Subgroups — #16016261

Order 3328.

696 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	C₂×#11718808	C₂	1	1	yes	maximal, characteristic
2	#13123713	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116710	C₂	1	2	yes	maximal, complemented
4	#13116700	C₂	1	2	yes	maximal, complemented
5	C₄×#88067	C₂	1	2	yes	maximal, complemented
6	C₄×#88066	C₂	1	2	yes	maximal, complemented
7	#13116625	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116541	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11718154	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116433	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
13	C₁₃×#518	—	2	1	no	—
14	C₂×#88158	C₂²	1	1	yes	characteristic
15	#11719228	—	2	1	no	—
16	C₁₃×#442	—	2	2	no	—
17	C₄×C₁₃×D₈	C₂²	1	2	yes	—
18	C₄×C₁₃×D₈	C₂²	1	2	yes	—
19	C₁₃×#427	C₂²	1	1	yes	characteristic
20	C₁₃×#418	—	2	1	no	—
21	C₁₃×#405	C₂²	1	1	yes	characteristic
22	C₂×#88067	C₂²	1	2	yes	—
23	#11718808	C₄	1	8	yes	complemented
24	C₂×#88066	C₂²	1	2	yes	—
25	#11718794	—	2	1	no	—
26	C₂×#88058	C₂²	1	1	yes	characteristic
27	C₂×#88052	C₂²	1	1	yes	characteristic
28	C₂²×#6090	C₂²	1	1	yes	characteristic
29	#11718371	—	2	2	no	—
30	C₄×Dic₅₂	C₂²	1	2	yes	complemented
31	C₄×D₁₀₄	C₂²	1	2	yes	complemented
32	#11718154	C₂²	1	2	yes	—
33	#11718154	C₂²	1	2	yes	—
34	#11718084	C₂²	1	1	yes	characteristic, complemented
35	#11718084	C₂²	1	1	yes	characteristic, complemented
36	#11718083	C₂²	1	1	yes	characteristic, complemented
37	#11718083	C₂²	1	1	yes	characteristic, complemented
38	#11718082	C₂²	1	2	yes	complemented
39	#11718081	C₂²	1	2	yes	complemented
40	#11718053	C₂²	1	2	yes	complemented
41	C₄×#6090	C₂²	1	2	yes	complemented
42	C₂²×D₄×C₁₃	—	2	1	no	—
43	C₂×#6127	—	2	1	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	—	2	2	no	—
46	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₃×D₈	C₂³	1	2	yes	—
48	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×D₄×C₁₃	—	2	2	no	—
51	C₄×D₄×C₁₃	—	2	2	no	—
52	C₄×D₄×C₁₃	—	4	1	no	—
53	C₄×D₄×C₁₃	—	4	1	no	—
54	C₁₃×#117	—	2	1	no	—
55	C₁₃×#117	D₄	1	1	yes	characteristic
56	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
57	C₂×C₄⋊Dic₁₃	—	2	1	no	—
58	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
59	#88158	—	2	2	no	—
60	#88158	C₂×C₄	1	4	yes	complemented
61	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
62	C₂×#6095	—	2	1	no	—
63	#88146	—	4	1	no	—
64	C₄⋊C₄×D₁₃	—	4	1	no	—
65	#88135	—	4	2	no	—
66	C₄×D₅₂	—	2	2	no	—
67	C₄×Dic₂₆	—	2	2	no	—
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#106	—	2	1	no	—
71	C₁₃×#102	—	2	2	no	—
72	C₁₃×#102	—	2	2	no	—
73	C₁₃×#97	D₄	1	2	yes	—
74	C₄×C₈×C₁₃	C₂³	1	2	yes	—
75	#88067	C₂×C₄	1	8	yes	complemented
76	#88066	C₂×C₄	1	8	yes	complemented
77	#88061	—	2	2	no	—
78	#88058	—	2	1	no	—
79	#88058	C₂³	1	2	yes	—
80	#88056	—	2	2	no	—
81	#88052	C₂×C₄	1	4	yes	complemented
82	C₂×#6090	—	2	2	no	—
83	C₂×#6090	C₂³	1	1	yes	characteristic
84	C₂×#6090	C₂³	1	1	yes	characteristic
85	C₂×#6090	C₂³	1	2	yes	—
86	#33454	—	13	1	no	maximal, Sylow
87	C₂⁴×C₁₃	—	4	1	no	—
