Subgroups — #16018958

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125486	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117103	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117093	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117081	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117067	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116888	C₂	1	2	yes	maximal, complemented
7	#13116878	C₂	1	2	yes	maximal, complemented
8	#13116759	C₂	1	2	yes	maximal, complemented
9	#13116753	C₂	1	2	yes	maximal, complemented
10	#13116643	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3156	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#137	—	2	1	no	—
13	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#536	—	2	1	no	—
16	#11719240	—	2	1	no	—
17	C₁₃×#497	—	2	1	no	—
18	C₁₃×#497	—	2	1	no	—
19	C₁₃×#495	C₂²	1	1	yes	characteristic
20	C₁₃×#495	C₂²	1	1	yes	characteristic
21	C₁₃×#494	C₂²	1	1	yes	characteristic
22	C₁₃×#494	C₂²	1	1	yes	characteristic
23	C₁₃×#405	C₂²	1	1	yes	characteristic
24	C₂×#88068	C₂²	1	2	yes	—
25	C₂×#88068	C₂²	1	2	yes	—
26	C₂×#88068	C₂²	1	2	yes	—
27	C₂×#88066	C₂²	1	2	yes	—
28	#11718796	—	2	1	no	—
29	#11718776	C₂²	1	2	yes	—
30	#11718374	—	2	1	no	—
31	#11718372	—	2	1	no	—
32	#11718356	C₂²	1	1	yes	characteristic
33	C₄×D₁₀₄	C₂²	1	2	yes	—
34	#11718344	C₂²	1	1	yes	characteristic
35	#11718156	C₂²	1	2	yes	—
36	#11718156	C₂²	1	2	yes	—
37	#11718156	C₂²	1	2	yes	—
38	#11718154	C₂²	1	2	yes	—
39	C₂²⋊#6090	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	2	yes	—
41	#11718053	C₂²	1	1	yes	characteristic
42	C₄×#6090	C₂²	1	1	yes	characteristic
43	C₂×C₁₃×#41	—	2	1	no	—
44	C₂²×Q₈×C₁₃	—	2	1	no	—
45	C₁₃×#137	—	4	4	no	—
46	C₁₃×#135	—	4	2	no	—
47	C₂×C₁₃×Q₁₆	—	2	1	no	—
48	C₂×C₁₃×Q₁₆	—	2	1	no	—
49	C₂×C₁₃×Q₁₆	—	2	1	no	—
50	C₂×C₁₃×Q₁₆	—	4	2	no	—
51	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₁₃×SD₁₆	—	2	1	no	—
55	C₂×C₁₃×SD₁₆	—	2	1	no	—
56	C₂×C₁₃×SD₁₆	—	2	1	no	—
57	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊Q₈	—	2	1	no	—
61	C₁₃×C₄⋊Q₈	—	2	1	no	—
62	C₁₃×#124	—	2	1	no	—
63	C₁₃×#124	—	2	1	no	—
64	C₁₃×#123	—	4	2	no	—
65	C₁₃×C₂²⋊Q₈	—	4	2	no	—
66	C₁₃×#117	D₄	1	1	yes	characteristic
67	#88184	—	2	2	no	—
68	C₂×D₁₀₄	—	2	1	no	—
69	C₂×D₁₀₄	C₂³	1	2	yes	—
70	#88148	—	4	2	no	—
71	#88138	—	4	2	no	—
72	#88128	—	2	1	no	—
73	C₄×D₅₂	—	2	2	no	—
74	#88124	—	2	1	no	—
75	C₁₃×#97	D₄	1	1	yes	characteristic
76	C₁₃×#97	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#88068	—	4	2	no	—
80	#88068	—	4	2	no	—
81	#88068	—	4	2	no	—
82	#88066	—	4	2	no	—
83	#88061	—	2	2	no	—
84	#88061	—	2	2	no	—
85	#88058	C₂³	1	2	yes	—
86	#88057	—	2	1	no	—
87	C₂×#6090	C₂³	1	2	yes	—
88	C₂×#6090	C₂³	1	2	yes	—
89	#33787	—	13	1	no	maximal, Sylow
90	C₁₃×#41	—	4	2	no	—
91	C₁₃×#41	—	4	2	no	—
92	C₂×Q₈×C₁₃	—	2	1	no	—
93	C₂×Q₈×C₁₃	—	2	1	no	—
