Subgroups — #15881813

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123787	C₂	1	2	yes	maximal
2	C₂×#11718362	C₂	1	1	yes	maximal, characteristic
3	#13115609	C₂	1	2	yes	maximal
4	#13115608	C₂	1	2	yes	maximal
5	#13115608	C₂	1	2	yes	maximal
6	#13115009	C₂	1	1	yes	maximal, characteristic
7	#13115006	C₂	1	1	yes	maximal, characteristic
8	#13114744	C₂	1	1	yes	maximal, characteristic
9	#13114742	C₂	1	1	yes	maximal, characteristic
10	#108×Dic₁₃	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1849	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#108	C₂²	1	2	yes	—
16	C₁₃×#423	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×#88062	C₂²	1	2	yes	—
19	C₂×#88062	C₂²	1	2	yes	—
20	#11718779	C₂²	1	2	yes	—
21	#11718779	C₂²	1	2	yes	—
22	M₁₆×Dic₁₃	C₂²	1	2	yes	—
23	M₁₆×Dic₁₃	C₂²	1	2	yes	—
24	C₂×#88059	C₂²	1	1	yes	characteristic
25	C₂×#88059	C₂²	1	1	yes	characteristic
26	C₂×#88055	C₂²	1	1	yes	characteristic
27	C₂×#88054	C₂²	1	1	yes	characteristic
28	C₂×#88054	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
31	C₄×#6099	C₂²	1	2	yes	—
32	#11718492	—	2	2	no	—
33	#11718491	—	2	2	no	—
34	#11718489	—	2	2	no	—
35	#11718489	—	2	2	no	—
36	#11718362	C₄	1	4	yes	—
37	#11718362	C₄	1	4	yes	—
38	#11718150	C₂²	1	2	yes	—
39	#11718150	C₂²	1	2	yes	—
40	#11718144	C₂²	1	2	yes	—
41	C₂²×Dic₂₆	Q₈	1	1	yes	characteristic
42	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
48	C₂×#6101	Q₈	1	1	yes	characteristic
49	C₂×#6101	Q₈	1	1	yes	characteristic
50	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂×#6099	C₂³	1	2	yes	—
53	C₂×#6099	C₂³	1	2	yes	—
54	C₂²×#1055	C₂³	1	1	yes	characteristic
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
57	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
58	C₄×Dic₂₆	D₄	1	2	yes	—
59	C₄×Dic₂₆	D₄	1	2	yes	—
60	C₁₃×#108	—	2	2	no	—
61	C₁₃×#108	—	2	2	no	—
62	C₁₃×#108	—	4	2	no	—
63	C₁₃×C₄⋊C₈	—	2	2	no	—
64	C₁₃×C₄⋊C₈	—	2	2	no	—
65	C₁₃×#97	C₂×C₄	1	4	yes	—
66	#88062	—	4	2	no	—
67	#88062	—	4	2	no	—
68	#88059	—	2	2	no	—
69	#88059	Q₈	1	4	yes	complemented
70	#88055	—	2	2	no	—
71	#88055	—	2	2	no	—
72	#88055	C₂×C₄	1	4	yes	—
73	#88054	—	2	2	no	—
74	#88054	—	2	2	no	—
75	#88054	—	2	2	no	—
76	#88054	—	2	2	no	—
77	#88054	C₂×C₄	1	4	yes	—
78	#88054	C₂×C₄	1	2	yes	—
79	#88054	C₂×C₄	1	2	yes	—
80	C₈×Dic₁₃	—	2	2	no	—
81	C₈×Dic₁₃	—	2	2	no	—
82	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
83	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
84	C₄⋊#1055	C₂×C₄	1	4	yes	—
85	#88043	—	2	2	no	—
86	C₄×#1055	—	2	2	no	—
87	#22733	—	13	1	no	maximal, Sylow
88	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
92	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
93	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×Dic₂₆	—	4	2	no	—
96	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
97	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
98	C₁₃×M₁₆	—	2	2	no	—
99	C₁₃×M₁₆	—	2	2	no	—
100	C₁₃×M₁₆	—	2	2	no	—
101	C₁₃×M₁₆	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	—	2	1	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₁₃	#41	1	2	yes	—
110	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₁₃	C₂×D₄	1	2	yes	—
112	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
113	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
114	#6101	C₄⋊C₄	1	2	yes	—
115	#6101	C₄⋊C₄	1	2	yes	—
116	#6101	C₄⋊C₄	1	2	yes	—
117	#6101	C₄⋊C₄	1	2	yes	—
118	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
