Subgroups — #16017792

Order 3328.

568 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	#13124502	C₂	1	1	yes	maximal, characteristic
2	#13123433	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117042	C₂	1	2	yes	maximal, complemented
4	#13117032	C₂	1	2	yes	maximal, complemented
5	#13117026	C₂	1	2	yes	maximal, complemented
6	#13117020	C₂	1	2	yes	maximal, complemented
7	#13116625	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11718097	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116548	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116449	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	#11719538	—	2	1	no	—
14	C₂×#88155	C₂²	1	1	yes	characteristic
15	#11719228	—	2	1	no	—
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	#11718893	—	2	1	no	—
19	#11718849	C₂²	1	2	yes	—
20	#11718835	—	2	2	no	—
21	#11718818	C₂²	1	2	yes	—
22	C₂×#88058	C₂²	1	1	yes	characteristic
23	C₂×#88058	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	#11718656	—	2	1	no	—
26	#11718541	C₂²	1	2	yes	—
27	D₁₃×#107	C₂²	1	2	yes	—
28	#11718522	—	2	2	no	—
29	C₁₃×#358	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	1	yes	characteristic
31	C₁₃×#357	C₂²	1	1	yes	characteristic
32	C₁₃×#357	C₂²	1	1	yes	characteristic
33	#11718156	C₂²	1	2	yes	—
34	#11718154	C₂²	1	2	yes	—
35	#11718099	C₂²	1	2	yes	—
36	#11718099	C₂²	1	2	yes	—
37	#11718097	C₂²	1	2	yes	—
38	#11718097	C₂²	1	2	yes	—
39	#11718082	C₂²	1	2	yes	—
40	#11718081	C₂²	1	2	yes	—
41	C₂×C₁₃×#41	—	2	1	no	—
42	C₂×#6127	D₄	1	1	yes	characteristic
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	#88200	—	2	2	no	—
51	#88193	—	4	2	no	—
52	#88192	—	2	2	no	—
53	#88184	—	4	1	no	—
54	C₂×C₄⋊Dic₁₃	—	2	1	no	—
55	C₂×C₄⋊Dic₁₃	—	2	1	no	—
56	#88178	—	4	1	no	—
57	C₂×#6099	D₄	1	1	yes	characteristic
58	#88155	—	2	2	no	—
59	#88155	—	2	2	no	—
60	C₂×#6094	D₄	1	2	yes	—
61	C₂×C₈×D₁₃	C₂³	1	2	yes	—
62	#88146	—	2	2	no	—
63	C₄⋊C₄×D₁₃	—	2	2	no	—
64	#88135	—	4	2	no	—
65	C₄×D₅₂	—	4	1	no	—
66	C₄×Dic₂₆	—	4	1	no	—
67	C₂×C₁₃×C₁₆	—	2	2	no	—
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	1	yes	characteristic
71	C₁₃×#107	C₂³	1	1	yes	characteristic
72	C₁₃×#106	—	2	1	no	—
73	#88075	—	2	2	no	—
74	#88072	—	2	2	no	—
75	#88058	—	2	1	no	—
76	#88058	—	2	1	no	—
77	#88058	C₂³	1	2	yes	—
78	#88058	C₂³	1	2	yes	—
79	#88052	—	2	2	no	—
80	C₂×#6090	C₂³	1	2	yes	—
81	#88048	—	2	2	no	—
82	#88047	—	2	2	no	—
83	#33653	—	13	1	no	maximal, Sylow
84	C₁₃×#41	—	4	2	no	—
85	C₁₃×#41	—	4	2	no	—
86	C₂×Q₈×C₁₃	—	2	1	no	—
87	C₂×D₄×C₁₃	—	2	1	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₃	—	4	2	no	—
92	C₂×C₂²⋊D₁₃	D₈	1	2	yes	—
93	C₂²×Dic₁₃	—	4	1	no	—
94	C₂²×Dic₁₃	—	4	1	no	—
95	#6127	—	2	2	no	—
96	#6127	—	2	2	no	—
97	C₂×D₅₂	D₈	1	1	yes	characteristic
98	C₂×C₄×D₁₃	—	4	2	no	—
99	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
100	C₂×Dic₂₆	D₈	1	1	yes	characteristic
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	—	4	2	no	—
115	#6103	—	4	2	no	—
116	#6103	—	4	2	no	—
117	C₄⋊Dic₁₃	—	2	2	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	—	4	1	no	—
120	C₄⋊Dic₁₃	—	4	1	no	—
121	#6101	—	4	2	no	—
122	#6101	—	4	2	no	—
123	C₄×Dic₁₃	—	4	2	no	—
124	#6099	—	2	2	no	—
125	C₂×#1055	C₂×D₄	1	2	yes	—
126	#6094	—	2	4	no	—
127	C₈×D₁₃	—	2	2	no	—
128	C₈×D₁₃	—	2	2	no	—
129	C₁₃×C₁₆	—	2	1	no	—
130	C₁₃×C₁₆	—	2	1	no	—
131	#6090	—	2	2	no	—
132	#3061	—	13	1	no	—
133	#2913	—	13	1	no	—
134	#2237	—	13	2	no	—
135	#2235	—	13	2	no	—
136	#2234	—	13	2	no	—
137	#2233	—	13	2	no	—
138	#2159	—	13	1	no	—
139	C₂×#367	—	13	1	no	—
140	#2139	—	13	1	no	—
141	#2138	D₁₃	1	1	yes	characteristic, complemented
142	#2113	—	13	1	no	—
143	C₂³×C₁₃	—	4	1	no	—
144	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂²×D₁₃	C₂×D₈	1	2	yes	—
146	Q₈×C₁₃	—	4	1	no	—
147	Q₈×C₁₃	—	4	1	no	—
148	D₄×C₁₃	—	4	1	no	—