94	C₂×Q₈×C₁₃	—	2	1	no	—
95	C₂×Q₈×C₁₃	—	4	2	no	—
96	C₂×Q₈×C₁₃	—	4	2	no	—
97	C₂×Q₈×C₁₃	—	4	1	no	—
98	C₂×D₄×C₁₃	—	2	1	no	—
99	C₂×D₄×C₁₃	—	4	1	no	—
100	C₂²×C₄×C₁₃	—	4	1	no	—
101	C₂²×C₄×C₁₃	—	4	1	no	—
102	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₂²⋊D₁₃	—	4	2	no	—
104	C₂×D₅₂	—	2	2	no	—
105	C₂×D₅₂	—	4	2	no	—
106	C₂×C₄×D₁₃	—	4	2	no	—
107	C₁₃×Q₁₆	—	2	2	no	—
108	C₁₃×Q₁₆	—	2	2	no	—
109	C₁₃×Q₁₆	—	2	2	no	—
110	C₁₃×Q₁₆	—	4	2	no	—
111	C₁₃×Q₁₆	—	4	2	no	—
112	C₁₃×Q₁₆	—	4	2	no	—
113	C₁₃×SD₁₆	—	4	2	no	—
114	C₁₃×SD₁₆	—	4	2	no	—
115	C₁₃×SD₁₆	—	4	2	no	—
116	C₁₃×D₈	—	2	2	no	—
117	C₁₃×M₁₆	—	2	2	no	—
118	C₂×C₈×C₁₃	—	2	2	no	—
119	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
123	C₁₃×C₄⋊C₄	—	4	2	no	—
124	C₁₃×C₄⋊C₄	—	4	2	no	—
125	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
126	C₁₃×C₂²⋊C₄	—	4	2	no	—
127	C₁₃×C₂²⋊C₄	—	4	2	no	—
128	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
129	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
131	#6103	—	4	2	no	—
132	#6103	—	4	2	no	—
133	C₄⋊Dic₁₃	—	2	2	no	—
134	#6101	—	4	2	no	—
135	D₁₀₄	—	2	2	no	—
136	D₁₀₄	—	2	2	no	—
137	D₁₀₄	—	4	2	no	—
138	#6090	—	2	2	no	—
139	#6090	—	2	2	no	—
140	#3176	—	13	1	no	—
141	#3156	D₁₃	1	1	yes	characteristic, complemented
142	#2250	—	13	1	no	—
143	#2248	—	13	1	no	—
144	#2245	—	13	1	no	—
145	#2242	—	13	1	no	—
146	#2210	—	13	2	no	—
147	#2208	—	13	2	no	—
148	#2186	—	13	2	no	—
149	#2185	—	13	2	no	—
150	#2164	—	13	1	no	—
151	C₂³×C₁₃	—	4	1	no	—
152	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂²×D₁₃	—	4	2	no	—
154	Q₈×C₁₃	—	4	1	no	—
155	Q₈×C₁₃	—	4	1	no	—
156	Q₈×C₁₃	—	4	1	no	—
157	Q₈×C₁₃	—	4	1	no	—
158	Q₈×C₁₃	—	4	1	no	—
159	Q₈×C₁₃	—	4	1	no	—
160	Q₈×C₁₃	—	8	2	no	—
161	D₄×C₁₃	—	4	1	no	—
162	D₄×C₁₃	—	4	1	no	—
163	D₄×C₁₃	—	8	2	no	—
164	C₂×C₄×C₁₃	—	2	1	no	—
165	C₂×C₄×C₁₃	—	2	1	no	—
166	C₂×C₄×C₁₃	—	4	1	no	—
167	C₂×C₄×C₁₃	—	4	1	no	—
168	C₂×C₄×C₁₃	—	4	1	no	—
169	C₂×C₄×C₁₃	—	8	2	no	—
170	C₂×C₄×C₁₃	—	8	2	no	—
171	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
172	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
174	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
175	C₂²⋊D₁₃	—	16	2	no	—
176	C₂×Dic₁₃	—	4	2	no	—
177	D₅₂	—	4	2	no	—
178	D₅₂	—	4	2	no	—
179	D₅₂	—	16	2	no	—
180	C₄×D₁₃	—	16	2	no	—
181	C₈×C₁₃	—	2	1	no	—
182	C₈×C₁₃	—	2	1	no	—
183	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
184	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
185	C₂×#137	—	2	1	no	—
186	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
187	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
188	#540	—	26	1	no	—
189	#536	—	2	1	no	—
190	C₂×SD₃₂	—	13	2	no	—
191	C₂×SD₃₂	—	13	2	no	—
192	C₂×SD₃₂	—	13	2	no	—
193	C₂×D₁₆	—	13	2	no	—
194	#499	—	13	1	no	—
195	#497	—	2	1	no	—
196	#497	—	2	1	no	—
197	#495	C₂×D₁₃	1	1	yes	characteristic
198	#495	C₂×D₁₃	1	1	yes	characteristic
