Subgroups — #15846704

Order 3328.

798 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124296	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114270	C₂	1	4	yes	maximal, complemented
4	#13114222	C₂	1	2	yes	maximal, complemented
5	#13114188	C₂	1	2	yes	maximal, complemented
6	#13113595	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113513	C₂	1	2	yes	maximal, complemented
8	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#532	C₂²	1	2	yes	—
10	C₂×#88203	—	2	2	no	—
11	C₂²×C₈×D₁₃	—	2	1	no	—
12	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
13	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#486	C₂²	1	2	yes	complemented
16	C₁₃×#466	—	2	1	no	—
17	C₁₃×#466	—	2	1	no	—
18	C₄×C₁₃×D₈	—	2	1	no	—
19	C₄×C₁₃×D₈	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#11718893	—	2	2	no	—
22	C₂×#88060	—	2	1	no	—
23	#11718744	C₂²	1	4	yes	—
24	#11718733	C₂²	1	4	yes	—
25	#11718718	—	2	4	no	—
26	#11718585	C₂²	1	2	yes	—
27	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₈×D₅₂	—	2	1	no	—
29	C₈×D₅₂	—	2	1	no	—
30	C₈×D₅₂	C₂²	1	2	yes	—
31	C₈×D₅₂	C₂²	1	2	yes	—
32	#11718139	C₂²	1	4	yes	complemented
33	#11718136	C₂²	1	4	yes	complemented
34	#11718050	C₂²	1	2	yes	complemented
35	#11718050	C₂²	1	2	yes	complemented
36	C₂×C₁₃×#41	—	2	2	no	—
37	C₂²×D₅₂	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₁₃	—	2	1	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	—	2	1	no	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
44	C₁₃×#127	C₂³	1	2	yes	—
45	C₁₃×#124	—	2	2	no	—
46	C₁₃×#121	—	2	2	no	—
47	C₁₃×#121	—	2	2	no	—
48	C₄×D₄×C₁₃	—	2	2	no	—
49	C₄×D₄×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	#88203	—	4	8	no	—
52	C₂×#6106	—	2	4	no	—
53	C₂×#6104	—	2	4	no	—
54	C₂×#6103	—	2	1	no	—
55	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
56	C₂×#6099	—	2	2	no	—
57	C₂²×#1055	—	2	1	no	—
58	C₂×C₈×D₁₃	—	2	2	no	—
59	C₂×C₈×D₁₃	—	2	2	no	—
60	C₂×C₈×D₁₃	—	4	4	no	—
61	C₄×D₅₂	—	2	1	no	—
62	C₄×D₅₂	—	2	1	no	—
63	C₄×D₅₂	D₄	1	2	yes	—
64	C₄×D₅₂	C₂³	1	2	yes	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#102	—	2	2	no	—
68	C₁₃×#102	—	2	2	no	—
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#88075	—	2	4	no	—
74	#88072	—	2	4	no	—
75	#88060	—	2	2	no	—
76	#88060	—	2	2	no	—
77	C₄⋊#1055	—	2	2	no	—
78	C₄⋊#1055	—	2	1	no	—
79	C₄⋊#1055	C₂³	1	4	yes	—
80	C₄⋊#1055	C₂³	1	2	yes	—
81	#19255	—	13	1	no	maximal, Sylow
82	C₂³×D₁₃	—	2	1	no	—
83	C₁₃×#41	—	4	4	no	—
84	C₁₃×#41	—	4	4	no	—
85	C₂×Q₈×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	—	4	2	no	—
90	C₂²×C₄×C₁₃	—	4	2	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
94	C₂²×Dic₁₃	—	2	1	no	—
95	C₂×D₅₂	—	2	1	no	—
96	C₂×D₅₂	—	2	2	no	—
97	C₂×D₅₂	D₈	1	4	yes	complemented
98	C₂×D₅₂	C₂×D₄	1	2	yes	—
99	C₂×C₄×D₁₃	—	2	2	no	—
100	C₂×C₄×D₁₃	—	2	2	no	—
101	C₂×C₄×D₁₃	—	4	2	no	—
102	C₂×C₄×D₁₃	—	4	2	no	—
