Subgroups — #16018255

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125376	C₂	1	1	yes	maximal, characteristic, complemented
2	#13125323	C₂	1	1	yes	maximal, characteristic
3	#13117076	C₂	1	2	yes	maximal, complemented
4	#13117074	C₂	1	2	yes	maximal, complemented
5	#13117069	C₂	1	2	yes	maximal, complemented
6	#13117058	C₂	1	2	yes	maximal, complemented
7	#13116551	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11718101	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2138	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
13	#11719535	—	2	1	no	—
14	C₂×#88203	—	2	1	no	—
15	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
16	#11719224	—	2	1	no	—
17	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₁₃×#427	C₂²	1	1	yes	characteristic
19	#11718852	C₂²	1	2	yes	—
20	#11718837	—	2	2	no	—
21	#11718812	C₂²	1	2	yes	—
22	C₂×#88058	C₂²	1	1	yes	characteristic
23	#11718762	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	#11718656	—	2	1	no	—
26	#11718539	C₂²	1	2	yes	complemented
27	#11718535	C₂²	1	2	yes	complemented
28	#11718517	—	2	2	no	—
29	C₁₃×#358	C₂²	1	1	yes	characteristic, complemented
30	C₁₃×#358	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×#357	C₂²	1	1	yes	characteristic, complemented
32	C₁₃×#357	C₂²	1	1	yes	characteristic, complemented
33	#11718156	C₂²	1	2	yes	complemented
34	#11718154	C₂²	1	2	yes	complemented
35	#11718101	C₂²	1	2	yes	—
36	#11718101	C₂²	1	2	yes	—
37	#11718094	C₂²	1	2	yes	complemented
38	C₁₆×Dic₁₃	C₂²	1	2	yes	complemented
39	#11718083	C₂²	1	2	yes	—
40	#11718083	C₂²	1	2	yes	—
41	C₂×C₁₃×#41	—	2	1	no	—
42	C₂²×D₅₂	—	2	1	no	—
43	C₁₃×#135	—	2	2	no	—
44	C₁₃×#135	—	2	2	no	—
45	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×SD₁₆	—	2	1	no	—
47	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#117	—	2	1	no	—
50	#88203	—	4	4	no	—
51	#88201	—	2	2	no	—
52	C₂×#6106	—	2	2	no	—
53	#88194	—	2	2	no	—
54	#88191	—	4	2	no	—
55	C₂×#6104	—	2	2	no	—
56	#88184	—	4	1	no	—
57	#88184	—	4	1	no	—
58	#88180	D₄	1	1	yes	characteristic
59	C₂×C₄⋊Dic₁₃	—	2	1	no	—
60	C₂×#6099	D₄	1	1	yes	characteristic
61	C₂×D₁₀₄	—	2	2	no	—
62	C₂×D₁₀₄	—	2	2	no	—
63	C₂×D₁₀₄	—	2	1	no	—
64	C₂×D₁₀₄	C₂³	1	2	yes	—
65	#88144	—	2	2	no	—
66	#88142	—	2	2	no	—
67	#88132	—	4	2	no	—
68	#88124	—	4	1	no	—
69	#88123	—	4	1	no	—
70	C₂×C₁₃×C₁₆	—	2	2	no	—
71	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
73	C₁₃×#107	C₂³	1	1	yes	characteristic
74	C₁₃×#107	C₂³	1	1	yes	characteristic
75	C₁₃×#106	—	2	1	no	—
76	#88058	—	2	1	no	—
77	#88058	C₂³	1	2	yes	—
78	#88055	D₄	1	2	yes	—
79	C₈×Dic₁₃	C₂³	1	2	yes	—
80	#88052	—	2	2	no	—
81	C₂×#6090	C₂³	1	2	yes	—
82	#88048	—	2	2	no	—
83	#88047	—	2	2	no	—
84	#33702	—	13	1	no	maximal, Sylow
85	C₂³×D₁₃	—	4	1	no	—
86	C₁₃×#41	—	4	2	no	—
87	C₁₃×#41	—	4	2	no	—
88	C₂×Q₈×C₁₃	—	2	1	no	—
89	C₂×D₄×C₁₃	—	2	1	no	—
90	C₂×D₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	—	4	1	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₃	—	4	2	no	—
94	C₂×C₂²⋊D₁₃	—	4	2	no	—
95	C₂²×Dic₁₃	—	4	1	no	—
96	C₂×D₅₂	—	2	2	no	—
97	C₂×D₅₂	—	4	2	no	—
98	C₂×D₅₂	—	4	2	no	—
99	C₂×D₅₂	—	4	1	no	—
100	C₂×Dic₂₆	—	4	2	no	—
101	C₁₃×Q₁₆	—	2	1	no	—
102	C₁₃×Q₁₆	—	2	1	no	—
103	C₁₃×SD₁₆	—	4	2	no	—
104	C₁₃×D₈	—	2	1	no	—
105	C₁₃×D₈	—	2	1	no	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	#41	1	2	yes	—
108	C₂×C₈×C₁₃	#41	1	2	yes	—
109	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	1	no	—
113	C₄²×C₁₃	—	4	1	no	—
114	C₂²⋊Dic₁₃	D₈	1	2	yes	—
115	#6106	—	4	4	no	—
116	#6104	—	4	4	no	—
117	#6103	—	4	2	no	—
118	#6103	—	4	2	no	—
119	C₄⋊Dic₁₃	—	2	2	no	—
120	C₄⋊Dic₁₃	—	4	1	no	—
121	C₄⋊Dic₁₃	—	4	2	no	—