199	#494	C₂×D₁₃	1	1	yes	characteristic
200	#494	C₂×D₁₃	1	1	yes	characteristic
201	#493	—	13	1	no	—
202	#490	—	26	1	no	—
203	#469	—	26	1	no	—
204	#466	—	13	2	no	—
205	#442	—	26	1	no	—
206	C₄×D₈	—	13	2	no	—
207	#405	C₂×D₁₃	1	1	yes	characteristic
208	C₄⋊C₁₆	—	13	2	no	—
209	#358	—	13	2	no	—
210	#358	—	13	2	no	—
211	#358	—	13	2	no	—
212	#357	—	13	2	no	—
213	C₂²⋊C₁₆	—	13	2	no	—
214	#346	—	13	1	no	—
215	C₄×C₁₆	—	13	1	no	—
216	C₂²×C₁₃	—	2	2	no	—
217	C₂²×C₁₃	—	2	1	no	—
218	C₂²×C₁₃	—	8	2	no	—
219	C₂²×C₁₃	—	8	1	no	—
220	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
221	C₂×D₁₃	—	8	2	no	—
222	C₂×D₁₃	—	8	2	no	—
223	C₂×D₁₃	—	8	2	no	—
224	C₄×C₁₃	—	2	2	no	—
225	C₄×C₁₃	—	2	2	no	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	—	8	1	no	—
228	C₄×C₁₃	—	8	1	no	—
229	C₄×C₁₃	—	8	1	no	—
230	C₄×C₁₃	#535	1	1	yes	characteristic
231	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	Dic₁₃	—	8	2	no	—
233	C₂×#41	—	2	1	no	—
234	C₂²×Q₈	—	2	1	no	—
235	#137	—	4	4	no	—
236	#135	—	4	2	no	—
237	C₂×Q₁₆	—	2	1	no	—
238	C₂×Q₁₆	—	2	1	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×Q₁₆	—	4	2	no	—
241	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
242	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
243	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×SD₁₆	—	2	1	no	—
245	C₂×SD₁₆	—	2	1	no	—
246	C₂×SD₁₆	—	2	1	no	—
247	C₂×D₈	—	13	2	no	—
248	C₂×D₈	—	26	1	no	—
249	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
250	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	#127	—	26	1	no	—
255	#125	—	26	1	no	—
256	#124	—	2	1	no	—
257	#124	—	2	1	no	—
258	#123	—	4	2	no	—
259	#123	—	52	2	no	—
260	C₂²⋊Q₈	—	4	2	no	—
261	#121	—	26	2	no	—
262	#121	—	52	2	no	—
263	C₄×D₄	—	26	2	no	—
264	#117	C₂²⋊D₁₃	1	1	yes	characteristic
265	SD₃₂	—	52	2	no	—
266	SD₃₂	—	52	2	no	—
267	SD₃₂	—	52	2	no	—
268	D₁₆	—	52	2	no	—
269	C₂×C₁₆	—	13	2	no	—
270	C₂×C₁₆	—	13	2	no	—
271	#107	—	13	2	no	—
272	#106	—	26	1	no	—
273	#102	—	26	2	no	—
274	#102	—	26	2	no	—
275	#97	C₂²⋊D₁₃	1	1	yes	characteristic
276	#97	C₂²⋊D₁₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
279	C₂×C₁₃	—	4	1	no	—
280	C₂×C₁₃	—	16	1	no	—
281	C₂×C₁₃	#3205	1	1	yes	characteristic
282	C₂×C₁₃	#3416	1	2	yes	—
283	D₁₃	—	16	2	no	—
284	#41	—	4	2	no	—
285	#41	—	4	2	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	2	1	no	—
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	—	4	2	no	—
290	C₂×Q₈	—	4	2	no	—
291	C₂×Q₈	—	4	1	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	26	2	no	—
295	C₂×D₄	—	52	2	no	—
296	C₂×D₄	—	52	2	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	—	52	2	no	—
300	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
301	Q₁₆	—	2	2	no	—
302	Q₁₆	—	2	2	no	—
303	Q₁₆	—	2	2	no	—
304	Q₁₆	—	4	2	no	—
305	Q₁₆	—	4	2	no	—
306	Q₁₆	—	4	2	no	—
307	SD₁₆	—	4	2	no	—
308	SD₁₆	—	4	2	no	—
309	SD₁₆	—	4	2	no	—