121	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
123	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
124	#6099	—	2	2	no	—
125	#6099	—	2	2	no	—
126	#6099	—	2	2	no	—
127	#6099	C₂×Q₈	1	4	yes	—
128	C₂×#1055	—	2	2	no	—
129	C₂×#1055	—	2	1	no	—
130	C₂×#1055	—	2	1	no	—
131	C₂×#1055	—	2	1	no	—
132	C₂×#1055	—	2	1	no	—
133	C₂×#1055	—	2	2	no	—
134	C₂×#1055	C₂²×C₄	1	2	yes	—
135	C₂×#1055	C₂²×C₄	1	2	yes	—
136	Q₈×M₁₆	—	13	2	no	—
137	C₂×#446	—	13	1	no	—
138	#1950	—	13	2	no	—
139	#1950	—	13	2	no	—
140	#1950	—	13	2	no	—
141	#1849	—	13	1	no	—
142	#1849	D₁₃	1	1	yes	characteristic
143	#1848	—	13	1	no	—
144	#1778	—	13	1	no	—
145	#1778	—	13	1	no	—
146	C₄×#108	—	13	1	no	—
147	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	—	4	1	no	—
160	C₂×Dic₁₃	—	4	1	no	—
161	C₂×Dic₁₃	—	4	1	no	—
162	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
168	Dic₂₆	—	2	4	no	—
169	Dic₂₆	—	4	2	no	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	—	4	2	no	—
172	C₈×C₁₃	—	4	2	no	—
173	C₈×C₁₃	C₄×Q₈	1	4	yes	—
174	#1055	—	2	2	no	—
175	#1055	—	2	2	no	—
176	#1055	—	2	4	no	—
177	#1055	—	4	2	no	—
178	C₂×C₄×Q₈	—	13	1	no	—
179	#446	—	13	4	no	—
180	#446	—	13	4	no	—
181	#446	—	26	2	no	—
182	#446	—	26	2	no	—
183	#446	—	26	2	no	—
184	C₈×Q₈	—	26	2	no	—
185	C₂×#108	—	13	1	no	—
186	C₂×#108	—	13	1	no	—
187	C₂×#108	—	13	2	no	—
188	C₂×#108	—	13	2	no	—
189	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
190	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
191	C₂×#108	C₂×D₁₃	1	2	yes	—
192	#423	—	13	2	no	—
193	#423	—	13	2	no	—
194	#423	C₂×D₁₃	1	2	yes	—
195	C₂×C₄⋊C₈	—	13	1	no	—
196	C₂×C₄⋊C₈	—	13	1	no	—
197	C₂×C₄⋊C₈	—	13	1	no	—
198	C₄×M₁₆	—	13	2	no	—
199	C₄×M₁₆	—	13	2	no	—
200	C₄×M₁₆	—	13	2	no	—
201	C₂×#97	—	13	1	no	—
202	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
203	C₂×C₄×C₈	—	13	1	no	—
204	#341	—	13	2	no	—
205	#341	—	13	2	no	—
206	#341	—	13	2	no	—
207	C₂²×C₁₃	—	2	1	no	—
208	C₂²×C₁₃	—	2	1	no	—
209	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
210	C₂²×C₁₃	#557	1	1	yes	characteristic
211	C₂²×C₁₃	Q₈²	1	1	yes	characteristic
212	C₄×C₁₃	—	2	2	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
215	C₄×C₁₃	#520	1	1	yes	characteristic
216	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
217	C₄×C₁₃	#550	1	1	yes	characteristic
218	Dic₁₃	—	2	2	no	—
219	Dic₁₃	—	2	1	no	—
220	Dic₁₃	—	2	1	no	—
221	Dic₁₃	—	2	1	no	—
222	Dic₁₃	—	2	2	no	—
223	Dic₁₃	#430	1	2	yes	—
224	C₂²×Q₈	—	13	1	no	—
225	C₂×M₁₆	—	13	2	no	—
226	C₂×M₁₆	—	13	2	no	—
227	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
228	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
229	C₂²×C₈	—	13	1	no	—
230	C₂²×C₈	—	13	1	no	—
231	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₄×Q₈	—	13	2	no	—
234	C₄×Q₈	—	13	2	no	—
235	C₄×Q₈	—	13	2	no	—
236	C₄×Q₈	—	13	2	no	—
237	C₂×C₄⋊C₄	—	13	1	no	—
238	C₂×C₄⋊C₄	—	13	1	no	—
239	C₂×C₄⋊C₄	—	13	1	no	—
240	C₂×C₄²	—	13	1	no	—
241	C₂×C₄²	—	13	1	no	—
242	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
243	#108	—	2	2	no	—
244	#108	—	2	2	no	—
245	#108	—	4	2	no	—
246	#108	—	13	4	no	—
247	#108	—	26	2	no	—
248	#108	—	52	2	no	—
249	#108	—	52	2	no	—
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	13	2	no	—
253	C₄⋊C₈	—	13	2	no	—
254	C₄⋊C₈	—	13	4	no	—
255	C₄⋊C₈	—	13	4	no	—
256	C₄⋊C₈	—	26	2	no	—
257	C₄⋊C₈	—	26	2	no	—
258	C₄⋊C₈	—	26	2	no	—
259	C₄⋊C₈	—	26	2	no	—
260	#97	—	13	4	no	—
261	#97	—	26	2	no	—