103	C₁₃×D₈	—	4	2	no	—
104	C₁₃×D₈	—	4	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	4	1	no	—
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	#41	1	1	yes	characteristic
118	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
119	#6106	—	4	8	no	—
120	#6104	—	4	8	no	—
121	#6103	—	2	2	no	—
122	#6103	—	2	2	no	—
123	C₄⋊Dic₁₃	—	2	1	no	—
124	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
125	#6099	—	4	4	no	—
126	C₂×#1055	—	2	4	no	—
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	—	4	2	no	—
129	C₈×D₁₃	—	4	4	no	—
130	C₈×D₁₃	—	4	4	no	—
131	#3169	D₁₃	1	1	yes	characteristic, complemented
132	#3036	—	13	2	no	—
133	C₂×C₈×D₄	—	13	1	no	—
134	#1681	—	13	4	no	—
135	#1673	—	13	2	no	—
136	#1667	—	13	2	no	—
137	#1567	—	13	1	no	—
138	#1547	—	13	2	no	—
139	C₂³×C₁₃	—	4	2	no	—
140	C₂³×C₁₃	#121	1	1	yes	characteristic
141	C₂²×D₁₃	—	2	2	no	—
142	C₂²×D₁₃	—	2	4	no	—
143	C₂²×D₁₃	—	4	2	no	—
144	C₂²×D₁₃	—	4	2	no	—
145	Q₈×C₁₃	—	4	2	no	—
146	Q₈×C₁₃	—	4	2	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	4	2	no	—
149	D₄×C₁₃	—	4	2	no	—
150	D₄×C₁₃	—	4	2	no	—
151	D₄×C₁₃	—	8	2	no	—
152	D₄×C₁₃	—	8	4	no	—
153	C₂×C₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	1	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	8	4	no	—
158	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
159	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
160	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
162	C₂×C₄×C₁₃	#135	1	2	yes	—
163	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
166	C₂×Dic₁₃	—	2	2	no	—
167	C₂×Dic₁₃	—	4	2	no	—
168	D₅₂	—	2	2	no	—
169	D₅₂	—	2	4	no	—
170	D₅₂	C₂×D₈	1	4	yes	—
171	C₄×D₁₃	—	4	4	no	—
172	C₄×D₁₃	—	4	2	no	—
173	C₄×D₁₃	—	4	2	no	—
174	C₈×C₁₃	—	2	1	no	—
175	C₈×C₁₃	—	2	1	no	—
176	C₈×C₁₃	—	2	1	no	—
177	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	complemented
178	#1055	—	4	4	no	—
179	#1055	—	4	2	no	—
180	C₂×AGL(1,ℤ/8)	—	26	2	no	—
181	C₂³×C₈	—	26	1	no	—
182	#532	C₂×D₁₃	1	2	yes	—
183	C₂×C₄×D₄	—	13	1	no	—
184	#500	C₂×D₁₃	1	1	yes	characteristic, complemented
185	#493	C₂×D₁₃	1	1	yes	characteristic, complemented
186	#486	C₂×D₁₃	1	2	yes	complemented
187	#466	—	2	1	no	—
188	#466	—	2	1	no	—
189	#463	—	26	4	no	—
190	#461	—	13	4	no	—
191	#459	—	13	4	no	—
192	C₄×D₈	—	2	1	no	—
193	C₄×D₈	—	2	1	no	—
194	C₈×D₄	—	13	2	no	—
195	C₈×D₄	—	13	2	no	—
196	C₈×D₄	—	26	1	no	—
197	C₈×D₄	—	26	1	no	—
198	#423	—	13	2	no	—
199	C₂×C₄⋊C₈	—	13	1	no	—
200	#417	—	26	2	no	—
201	C₂×C₂²⋊C₈	—	26	1	no	—
202	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
203	#335	—	13	2	no	—
204	#335	—	13	2	no	—
205	#332	—	13	4	no	—
206	#331	—	13	4	no	—
207	C₂²×C₁₃	—	2	1	no	—
208	C₂²×C₁₃	—	2	1	no	—
209	C₂²×C₁₃	—	8	2	no	—
210	C₂²×C₁₃	—	8	2	no	—
211	C₂²×C₁₃	—	8	2	no	—
212	C₂²×C₁₃	#466	1	2	yes	—
213	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