88	C₂×D₄×C₁₃	—	2	4	no	—
89	C₂×D₄×C₁₃	—	2	2	no	—
90	C₂×D₄×C₁₃	—	4	1	no	—
91	C₂×D₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	—	4	2	no	—
93	C₂²×C₄×C₁₃	—	4	2	no	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₂²⋊D₁₃	—	4	1	no	—
96	C₂²×Dic₁₃	—	4	1	no	—
97	#6127	—	2	4	no	—
98	#6127	—	4	2	no	—
99	C₂×D₅₂	—	2	1	no	—
100	C₂×C₄×D₁₃	—	4	1	no	—
101	C₂×C₄×D₁₃	—	4	2	no	—
102	C₂×Dic₂₆	—	2	1	no	—
103	C₁₃×D₈	—	2	4	no	—
104	C₁₃×D₈	—	2	2	no	—
105	C₁₃×D₈	C₂²×C₄	1	4	yes	—
106	C₁₃×M₁₆	—	2	2	no	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	2	yes	—
111	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	—	2	1	no	—
113	C₁₃×C₄⋊C₄	—	2	1	no	—
114	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
115	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
116	C₁₃×C₂²⋊C₄	—	4	1	no	—
117	C₁₃×C₂²⋊C₄	—	4	2	no	—
118	C₁₃×C₂²⋊C₄	—	4	2	no	—
119	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
120	C₄²×C₁₃	—	4	1	no	—
121	C₄²×C₁₃	C₂×D₄	1	2	yes	—
122	#6103	—	4	2	no	—
123	C₄⋊Dic₁₃	—	2	2	no	—
124	C₄⋊Dic₁₃	—	4	1	no	—
125	#6101	—	4	2	no	—
126	C₄×Dic₁₃	—	4	2	no	—
127	Dic₅₂	—	2	1	no	—
128	Dic₅₂	C₂²×C₄	1	2	yes	—
129	D₁₀₄	—	2	1	no	—
130	D₁₀₄	C₂²×C₄	1	2	yes	—
131	#6095	—	2	4	no	—
132	#6090	—	2	1	no	—
133	#6090	—	2	1	no	—
134	#6090	C₂²×C₄	1	4	yes	—
135	C₂×#509	—	13	1	no	—
136	#2951	—	13	1	no	—
137	#2948	D₁₃	1	1	yes	characteristic, complemented
138	#2177	—	13	2	no	—
139	#2175	—	13	2	no	—
140	C₄×SD₃₂	—	13	2	no	—
141	C₄×D₁₆	—	13	2	no	—
142	#2159	—	13	1	no	—
143	#2138	—	13	1	no	—
144	C₂×#357	—	13	1	no	—
145	#2108	—	13	1	no	—
146	C₂³×C₁₃	—	4	2	no	—
147	C₂³×C₁₃	—	4	4	no	—
148	C₂³×C₁₃	—	16	1	no	—
149	C₂³×C₁₃	—	16	1	no	—
150	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂²×D₁₃	—	4	1	no	—
152	D₄×C₁₃	—	2	4	no	—
153	D₄×C₁₃	—	4	2	no	—
154	D₄×C₁₃	—	4	4	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	1	no	—
157	C₂×C₄×C₁₃	—	16	2	no	—
158	C₂×C₄×C₁₃	—	16	2	no	—
159	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
160	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
162	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#135	1	2	yes	—
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₂²⋊D₁₃	—	4	4	no	—
166	C₂×Dic₁₃	—	4	1	no	—
167	C₂×Dic₁₃	—	4	2	no	—
168	C₂×Dic₁₃	—	16	1	no	—
169	D₅₂	—	2	2	no	—
170	D₅₂	—	4	1	no	—
171	C₄×D₁₃	—	4	4	no	—
172	C₄×D₁₃	—	16	2	no	—
173	Dic₂₆	—	2	2	no	—
174	Dic₂₆	—	4	1	no	—
175	C₈×C₁₃	—	2	2	no	—
176	C₈×C₁₃	C₄×D₄	1	2	yes	—
177	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
178	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
179	C₂×#135	—	13	1	no	—
180	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
181	#520	—	26	1	no	—
182	#518	—	2	1	no	—
183	#509	—	13	8	no	—
184	C₂×SD₃₂	—	13	2	no	—
185	C₂×D₁₆	—	13	2	no	—
186	C₂×M₃₂	—	13	1	no	—
187	C₂²×C₁₆	—	13	1	no	—
188	#442	—	2	2	no	—
189	#440	—	26	2	no	—
190	C₄×Q₁₆	—	13	2	no	—
191	C₄×D₈	—	13	2	no	—
192	C₄×D₈	C₂×D₁₃	1	2	yes	—
193	C₄×D₈	C₂×D₁₃	1	2	yes	—
194	#427	C₂×D₁₃	1	1	yes	characteristic
195	C₂×#107	—	13	1	no	—
196	#418	—	2	1	no	—
197	#417	—	26	1	no	—
198	#405	C₂×D₁₃	1	1	yes	characteristic
