Subgroups — #16016302

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718855	C₂	1	1	yes	maximal, characteristic
2	#13123713	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116700	C₂	1	2	yes	maximal, complemented
4	#13116698	C₂	1	2	yes	maximal, complemented
5	C₄×#88069	C₂	1	2	yes	maximal, complemented
6	C₄×#88066	C₂	1	2	yes	maximal, complemented
7	C₂×#11718081	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116548	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116546	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116433	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2951	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
13	C₁₃×#520	—	2	1	no	—
14	C₂×#88158	C₂²	1	1	yes	characteristic
15	#11719228	—	2	1	no	—
16	C₁₃×#440	—	2	2	no	—
17	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
18	C₄×C₁₃×D₈	C₂²	1	2	yes	—
19	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₁₃×#417	—	2	1	no	—
21	C₁₃×#405	C₂²	1	1	yes	characteristic
22	#11718855	C₄	1	8	yes	complemented
23	C₂×#88069	C₂²	1	1	yes	characteristic
24	C₂×#88068	C₂²	1	1	yes	characteristic
25	C₂×#88067	C₂²	1	1	yes	characteristic
26	C₂×#88066	C₂²	1	1	yes	characteristic
27	#11718794	—	2	1	no	—
28	C₂×#88058	C₂²	1	1	yes	characteristic
29	C₂×#88052	C₂²	1	1	yes	characteristic
30	C₂²×#6090	C₂²	1	1	yes	characteristic
31	#11718371	—	2	2	no	—
32	C₄×Dic₅₂	C₂²	1	2	yes	—
33	C₄×D₁₀₄	C₂²	1	2	yes	—
34	#11718156	C₂²	1	2	yes	—
35	#11718154	C₂²	1	2	yes	—
36	#11718084	C₂²	1	2	yes	—
37	#11718083	C₂²	1	2	yes	—
38	#11718081	C₂²	1	2	yes	—
39	#11718081	C₂²	1	2	yes	—
40	#11718053	C₂²	1	2	yes	—
41	C₄×#6090	C₂²	1	2	yes	—
42	C₂×C₁₃×#41	—	2	1	no	—
43	C₂×#6127	—	2	1	no	—
44	C₁₃×#135	—	2	2	no	—
45	C₁₃×#135	C₂×C₄	1	4	yes	complemented
46	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×SD₁₆	—	2	1	no	—
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₄×D₄×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	4	2	no	—
54	C₁₃×#117	—	4	1	no	—
55	C₁₃×#117	D₄	1	1	yes	characteristic
56	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
58	C₂×C₄⋊Dic₁₃	—	2	1	no	—
59	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
60	#88158	—	2	2	no	—
61	#88158	C₂×C₄	1	4	yes	complemented
62	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
63	C₂×#6095	—	2	1	no	—
64	#88146	—	4	1	no	—
65	C₄⋊C₄×D₁₃	—	4	1	no	—
66	#88135	—	4	2	no	—
67	C₄×D₅₂	—	2	2	no	—
68	C₄×Dic₂₆	—	2	2	no	—
69	C₁₃×#107	—	2	1	no	—
70	C₁₃×#107	C₂³	1	2	yes	—
71	C₁₃×#103	—	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	#88069	C₂×C₄	1	4	yes	complemented
76	#88068	C₂×C₄	1	4	yes	complemented
77	#88067	C₂×C₄	1	4	yes	complemented
78	#88066	C₂×C₄	1	4	yes	complemented
79	#88061	—	2	2	no	—
80	#88058	—	2	1	no	—
81	#88058	C₂³	1	2	yes	—
82	#88056	—	2	2	no	—
83	#88052	—	2	2	no	—
84	C₂×#6090	C₂×C₄	1	4	yes	complemented
85	C₂×#6090	C₂³	1	2	yes	—
86	C₂×#6090	C₂³	1	1	yes	characteristic
87	C₂×#6090	C₂³	1	1	yes	characteristic
88	#33459	—	13	1	no	maximal, Sylow
89	C₁₃×#41	—	2	4	no	—
90	C₁₃×#41	—	4	2	no	—
91	C₂×Q₈×C₁₃	—	2	1	no	—
92	C₂×D₄×C₁₃	—	2	1	no	—
93	C₂×D₄×C₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	—	4	1	no	—