262	#97	—	26	2	no	—
263	#97	—	26	2	no	—
264	#97	C₄×D₁₃	1	4	yes	—
265	C₄×C₈	—	13	2	no	—
266	C₄×C₈	—	13	2	no	—
267	C₄×C₈	—	26	2	no	—
268	C₄×C₈	—	26	2	no	—
269	C₄×C₈	—	26	2	no	—
270	C₂×C₁₃	—	2	1	no	—
271	C₂×C₁₃	—	2	1	no	—
272	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
274	C₂×C₁₃	#2916	1	1	yes	characteristic
275	C₂×Q₈	—	13	2	no	—
276	C₂×Q₈	—	13	2	no	—
277	C₂×Q₈	—	52	2	no	—
278	C₂²×C₄	—	13	1	no	—
279	C₂²×C₄	—	13	1	no	—
280	C₂²×C₄	—	13	1	no	—
281	C₂²×C₄	—	13	1	no	—
282	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
283	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
284	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	13	4	no	—
290	M₁₆	—	26	2	no	—
291	M₁₆	—	26	2	no	—
292	M₁₆	—	26	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	13	2	no	—
298	C₂×C₈	—	13	2	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	26	1	no	—
301	C₂×C₈	—	26	1	no	—
302	C₂×C₈	—	26	1	no	—
303	C₂×C₈	—	26	1	no	—
304	C₂×C₈	—	26	2	no	—
305	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
306	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
307	C₂×C₈	#6129	1	2	yes	—
308	C₂×C₈	Q₈×D₁₃	1	2	yes	—
309	C₄⋊C₄	—	13	2	no	—
310	C₄⋊C₄	—	13	2	no	—
311	C₄⋊C₄	—	13	2	no	—
312	C₄⋊C₄	—	13	2	no	—
313	C₄⋊C₄	—	13	2	no	—
314	C₄⋊C₄	—	13	2	no	—
315	C₄²	—	13	1	no	—
316	C₄²	—	13	1	no	—
317	C₄²	—	13	2	no	—
318	C₄²	—	13	1	no	—
319	C₄²	—	13	2	no	—
320	C₄²	—	13	1	no	—
321	C₄²	C₂×C₄×D₁₃	1	2	yes	—
322	C₄²	D₄×D₁₃	1	2	yes	—
323	C₁₃	#22733	1	1	yes	characteristic, complemented, Sylow
324	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
325	Q₈	—	26	4	no	—
326	Q₈	—	52	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	13	1	no	—
331	C₂×C₄	—	13	1	no	—
332	C₂×C₄	—	13	2	no	—
333	C₂×C₄	—	13	1	no	—
334	C₂×C₄	—	13	2	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	52	1	no	—
337	C₂×C₄	—	52	1	no	—
338	C₂×C₄	—	52	1	no	—
339	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
340	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
341	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
343	C₂×C₄	C₂×#6129	1	1	yes	characteristic
344	C₂×C₄	#88251	1	1	yes	characteristic
345	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
346	C₂×C₄	#88254	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	26	4	no	—
351	C₈	—	26	2	no	—
352	C₈	—	26	2	no	—
353	C₈	—	52	2	no	—
354	C₈	#88141	1	4	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
358	C₂²	#11719426	1	1	yes	characteristic
359	C₂²	#11719427	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	13	2	no	—
363	C₄	—	26	1	no	—
364	C₄	—	26	2	no	—
365	C₄	—	26	1	no	—
366	C₄	—	26	1	no	—
367	C₄	—	26	2	no	—
368	C₄	C₂×#88141	1	1	yes	characteristic
369	C₄	#11719209	1	1	yes	characteristic
370	C₄	#11719423	1	1	yes	characteristic
371	C₄	#11719432	1	1	yes	characteristic
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	#13117748	1	1	yes	characteristic
375	C₂	D₁₃×#430	1	1	yes	characteristic
376	C₂	#13123446	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1849	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1849	D₁₃
hypocenter	C₁₃	#22733
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
69. #88059	Q₈	semidirect
323. C₁₃	#22733	semidirect

Maximal subgroups

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

subgroup	index
#13123787	2
C₂×#11718362	2
#13115609	2
#13115608	2
#13115009	2
#13115006	2
#13114744	2
#13114742	2
#108×Dic₁₃	2
C₁₃×#1849	2
#22733	13