199	#367	—	13	2	no	—
200	#366	—	13	2	no	—
201	#358	—	13	1	no	—
202	#358	—	13	1	no	—
203	#357	—	13	2	no	—
204	#357	—	13	2	no	—
205	#357	—	13	1	no	—
206	#357	—	13	1	no	—
207	#346	—	13	2	no	—
208	C₄×C₁₆	—	13	2	no	—
209	C₂²×C₁₃	—	2	1	no	—
210	C₂²×C₁₃	—	2	1	no	—
211	C₂²×C₁₃	—	4	4	no	—
212	C₂²×C₁₃	—	8	2	no	—
213	C₂²×C₁₃	—	8	2	no	—
214	C₂²×C₁₃	—	16	4	no	—
215	C₂²×C₁₃	C₄×D₈	1	2	yes	—
216	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
217	C₂×D₁₃	—	4	2	no	—
218	C₂×D₁₃	—	8	1	no	—
219	C₂×D₁₃	—	8	1	no	—
220	C₄×C₁₃	—	2	2	no	—
221	C₄×C₁₃	—	2	2	no	—
222	C₄×C₁₃	—	8	1	no	—
223	C₄×C₁₃	—	8	1	no	—
224	C₄×C₁₃	C₄×D₈	1	2	yes	—
225	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
227	Dic₁₃	—	4	2	no	—
228	Dic₁₃	—	8	2	no	—
229	C₂×#41	—	26	1	no	—
230	C₂²×D₄	—	2	1	no	—
231	#135	—	13	4	no	—
232	#135	—	26	2	no	—
233	C₂×Q₁₆	—	13	1	no	—
234	C₂×SD₁₆	—	26	1	no	—
235	C₂×D₈	—	2	2	no	—
236	C₂×D₈	—	2	1	no	—
237	C₂×D₈	—	13	1	no	—
238	C₂×D₈	C₄×D₁₃	1	4	yes	complemented
239	C₂×D₈	C₂²×D₁₃	1	2	yes	—
240	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
241	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₄×Q₈	—	26	2	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	2	2	no	—
245	C₄×D₄	—	4	1	no	—
246	C₄×D₄	—	4	1	no	—
247	C₄×D₄	—	26	2	no	—
248	C₄×D₄	—	52	2	no	—
249	#117	—	2	1	no	—
250	#117	—	52	1	no	—
251	#117	C₂²⋊D₁₃	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	26	1	no	—
253	C₂×C₄⋊C₄	—	52	1	no	—
254	C₂×C₂²⋊C₄	—	4	2	no	—
255	SD₃₂	—	13	8	no	—
256	D₁₆	—	13	8	no	—
257	M₃₂	—	13	4	no	—
258	C₂×C₁₆	—	13	2	no	—
259	C₂×C₁₆	—	13	1	no	—
260	C₂×C₁₆	—	13	1	no	—
261	C₂×C₁₆	—	26	2	no	—
262	#107	—	13	2	no	—
263	#107	—	26	1	no	—
264	#107	C₂²×D₁₃	1	1	yes	characteristic
265	#107	C₂²×D₁₃	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#103	—	26	2	no	—
268	#102	—	2	2	no	—
269	#102	—	2	2	no	—
270	#102	—	26	2	no	—
271	#97	C₂²⋊D₁₃	1	2	yes	—
272	C₄×C₈	C₂²×D₁₃	1	2	yes	—
273	C₂×C₁₃	—	2	2	no	—
274	C₂×C₁₃	—	8	4	no	—
275	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
276	C₂×C₁₃	#3415	1	1	yes	characteristic
277	C₂×C₁₃	#3416	1	1	yes	characteristic
278	D₁₃	—	8	2	no	—
279	C₂⁴	—	4	1	no	—
280	#41	—	26	4	no	—
281	#41	—	52	2	no	—
282	C₂×Q₈	—	26	1	no	—
283	C₂×D₄	—	2	4	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	4	1	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	26	1	no	—
288	C₂×D₄	—	52	1	no	—
289	C₂²×C₄	—	4	2	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	52	1	no	—
292	C₂²×C₄	—	52	1	no	—
293	C₂²×C₄	—	52	2	no	—
294	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
295	Q₁₆	—	13	2	no	—
296	Q₁₆	—	26	1	no	—
297	SD₁₆	—	26	4	no	—
298	D₈	—	2	4	no	—
299	D₈	—	2	2	no	—
300	D₈	—	13	2	no	—
301	D₈	—	26	1	no	—
302	D₈	C₂×C₄×D₁₃	1	4	yes	—
303	M₁₆	—	2	2	no	—
304	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
305	C₂×C₈	#6127	1	2	yes	—
306	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
307	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	26	2	no	—
312	C₄⋊C₄	—	52	1	no	—
313	C₄⋊C₄	—	52	2	no	—
314	C₄⋊C₄	#6104	1	1	yes	characteristic