95	C₂²×C₄×C₁₃	—	4	1	no	—
96	C₂²×C₄×C₁₃	—	4	2	no	—
97	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₂²⋊D₁₃	—	4	1	no	—
99	C₂²×Dic₁₃	—	4	1	no	—
100	#6127	—	2	4	no	—
101	#6127	—	4	2	no	—
102	C₂×D₅₂	—	2	1	no	—
103	C₂×C₄×D₁₃	—	4	1	no	—
104	C₂×C₄×D₁₃	—	4	2	no	—
105	C₂×Dic₂₆	—	2	1	no	—
106	C₁₃×Q₁₆	—	2	1	no	—
107	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
108	C₁₃×SD₁₆	—	2	4	no	—
109	C₁₃×D₈	—	2	1	no	—
110	C₁₃×D₈	C₂²×C₄	1	2	yes	—
111	C₁₃×M₁₆	—	2	2	no	—
112	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₁₃	#41	1	2	yes	—
116	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
117	C₁₃×C₄⋊C₄	—	2	2	no	—
118	C₁₃×C₄⋊C₄	—	4	1	no	—
119	C₁₃×C₄⋊C₄	—	4	2	no	—
120	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
121	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
122	C₁₃×C₂²⋊C₄	—	4	2	no	—
123	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
124	C₄²×C₁₃	—	4	2	no	—
125	C₄²×C₁₃	C₂×D₄	1	2	yes	—
126	#6103	—	4	2	no	—
127	C₄⋊Dic₁₃	—	2	2	no	—
128	C₄⋊Dic₁₃	—	4	1	no	—
129	#6101	—	4	2	no	—
130	C₄×Dic₁₃	—	4	2	no	—
131	Dic₅₂	—	2	1	no	—
132	Dic₅₂	C₂²×C₄	1	2	yes	—
133	D₁₀₄	—	2	1	no	—
134	D₁₀₄	C₂²×C₄	1	2	yes	—
135	#6095	—	2	4	no	—
136	#6090	—	2	2	no	—
137	#6090	C₂²×C₄	1	2	yes	—
138	#6090	C₂²×C₄	1	2	yes	—
139	C₂×#508	—	13	1	no	—
140	#2951	—	13	1	no	—
141	#2951	D₁₃	1	1	yes	characteristic, complemented
142	#2175	—	13	2	no	—
143	#2175	—	13	2	no	—
144	C₄×Q₃₂	—	13	2	no	—
145	C₄×D₁₆	—	13	2	no	—
146	C₂×#366	—	13	1	no	—
147	#2139	—	13	1	no	—
148	#2139	—	13	1	no	—
149	#2108	—	13	1	no	—
150	C₂³×C₁₃	—	4	1	no	—
151	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂²×D₁₃	—	4	1	no	—
153	Q₈×C₁₃	—	2	2	no	—
154	Q₈×C₁₃	—	4	1	no	—
155	D₄×C₁₃	—	2	2	no	—
156	D₄×C₁₃	—	4	4	no	—
157	D₄×C₁₃	—	4	1	no	—
158	C₂×C₄×C₁₃	—	4	1	no	—
159	C₂×C₄×C₁₃	—	4	4	no	—
160	C₂×C₄×C₁₃	—	4	2	no	—
161	C₂×C₄×C₁₃	—	16	2	no	—
162	C₂×C₄×C₁₃	—	16	1	no	—
163	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
164	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
165	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
166	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₁₃	#135	1	2	yes	—
168	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
169	C₂²⋊D₁₃	—	4	4	no	—
170	C₂×Dic₁₃	—	4	1	no	—
171	C₂×Dic₁₃	—	4	2	no	—
172	C₂×Dic₁₃	—	16	1	no	—
173	D₅₂	—	2	2	no	—
174	D₅₂	—	4	1	no	—
175	C₄×D₁₃	—	4	4	no	—
176	C₄×D₁₃	—	16	2	no	—
177	Dic₂₆	—	2	2	no	—
178	Dic₂₆	—	4	1	no	—
179	C₈×C₁₃	—	2	2	no	—
180	C₈×C₁₃	C₄×D₄	1	2	yes	—
181	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
182	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
183	C₂×#135	—	13	1	no	—
184	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
185	#520	—	2	1	no	—
186	#520	—	26	1	no	—
187	#508	—	13	8	no	—
188	C₂×Q₃₂	—	13	1	no	—
189	C₂×SD₃₂	—	13	1	no	—
190	C₂×SD₃₂	—	13	1	no	—
191	C₂×D₁₆	—	13	1	no	—
192	C₂×M₃₂	—	13	1	no	—
193	C₂²×C₁₆	—	13	1	no	—
194	#440	—	2	2	no	—
195	#440	—	26	2	no	—
196	C₄×Q₁₆	—	13	2	no	—
197	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
198	C₄×D₈	—	13	2	no	—
199	C₄×D₈	C₂×D₁₃	1	2	yes	—
200	C₂×#107	—	13	1	no	—
201	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