122	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
123	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
124	#6101	—	4	2	no	—
125	#6101	—	4	2	no	—
126	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
127	#6099	SD₁₆	1	4	yes	—
128	C₂×#1055	C₂×D₄	1	2	yes	—
129	D₁₀₄	—	2	2	no	—
130	D₁₀₄	—	2	2	no	—
131	D₁₀₄	—	4	2	no	—
132	C₁₃×C₁₆	—	2	1	no	—
133	C₁₃×C₁₆	—	2	1	no	—
134	#6090	—	2	2	no	—
135	#3164	—	13	1	no	—
136	#3155	—	13	1	no	—
137	#2244	—	13	2	no	—
138	#2243	—	13	2	no	—
139	#2242	—	13	2	no	—
140	#2240	—	13	2	no	—
141	#2140	—	13	1	no	—
142	#2139	—	13	1	no	—
143	#2138	D₁₃	1	1	yes	characteristic, complemented
144	C₂×#357	—	13	1	no	—
145	#2108	—	13	1	no	—
146	C₂³×C₁₃	—	4	1	no	—
147	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂²×D₁₃	—	4	2	no	—
149	C₂²×D₁₃	—	8	2	no	—
150	C₂²×D₁₃	—	16	1	no	—
151	C₂²×D₁₃	—	16	1	no	—
152	Q₈×C₁₃	—	4	1	no	—
153	Q₈×C₁₃	—	4	1	no	—
154	D₄×C₁₃	—	4	1	no	—
155	D₄×C₁₃	—	4	1	no	—
156	D₄×C₁₃	—	8	2	no	—
157	C₂×C₄×C₁₃	—	4	1	no	—
158	C₂×C₄×C₁₃	—	4	1	no	—
159	C₂×C₄×C₁₃	—	4	1	no	—
160	C₂×C₄×C₁₃	—	8	2	no	—
161	C₂×C₄×C₁₃	#124	1	2	yes	—
162	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₂²⋊D₁₃	—	16	2	no	—
166	C₂²⋊D₁₃	—	16	2	no	—
167	C₂×Dic₁₃	—	4	2	no	—
168	C₂×Dic₁₃	—	16	1	no	—
169	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
170	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
171	D₅₂	—	4	2	no	—
172	D₅₂	—	4	2	no	—
173	D₅₂	—	8	2	no	—
174	Dic₂₆	—	16	2	no	—
175	C₈×C₁₃	#124	1	2	yes	—
176	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
177	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
178	#1055	C₂×SD₁₆	1	4	yes	—
179	C₂×AGL(1,ℤ/8)	—	26	1	no	—
180	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
181	C₂²×D₈	—	13	1	no	—
182	#544	—	26	1	no	—
183	#535	—	26	1	no	—
184	C₂×M₃₂	—	13	1	no	—
185	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
186	#501	—	26	2	no	—
187	#500	—	13	2	no	—
188	#499	—	13	2	no	—
189	#496	—	26	2	no	—
190	#495	—	13	2	no	—
191	#493	—	13	2	no	—
192	#428	—	26	1	no	—
193	#427	C₂×D₁₃	1	1	yes	characteristic
194	C₂×#107	—	13	1	no	—
195	#405	—	13	1	no	—
196	#358	—	13	2	no	—
197	#358	C₂×D₁₃	1	1	yes	characteristic, complemented
198	#358	C₂×D₁₃	1	1	yes	characteristic, complemented
199	#357	—	13	2	no	—
200	#357	—	13	2	no	—
201	#357	—	13	2	no	—
202	#357	—	13	2	no	—
203	#357	—	13	2	no	—
204	#357	C₂×D₁₃	1	1	yes	characteristic, complemented
205	#357	C₂×D₁₃	1	1	yes	characteristic, complemented
206	#346	—	13	2	no	—
207	C₄×C₁₆	—	13	2	no	—
208	C₂²×C₁₃	—	2	1	no	—
209	C₂²×C₁₃	—	2	1	no	—
210	C₂²×C₁₃	—	8	1	no	—
211	C₂²×C₁₃	—	8	1	no	—
212	C₂²×C₁₃	—	8	1	no	—
213	C₂²×C₁₃	#486	1	2	yes	—
214	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
215	C₂×D₁₃	—	8	2	no	—
216	C₂×D₁₃	—	8	2	no	—
217	C₂×D₁₃	—	8	2	no	—
218	C₂×D₁₃	—	16	4	no	—
219	C₄×C₁₃	—	8	1	no	—
220	C₄×C₁₃	—	8	1	no	—
221	C₄×C₁₃	—	8	1	no	—
222	C₄×C₁₃	#486	1	2	yes	—
223	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
224	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
225	Dic₁₃	—	2	2	no	—
226	Dic₁₃	—	2	2	no	—
227	Dic₁₃	—	8	2	no	—
228	C₂×#41	—	2	1	no	—
229	C₂²×D₄	—	26	1	no	—
230	AGL(1,ℤ/8)	—	52	4	no	—
231	#135	—	2	2	no	—
232	#135	—	2	2	no	—
233	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
234	C₂×SD₁₆	—	2	1	no	—
235	C₂×SD₁₆	—	26	2	no	—
236	C₂×D₈	—	13	2	no	—
237	C₂×D₈	—	26	2	no	—
238	C₂×D₈	—	26	2	no	—
239	C₂×D₈	—	26	1	no	—
240	C₂×D₈	—	26	2	no	—
241	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₂×M₁₆	—	13	1	no	—
243	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₄⋊Q₈	—	26	2	no	—
245	C₄⋊Q₈	—	52	1	no	—
246	#127	—	26	2	no	—
247	#125	—	26	2	no	—