214	C₂×D₁₃	—	2	4	no	—
215	C₂×D₁₃	—	4	8	no	—
216	C₂×D₁₃	—	4	2	no	—
217	C₂×D₁₃	—	4	2	no	—
218	C₄×C₁₃	—	2	1	no	—
219	C₄×C₁₃	—	2	1	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	8	2	no	—
222	C₄×C₁₃	#466	1	2	yes	—
223	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
224	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
225	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₄×C₁₃	#575	1	1	yes	characteristic
227	Dic₁₃	—	4	2	no	—
228	C₂×#41	—	2	2	no	—
229	C₂²×D₄	—	13	1	no	—
230	C₂³×C₄	—	26	1	no	—
231	AGL(1,ℤ/8)	—	52	8	no	—
232	C₂×SD₁₆	—	26	4	no	—
233	C₂×D₈	—	2	1	no	—
234	C₂×D₈	—	2	1	no	—
235	C₂×D₈	—	26	4	no	—
236	C₂×M₁₆	—	26	2	no	—
237	C₂²×C₈	—	26	1	no	—
238	C₂²×C₈	—	26	2	no	—
239	C₂²×C₈	—	26	2	no	—
240	C₂²×C₈	—	52	4	no	—
241	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
242	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
243	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
244	#127	C₂²×D₁₃	1	2	yes	—
245	#124	—	2	2	no	—
246	#121	—	2	2	no	—
247	#121	—	2	2	no	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	—	13	2	no	—
251	C₄×D₄	—	13	2	no	—
252	C₄×D₄	—	26	1	no	—
253	C₄×D₄	—	26	1	no	—
254	C₂×C₄⋊C₄	—	13	1	no	—
255	C₂×C₂²⋊C₄	—	26	1	no	—
256	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
257	#107	—	2	1	no	—
258	#107	—	2	1	no	—
259	C₄⋊C₈	—	13	2	no	—
260	C₄⋊C₈	—	13	4	no	—
261	C₄⋊C₈	—	26	2	no	—
262	C₄⋊C₈	—	26	1	no	—
263	#103	—	26	4	no	—
264	#102	—	2	2	no	—
265	#102	—	2	2	no	—
266	#102	—	26	4	no	—
267	C₂²⋊C₈	—	26	2	no	—
268	C₂²⋊C₈	—	26	2	no	—
269	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
270	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
273	C₂×C₁₃	—	2	2	no	—
274	C₂×C₁₃	—	16	2	no	—
275	C₂×C₁₃	#3036	1	1	yes	characteristic
276	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
277	C₂×C₁₃	#3067	1	1	yes	characteristic
278	D₁₃	—	4	4	no	—
279	C₂⁴	—	26	1	no	—
280	#41	—	4	4	no	—
281	#41	—	4	4	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	13	4	no	—
288	C₂×D₄	—	13	2	no	—
289	C₂×D₄	—	26	2	no	—
290	C₂×D₄	—	26	1	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	26	1	no	—
293	C₂²×C₄	—	26	2	no	—
294	C₂²×C₄	—	26	2	no	—
295	C₂²×C₄	—	52	2	no	—
296	C₂²×C₄	—	52	2	no	—
297	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
298	C₂²×C₄	#6129	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
300	SD₁₆	—	52	8	no	—
301	D₈	—	4	2	no	—
302	D₈	—	4	2	no	—
303	D₈	—	52	8	no	—
304	M₁₆	—	52	4	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	26	4	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	52	4	no	—
310	C₂×C₈	—	52	2	no	—
311	C₂×C₈	—	52	4	no	—
312	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
314	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	13	2	no	—
319	C₄⋊C₄	—	26	1	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	—	4	2	no	—
322	C₂²⋊C₄	—	26	2	no	—