202	#417	—	2	1	no	—
203	#417	—	26	1	no	—
204	#405	C₂×D₁₃	1	1	yes	characteristic
205	#366	—	13	2	no	—
206	#366	—	13	2	no	—
207	#358	—	13	2	no	—
208	#358	—	13	2	no	—
209	#357	—	13	2	no	—
210	#357	—	13	2	no	—
211	#346	—	13	2	no	—
212	C₄×C₁₆	—	13	2	no	—
213	C₂²×C₁₃	—	2	1	no	—
214	C₂²×C₁₃	—	2	1	no	—
215	C₂²×C₁₃	—	4	2	no	—
216	C₂²×C₁₃	—	8	1	no	—
217	C₂²×C₁₃	—	8	1	no	—
218	C₂²×C₁₃	C₄×D₈	1	2	yes	—
219	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
220	C₂×D₁₃	—	4	2	no	—
221	C₂×D₁₃	—	8	1	no	—
222	C₂×D₁₃	—	8	1	no	—
223	C₄×C₁₃	—	2	2	no	—
224	C₄×C₁₃	—	2	2	no	—
225	C₄×C₁₃	—	4	2	no	—
226	C₄×C₁₃	—	8	2	no	—
227	C₄×C₁₃	C₄×D₈	1	2	yes	—
228	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
229	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
230	Dic₁₃	—	4	2	no	—
231	Dic₁₃	—	8	2	no	—
232	C₂×#41	—	2	1	no	—
233	C₂×#41	—	26	1	no	—
234	#135	—	2	2	no	—
235	#135	—	13	4	no	—
236	#135	—	26	2	no	—
237	#135	C₄×D₁₃	1	4	yes	complemented
238	C₂×Q₁₆	—	13	1	no	—
239	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×SD₁₆	—	2	1	no	—
241	C₂×SD₁₆	—	26	1	no	—
242	C₂×D₈	—	13	1	no	—
243	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
246	C₄×Q₈	—	2	2	no	—
247	C₄×Q₈	—	26	2	no	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	—	4	2	no	—
250	C₄×D₄	—	26	2	no	—
251	C₄×D₄	—	52	2	no	—
252	#117	—	4	1	no	—
253	#117	—	52	1	no	—
254	#117	C₂²⋊D₁₃	1	1	yes	characteristic
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	4	1	no	—
257	C₂×C₄⋊C₄	—	26	1	no	—
258	C₂×C₄⋊C₄	—	52	1	no	—
259	Q₃₂	—	13	4	no	—
260	SD₃₂	—	13	4	no	—
261	SD₃₂	—	13	4	no	—
262	D₁₆	—	13	4	no	—
263	M₃₂	—	26	2	no	—
264	C₂×C₁₆	—	13	4	no	—
265	C₂×C₁₆	—	13	1	no	—
266	C₂×C₁₆	—	13	1	no	—
267	C₂×C₁₆	—	13	2	no	—
268	#107	—	2	1	no	—
269	#107	—	13	2	no	—
270	#107	—	26	1	no	—
271	#107	C₂²×D₁₃	1	2	yes	—
272	#103	—	2	2	no	—
273	#103	—	26	2	no	—
274	#102	—	2	2	no	—
275	#102	—	26	2	no	—
276	#97	C₂²⋊D₁₃	1	2	yes	—
277	C₄×C₈	C₂²×D₁₃	1	2	yes	—
278	C₂×C₁₃	—	2	2	no	—
279	C₂×C₁₃	—	8	2	no	—
280	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
281	C₂×C₁₃	#3415	1	1	yes	characteristic
282	C₂×C₁₃	#3417	1	1	yes	characteristic
283	D₁₃	—	8	2	no	—
284	#41	—	2	4	no	—
285	#41	—	4	2	no	—
286	#41	—	26	4	no	—
287	#41	—	52	2	no	—
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	—	26	1	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	—	4	1	no	—
292	C₂×D₄	—	26	1	no	—
293	C₂×D₄	—	52	1	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	52	1	no	—
298	C₂²×C₄	—	52	1	no	—
299	C₂²×C₄	—	52	2	no	—
300	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
301	Q₁₆	—	2	1	no	—
302	Q₁₆	—	13	2	no	—
303	Q₁₆	—	26	1	no	—
304	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
305	SD₁₆	—	2	4	no	—
306	SD₁₆	—	26	4	no	—
307	D₈	—	2	1	no	—
308	D₈	—	13	2	no	—
309	D₈	—	26	1	no	—
310	D₈	C₂×C₄×D₁₃	1	2	yes	—
311	M₁₆	—	2	2	no	—
312	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
313	C₂×C₈	#6127	1	2	yes	—
314	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
315	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
316	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	4	2	no	—