310	D₈	—	2	2	no	—
311	D₈	—	26	2	no	—
312	D₈	—	26	2	no	—
313	D₈	—	52	2	no	—
314	M₁₆	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
317	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
318	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
319	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	4	2	no	—
322	C₄⋊C₄	—	26	2	no	—
323	C₄⋊C₄	—	52	2	no	—
324	C₄⋊C₄	#6104	1	2	yes	—
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	—	52	2	no	—
328	C₂²⋊C₄	—	52	2	no	—
329	C₂²⋊C₄	#6104	1	2	yes	—
330	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
331	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
332	C₁₆	—	26	2	no	—
333	C₁₆	—	26	2	no	—
334	C₁₃	#33787	1	1	yes	characteristic, complemented, Sylow
335	C₂³	—	4	1	no	—
336	C₂³	—	52	2	no	—
337	C₂³	C₂×#6104	1	1	yes	characteristic
338	Q₈	—	4	1	no	—
339	Q₈	—	4	1	no	—
340	Q₈	—	4	1	no	—
341	Q₈	—	4	1	no	—
342	Q₈	—	4	1	no	—
343	Q₈	—	4	1	no	—
344	Q₈	—	8	2	no	—
345	D₄	—	4	1	no	—
346	D₄	—	4	1	no	—
347	D₄	—	8	2	no	—
348	D₄	—	52	2	no	—
349	D₄	—	52	2	no	—
350	D₄	—	208	2	no	—
351	D₄	—	208	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	8	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	—	52	2	no	—
360	C₂×C₄	—	208	2	no	—
361	C₂×C₄	C₂×#6104	1	2	yes	—
362	C₂×C₄	C₂×#6104	1	2	yes	—
363	C₂×C₄	C₂×#6104	1	1	yes	characteristic
364	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	#88186	1	2	yes	—
368	C₈	#88251	1	2	yes	—
369	C₂²	—	2	2	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	8	2	no	—
372	C₂²	—	8	1	no	—
373	C₂²	—	104	2	no	—
374	C₂²	—	104	2	no	—
375	C₂²	—	104	2	no	—
376	C₂²	C₂²×#6104	1	1	yes	characteristic
377	C₄	—	2	2	no	—
378	C₄	—	2	2	no	—
379	C₄	—	2	1	no	—
380	C₄	—	8	1	no	—
381	C₄	—	8	1	no	—
382	C₄	—	8	1	no	—
383	C₄	—	104	2	no	—
384	C₄	C₂×#88186	1	1	yes	characteristic
385	C₄	#11719510	1	1	yes	characteristic
386	C₂	—	4	1	no	—
387	C₂	—	16	1	no	—
388	C₂	—	208	2	no	—
389	C₂	#13125754	1	1	yes	characteristic
390	C₂	#13127980	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3156	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3156	D₁₃
hypocenter	C₁₃	#33787
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3156

Complements

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

N	H ≅ G/N	type
1. #13125486	C₂	semidirect
2. #13117103	C₂	semidirect
3. #13117093	C₂	semidirect
4. #13117081	C₂	semidirect
5. #13117067	C₂	semidirect
6. #13116888	C₂	semidirect
7. #13116878	C₂	semidirect
8. #13116759	C₂	semidirect
9. #13116753	C₂	semidirect
10. #13116643	C₂	semidirect
11. C₁₃×#3156	C₂	semidirect
141. #3156	D₁₃	semidirect
334. C₁₃	#33787	semidirect

Maximal subgroups

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

subgroup	index
#13125486	2
#13117103	2
#13117093	2
#13117081	2
#13117067	2
#13116888	2
#13116878	2
#13116759	2
#13116753	2
#13116643	2
C₁₃×#3156	2
#33787	13