315	C₄⋊C₄	#6104	1	1	yes	characteristic
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	52	2	no	—
320	C₂²⋊C₄	#6104	1	2	yes	—
321	C₄²	—	4	1	no	—
322	C₄²	—	52	2	no	—
323	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
324	C₁₆	—	13	4	no	—
325	C₁₆	—	26	1	no	—
326	C₁₆	—	26	1	no	—
327	C₁₃	#33454	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	4	2	no	—
329	C₂³	—	4	4	no	—
330	C₂³	—	16	1	no	—
331	C₂³	—	16	1	no	—
332	C₂³	—	52	1	no	—
333	C₂³	C₂×#6104	1	1	yes	characteristic
334	Q₈	—	26	2	no	—
335	Q₈	—	52	1	no	—
336	D₄	—	2	4	no	—
337	D₄	—	4	4	no	—
338	D₄	—	4	2	no	—
339	D₄	—	26	2	no	—
340	D₄	—	52	4	no	—
341	D₄	—	52	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	16	2	no	—
345	C₂×C₄	—	16	2	no	—
346	C₂×C₄	—	52	1	no	—
347	C₂×C₄	—	52	4	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	208	2	no	—
350	C₂×C₄	—	208	1	no	—
351	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
352	C₂×C₄	C₂×#6104	1	2	yes	—
353	C₂×C₄	C₂×#6104	1	2	yes	—
354	C₂×C₄	C₂×#6104	1	1	yes	characteristic
355	C₂×C₄	#88204	1	2	yes	—
356	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
359	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
360	C₈	C₂×#6127	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	16	4	no	—
367	C₂²	—	52	2	no	—
368	C₂²	—	104	1	no	—
369	C₂²	—	104	1	no	—
370	C₂²	C₄×#6104	1	2	yes	—
371	C₂²	C₂²×#6104	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	8	1	no	—
375	C₄	—	8	1	no	—
376	C₄	—	52	2	no	—
377	C₄	—	104	2	no	—
378	C₄	C₄×#6104	1	2	yes	—
379	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
380	C₄	C₂×#88204	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	8	4	no	—
383	C₂	—	104	2	no	—
384	C₂	C₂×C₄×#6104	1	1	yes	characteristic
385	C₂	#13127975	1	1	yes	characteristic
386	C₂	#13127983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2948	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718808	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2948	D₁₃
hypocenter	C₁₃	#33454
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
2. #13123713	C₂	semidirect
3. #13116710	C₂	semidirect
4. #13116700	C₂	semidirect
5. C₄×#88067	C₂	semidirect
6. C₄×#88066	C₂	semidirect
7. #13116625	C₂	semidirect
8. #13116541	C₂	semidirect
9. C₂×#11718154	C₂	semidirect
10. #13116433	C₂	semidirect
11. C₁₃×#2948	C₂	semidirect
23. #11718808	C₄	semidirect
30. C₄×Dic₅₂	C₂²	semidirect
31. C₄×D₁₀₄	C₂²	semidirect
34. #11718084	C₂²	semidirect
35. #11718084	C₂²	semidirect
36. #11718083	C₂²	semidirect
37. #11718083	C₂²	semidirect
38. #11718082	C₂²	semidirect
39. #11718081	C₂²	semidirect
40. #11718053	C₂²	semidirect
41. C₄×#6090	C₂²	semidirect
46. C₂×C₁₃×D₈	C₂×C₄	semidirect
60. #88158	C₂×C₄	semidirect
75. #88067	C₂×C₄	semidirect
76. #88066	C₂×C₄	semidirect
81. #88052	C₂×C₄	semidirect
137. #2948	D₁₃	semidirect
238. C₂×D₈	C₄×D₁₃	semidirect
327. C₁₃	#33454	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718808	2
#13123713	2
#13116710	2
#13116700	2
C₄×#88067	2
C₄×#88066	2
#13116625	2
#13116541	2
C₂×#11718154	2
#13116433	2
C₁₃×#2948	2
#33454	13