248	#125	—	52	1	no	—
249	#124	—	26	2	no	—
250	C₂²⋊Q₈	—	52	2	no	—
251	#121	—	52	1	no	—
252	#121	—	52	1	no	—
253	#120	—	52	2	no	—
254	#117	—	2	1	no	—
255	#117	—	13	1	no	—
256	C₂×C₄⋊C₄	—	26	1	no	—
257	M₃₂	—	26	2	no	—
258	C₂×C₁₆	—	2	2	no	—
259	C₂×C₁₆	—	13	2	no	—
260	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
262	#107	—	13	2	no	—
263	#107	—	26	2	no	—
264	#107	—	26	1	no	—
265	#107	C₂²×D₁₃	1	1	yes	characteristic
266	#107	C₂²×D₁₃	1	1	yes	characteristic
267	#106	—	2	1	no	—
268	#106	—	26	2	no	—
269	#97	—	13	2	no	—
270	C₄×C₈	—	13	2	no	—
271	C₂×C₁₃	—	2	2	no	—
272	C₂×C₁₃	—	16	1	no	—
273	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
274	C₂×C₁₃	#3415	1	1	yes	characteristic
275	C₂×C₁₃	#3416	1	1	yes	characteristic
276	D₁₃	—	16	2	no	—
277	C₂⁴	—	52	1	no	—
278	#41	—	4	2	no	—
279	#41	—	4	2	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	52	2	no	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	4	1	no	—
284	C₂×D₄	—	26	2	no	—
285	C₂×D₄	—	52	2	no	—
286	C₂×D₄	—	52	2	no	—
287	C₂×D₄	—	52	2	no	—
288	C₂×D₄	—	52	1	no	—
289	C₂×D₄	—	52	2	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	52	1	no	—
292	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
293	Q₁₆	—	2	1	no	—
294	Q₁₆	—	2	1	no	—
295	SD₁₆	—	4	2	no	—
296	SD₁₆	—	52	4	no	—
297	D₈	—	2	1	no	—
298	D₈	—	2	1	no	—
299	D₈	—	26	2	no	—
300	D₈	—	26	2	no	—
301	D₈	—	52	4	no	—
302	D₈	—	52	2	no	—
303	M₁₆	—	13	4	no	—
304	C₂×C₈	—	13	2	no	—
305	C₂×C₈	#6127	1	2	yes	—
306	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
307	C₂×C₈	#6129	1	2	yes	—
308	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	13	1	no	—
312	C₄⋊C₄	—	13	1	no	—
313	C₄⋊C₄	—	26	2	no	—
314	C₄⋊C₄	—	52	2	no	—
315	C₄⋊C₄	—	52	2	no	—
316	C₄⋊C₄	—	52	2	no	—
317	C₄⋊C₄	—	52	1	no	—
318	C₂²⋊C₄	—	4	1	no	—
319	C₂²⋊C₄	—	13	2	no	—
320	C₂²⋊C₄	—	52	2	no	—
321	C₂²⋊C₄	—	52	2	no	—
322	C₄²	—	4	1	no	—
323	C₄²	—	13	2	no	—
324	C₁₆	—	2	1	no	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	26	2	no	—
327	C₁₃	#33702	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	4	1	no	—
329	C₂³	—	52	2	no	—
330	C₂³	—	104	2	no	—
331	C₂³	—	208	1	no	—
332	C₂³	—	208	1	no	—
333	C₂³	D₈×D₁₃	1	1	yes	characteristic
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	208	2	no	—
337	D₄	—	4	1	no	—
338	D₄	—	4	1	no	—
339	D₄	—	8	2	no	—
340	D₄	—	52	2	no	—
341	D₄	—	52	2	no	—
342	D₄	—	104	2	no	—
343	D₄	—	208	2	no	—
344	D₄	—	208	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	13	2	no	—
350	C₂×C₄	—	13	2	no	—
351	C₂×C₄	—	52	2	no	—
352	C₂×C₄	—	208	1	no	—
353	C₂×C₄	#88139	1	2	yes	—
354	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
355	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
357	C₈	—	13	4	no	—
358	C₈	#88139	1	2	yes	—
359	C₈	C₂×#6127	1	1	yes	characteristic
360	C₈	C₂×#6129	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	104	2	no	—
367	C₂²	—	104	2	no	—
368	C₂²	—	104	2	no	—
369	C₂²	—	208	4	no	—
370	C₂²	#11718421	1	2	yes	—
371	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
372	C₄	—	8	1	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	1	no	—
375	C₄	—	26	2	no	—
376	C₄	—	26	2	no	—
377	C₄	—	104	2	no	—
378	C₄	#11718421	1	2	yes	—
379	C₄	C₂×#88139	1	1	yes	characteristic
380	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	16	1	no	—
383	C₂	—	208	2	no	—
384	C₂	C₂×#11718421	1	1	yes	characteristic
385	C₂	#13127974	1	1	yes	characteristic
386	C₂	#13127979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2138	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13125323	C₂
hypercenter	#2138	D₁₃
hypocenter	C₁₃	#33702
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

