C₄	—	4	1	no	—
488	C₄⋊C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
489	C₄⋊C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
490	C₄⋊C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
491	C₄⋊C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	4	1	no	—
494	C₂²⋊C₄	—	4	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	4	1	no	—
499	C₄²	—	4	1	no	—
500	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
501	C₁₃	#19809	1	1	yes	characteristic, direct product factor, Sylow
502	C₂³	—	2	1	no	—
503	C₂³	—	4	1	no	—
504	Q₈	—	2	1	no	—
505	Q₈	—	2	1	no	—
506	Q₈	—	2	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	—	4	1	no	—
509	Q₈	—	8	2	no	—
510	Q₈	—	8	1	no	—
511	Q₈	—	8	1	no	—
512	Q₈	—	16	1	no	—
513	Q₈	—	16	1	no	—
514	Q₈	C₂×C₁₃×SD₁₆	1	2	yes	—
515	Q₈	C₂×C₁₃×SD₁₆	1	2	yes	—
516	Q₈	C₁₃×#137	1	2	yes	complemented
517	D₄	—	2	1	no	—
518	D₄	—	2	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	8	2	no	—
524	D₄	—	8	1	no	—
525	D₄	—	16	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
542	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
543	C₈	—	2	1	no	—
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	—	8	1	no	—
558	C₂²	D₄²×C₁₃	1	1	yes	characteristic
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	1	no	—
563	C₄	—	4	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	8	1	no	—
568	C₄	—	8	1	no	—
569	C₄	—	8	1	no	—
570	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
571	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
572	C₄	C₁₃×#577	1	1	yes	characteristic
573	C₄	C₁₃×#578	1	1	yes	characteristic
574	C₂	—	8	1	no	—
575	C₂	—	16	1	no	—
576	C₂	C₁₃×#3270	1	1	yes	characteristic
577	C₂	D₄×C₁₃×SD₁₆	1	1	yes	characteristic
578	C₂	C₁₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#550	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×C₁₃×Q₁₆	C₂	semidirect
2. Q₈×C₁₃×SD₁₆	C₂	semidirect
5. C₁₃×#1714	C₂	semidirect
6. C₁₃×#1695	C₂	semidirect
7. C₁₃×#1678	C₂	semidirect
8. C₁₃×#1672	C₂	semidirect
9. C₁₃×#1666	C₂	semidirect
10. C₁₃×#1645	C₂	semidirect
11. C₁₃×#1637	C₂	semidirect
13. C₁₃×#1623	C₂	semidirect
14. C₁₃×#1583	C₂	semidirect
15. C₁₃×#1579	C₂	semidirect
145. #19809	C₁₃	direct (N × H)
169. C₁₃×Q₁₆	SD₁₆	semidirect
213. Q₈×Q₁₆	C₂×C₁₃	semidirect
214. Q₈×SD₁₆	C₂×C₁₃	semidirect
217. #1714	C₂×C₁₃	semidirect
218. #1695	C₂×C₁₃	semidirect
219. #1678	C₂×C₁₃	semidirect
220. #1672	C₂×C₁₃	semidirect
221. #1666	C₂×C₁₃	semidirect
222. #1645	C₂×C₁₃	semidirect
223. #1637	C₂×C₁₃	semidirect
225. #1623	C₂×C₁₃	semidirect
226. #1583	C₂×C₁₃	semidirect
227. #1579	C₂×C₁₃	semidirect
242. Q₈×C₁₃	#137	semidirect
457. Q₁₆	C₁₃×SD₁₆	semidirect
501. C₁₃	#19809	direct (N × H)
516. Q₈	C₁₃×#137	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₁₃×Q₁₆	2
Q₈×C₁₃×SD₁₆	2
C₁₃×#3362	2
C₁₃×#3278	2
C₁₃×#1714	2
C₁₃×#1695	2
C₁₃×#1678	2
C₁₃×#1672	2
C₁₃×#1666	2
C₁₃×#1645	2
C₁₃×#1637	2
C₁₃×#1631	2
C₁₃×#1623	2
C₁₃×#1583	2
C₁₃×#1579	2
#19809	13

Subgroups — C13×#19809 #15706148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#19809 #15706148