323	C₂²⋊C₄	—	26	2	no	—
324	C₄²	D₄×D₁₃	1	1	yes	characteristic
325	C₄²	#6129	1	1	yes	characteristic
326	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
327	C₄²	C₂³×D₁₃	1	1	yes	characteristic
328	C₁₃	#19255	1	1	yes	characteristic, complemented, Sylow
329	C₂³	—	4	2	no	—
330	C₂³	—	26	2	no	—
331	C₂³	—	26	4	no	—
332	C₂³	—	52	2	no	—
333	C₂³	—	52	2	no	—
334	C₂³	#88192	1	1	yes	characteristic
335	Q₈	—	4	2	no	—
336	Q₈	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	4	no	—
343	D₄	—	13	4	no	—
344	D₄	—	26	2	no	—
345	D₄	—	26	4	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	4	no	—
351	C₂×C₄	—	26	2	no	—
352	C₂×C₄	—	52	4	no	—
353	C₂×C₄	—	52	2	no	—
354	C₂×C₄	—	52	2	no	—
355	C₂×C₄	—	52	2	no	—
356	C₂×C₄	D₈×D₁₃	1	2	yes	—
357	C₂×C₄	#88166	1	2	yes	—
358	C₂×C₄	#88192	1	1	yes	characteristic
359	C₂×C₄	#88192	1	1	yes	characteristic
360	C₂×C₄	#88192	1	1	yes	characteristic
361	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
362	C₂×C₄	C₂×#6129	1	1	yes	characteristic
363	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	—	52	4	no	—
368	C₈	—	52	2	no	—
369	C₈	#88203	1	2	yes	complemented
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	8	2	no	—
373	C₂²	—	8	2	no	—
374	C₂²	—	8	2	no	—
375	C₂²	—	26	4	no	—
376	C₂²	—	52	8	no	—
377	C₂²	—	52	2	no	—
378	C₂²	—	52	2	no	—
379	C₂²	#11718818	1	2	yes	—
380	C₂²	C₂×#88192	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	8	2	no	—
385	C₄	—	52	2	no	—
386	C₄	#11718818	1	2	yes	—
387	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
388	C₄	C₂×#88166	1	1	yes	characteristic
389	C₄	C₂×#88203	1	1	yes	characteristic
390	C₄	#11719527	1	1	yes	characteristic
391	C₂	—	2	2	no	—
392	C₂	—	16	2	no	—
393	C₂	—	52	4	no	—
394	C₂	#13124296	1	1	yes	characteristic
395	C₂	C₂×#11718818	1	1	yes	characteristic
396	C₂	#13124552	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂×#121
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#19255
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13124296	C₂	semidirect
2. C₂×C₈×D₅₂	C₂	semidirect
3. #13114270	C₂	semidirect
4. #13114222	C₂	semidirect
5. #13114188	C₂	semidirect
6. #13113595	C₂	semidirect
7. #13113513	C₂	semidirect
8. C₁₃×#3169	C₂	semidirect
13. C₁₃×#500	C₂²	semidirect
14. C₁₃×#493	C₂²	semidirect
15. C₁₃×#486	C₂²	semidirect
32. #11718139	C₂²	semidirect
33. #11718136	C₂²	semidirect
34. #11718050	C₂²	semidirect
35. #11718050	C₂²	semidirect
97. C₂×D₅₂	D₈	semidirect
131. #3169	D₁₃	semidirect
177. C₈×C₁₃	AGL(1,ℤ/8)	semidirect
184. #500	C₂×D₁₃	semidirect
185. #493	C₂×D₁₃	semidirect
186. #486	C₂×D₁₃	semidirect
328. C₁₃	#19255	semidirect
369. C₈	#88203	semidirect

Maximal subgroups

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

subgroup	index
#13124296	2
C₂×C₈×D₅₂	2
#13114270	2
#13114222	2
#13114188	2
#13113595	2
#13113513	2
C₁₃×#3169	2
#19255	13