Complements

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

N	H ≅ G/N	type
1. #13125376	C₂	semidirect
3. #13117076	C₂	semidirect
4. #13117074	C₂	semidirect
5. #13117069	C₂	semidirect
6. #13117058	C₂	semidirect
7. #13116551	C₂	semidirect
8. #13116548	C₂	semidirect
9. C₂×#11718101	C₂	semidirect
10. #13116432	C₂	semidirect
11. C₁₃×#2138	C₂	semidirect
26. #11718539	C₂²	semidirect
27. #11718535	C₂²	semidirect
29. C₁₃×#358	C₂²	semidirect
30. C₁₃×#358	C₂²	semidirect
31. C₁₃×#357	C₂²	semidirect
32. C₁₃×#357	C₂²	semidirect
33. #11718156	C₂²	semidirect
34. #11718154	C₂²	semidirect
37. #11718094	C₂²	semidirect
38. C₁₆×Dic₁₃	C₂²	semidirect
143. #2138	D₁₃	semidirect
197. #358	C₂×D₁₃	semidirect
198. #358	C₂×D₁₃	semidirect
204. #357	C₂×D₁₃	semidirect
205. #357	C₂×D₁₃	semidirect
327. C₁₃	#33702	semidirect

Maximal subgroups

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

subgroup	index
#13125376	2
#13125323	2
#13117076	2
#13117074	2
#13117069	2
#13117058	2
#13116551	2
#13116548	2
C₂×#11718101	2
#13116432	2
C₁₃×#2138	2
#33702	13