320	C₄⋊C₄	—	26	2	no	—
321	C₄⋊C₄	—	52	1	no	—
322	C₄⋊C₄	—	52	2	no	—
323	C₄⋊C₄	#6104	1	1	yes	characteristic
324	C₄⋊C₄	#6104	1	1	yes	characteristic
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	52	2	no	—
327	C₂²⋊C₄	#6104	1	2	yes	—
328	C₄²	—	4	2	no	—
329	C₄²	—	52	2	no	—
330	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
331	C₁₆	—	13	2	no	—
332	C₁₆	—	13	2	no	—
333	C₁₆	—	26	2	no	—
334	C₁₃	#33459	1	1	yes	characteristic, complemented, Sylow
335	C₂³	—	4	1	no	—
336	C₂³	—	52	1	no	—
337	C₂³	C₂×#6104	1	1	yes	characteristic
338	Q₈	—	2	2	no	—
339	Q₈	—	4	1	no	—
340	Q₈	—	26	2	no	—
341	Q₈	—	52	1	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	4	no	—
344	D₄	—	4	1	no	—
345	D₄	—	26	2	no	—
346	D₄	—	52	4	no	—
347	D₄	—	52	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	16	2	no	—
352	C₂×C₄	—	16	1	no	—
353	C₂×C₄	—	52	1	no	—
354	C₂×C₄	—	52	4	no	—
355	C₂×C₄	—	52	2	no	—
356	C₂×C₄	—	208	2	no	—
357	C₂×C₄	—	208	1	no	—
358	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
359	C₂×C₄	C₂×#6104	1	2	yes	—
360	C₂×C₄	C₂×#6104	1	2	yes	—
361	C₂×C₄	C₂×#6104	1	1	yes	characteristic
362	C₂×C₄	#88204	1	2	yes	—
363	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
364	C₈	—	2	2	no	—
365	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
366	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
367	C₈	C₂×#6127	1	1	yes	characteristic
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	8	1	no	—
372	C₂²	—	8	1	no	—
373	C₂²	—	52	2	no	—
374	C₂²	—	104	1	no	—
375	C₂²	—	104	1	no	—
376	C₂²	C₄×#6104	1	2	yes	—
377	C₂²	C₂²×#6104	1	1	yes	characteristic
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	4	2	no	—
381	C₄	—	8	2	no	—
382	C₄	—	52	2	no	—
383	C₄	—	104	2	no	—
384	C₄	C₄×#6104	1	2	yes	—
385	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
386	C₄	C₂×#88204	1	1	yes	characteristic
387	C₂	—	2	2	no	—
388	C₂	—	8	2	no	—
389	C₂	—	104	2	no	—
390	C₂	C₂×C₄×#6104	1	1	yes	characteristic
391	C₂	#13127975	1	1	yes	characteristic
392	C₂	#13127987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2951	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718855	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2951	D₁₃
hypocenter	C₁₃	#33459
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2951

Complements

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

N	H ≅ G/N	type
2. #13123713	C₂	semidirect
3. #13116700	C₂	semidirect
4. #13116698	C₂	semidirect
5. C₄×#88069	C₂	semidirect
6. C₄×#88066	C₂	semidirect
7. C₂×#11718081	C₂	semidirect
8. #13116548	C₂	semidirect
9. #13116546	C₂	semidirect
10. #13116433	C₂	semidirect
11. C₁₃×#2951	C₂	semidirect
22. #11718855	C₄	semidirect
45. C₁₃×#135	C₂×C₄	semidirect
61. #88158	C₂×C₄	semidirect
75. #88069	C₂×C₄	semidirect
76. #88068	C₂×C₄	semidirect
77. #88067	C₂×C₄	semidirect
78. #88066	C₂×C₄	semidirect
84. C₂×#6090	C₂×C₄	semidirect
141. #2951	D₁₃	semidirect
237. #135	C₄×D₁₃	semidirect
334. C₁₃	#33459	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718855	2
#13123713	2
#13116700	2
#13116698	2
C₄×#88069	2
C₄×#88066	2
C₂×#11718081	2
#13116548	2
#13116546	2
#13116433	2
C₁₃×#2951	2
#33459	13