149	D₄×C₁₃	—	4	1	no	—
150	D₄×C₁₃	—	8	2	no	—
151	C₂×C₄×C₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	8	2	no	—
155	C₂×C₄×C₁₃	#123	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	#137	1	2	yes	—
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²⋊D₁₃	—	2	4	no	—
160	C₂²⋊D₁₃	—	16	2	no	—
161	C₂×Dic₁₃	—	4	2	no	—
162	C₂×Dic₁₃	—	4	2	no	—
163	C₂×Dic₁₃	—	16	1	no	—
164	C₂×Dic₁₃	—	16	1	no	—
165	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
166	D₅₂	—	2	2	no	—
167	C₄×D₁₃	—	2	2	no	—
168	C₄×D₁₃	—	2	2	no	—
169	C₄×D₁₃	—	16	2	no	—
170	Dic₂₆	—	2	2	no	—
171	C₈×C₁₃	#123	1	2	yes	—
172	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
173	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
174	#1055	—	2	2	no	—
175	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
176	C₂×#131	—	13	1	no	—
177	#537	—	26	1	no	—
178	#520	—	26	1	no	—
179	C₂×M₃₂	—	13	1	no	—
180	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
181	#468	—	26	2	no	—
182	#467	—	13	2	no	—
183	#466	—	13	2	no	—
184	#428	—	26	2	no	—
185	#428	—	26	1	no	—
186	#427	—	13	2	no	—
187	#427	C₂×D₁₃	1	1	yes	characteristic
188	C₂×#107	—	13	1	no	—
189	C₂×#107	—	13	1	no	—
190	C₂×#107	—	13	2	no	—
191	#417	—	26	1	no	—
192	#367	—	13	2	no	—
193	#367	—	13	2	no	—
194	#367	—	13	2	no	—
195	#366	—	13	2	no	—
196	#358	—	13	2	no	—
197	#358	C₂×D₁₃	1	1	yes	characteristic
198	#358	C₂×D₁₃	1	1	yes	characteristic
199	#357	—	13	2	no	—
200	#357	C₂×D₁₃	1	1	yes	characteristic
201	#357	C₂×D₁₃	1	1	yes	characteristic
202	C₂²⋊C₁₆	—	13	2	no	—
203	C₂²⋊C₁₆	—	13	2	no	—
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	—	2	1	no	—
206	C₂²×C₁₃	—	8	1	no	—
207	C₂²×C₁₃	—	8	1	no	—
208	C₂²×C₁₃	—	8	1	no	—
209	C₂²×C₁₃	#480	1	2	yes	—
210	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
211	C₂×D₁₃	—	2	2	no	—
212	C₂×D₁₃	—	2	2	no	—
213	C₂×D₁₃	—	2	2	no	—
214	C₄×C₁₃	—	8	1	no	—
215	C₄×C₁₃	—	8	1	no	—
216	C₄×C₁₃	—	8	1	no	—
217	C₄×C₁₃	#480	1	2	yes	—
218	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
219	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
220	Dic₁₃	—	2	2	no	—
221	Dic₁₃	—	8	2	no	—
222	Dic₁₃	—	8	2	no	—
223	C₂×#41	—	2	1	no	—
224	C₂×#41	—	13	1	no	—
225	#135	—	2	2	no	—
226	#135	—	2	2	no	—
227	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×SD₁₆	—	2	1	no	—
229	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
230	#131	—	26	2	no	—
231	#131	—	26	2	no	—
232	C₂×M₁₆	—	13	1	no	—
233	C₂×M₁₆	—	13	2	no	—
234	C₂²×C₈	—	13	2	no	—
235	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₂²⋊Q₈	—	26	2	no	—
237	C₂²⋊Q₈	—	52	1	no	—
238	#121	—	26	2	no	—
239	#121	—	52	1	no	—
240	#121	—	52	2	no	—
241	C₄×Q₈	—	52	1	no	—
242	C₄×D₄	—	52	1	no	—
243	C₄×D₄	—	52	2	no	—
244	#117	—	2	1	no	—
245	#117	—	26	2	no	—
246	C₂×C₄⋊C₄	—	26	2	no	—
247	C₂×C₄⋊C₄	—	26	1	no	—
248	C₂×C₄⋊C₄	—	26	1	no	—
249	M₃₂	—	26	2	no	—
250	C₂×C₁₆	—	2	2	no	—
251	C₂×C₁₆	—	13	2	no	—
252	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
254	#107	—	13	2	no	—
255	#107	—	13	2	no	—
256	#107	—	26	2	no	—
257	#107	—	26	1	no	—
258	#107	—	26	1	no	—
259	#107	C₂²×D₁₃	1	1	yes	characteristic
260	#107	C₂²×D₁₃	1	1	yes	characteristic
261	#106	—	2	1	no	—
262	#106	—	26	2	no	—
263	#103	—	26	2	no	—
264	#102	—	26	2	no	—
265	C₂×C₁₃	—	2	2	no	—
266	C₂×C₁₃	—	16	1	no	—
267	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
268	C₂×C₁₃	#3416	1	1	yes	characteristic
269	C₂×C₁₃	#3416	1	1	yes	characteristic
270	D₁₃	—	4	2	no	—
271	#41	—	4	2	no	—
272	#41	—	4	2	no	—
273	#41	—	26	2	no	—
274	#41	—	26	2	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	13	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	13	2	no	—
280	C₂×D₄	—	13	1	no	—
281	C₂×D₄	—	52	2	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	13	2	no	—
284	C₂²×C₄	—	52	2	no	—
285	C₂²×C₄	—	52	1	no	—
286	C₂²×C₄	—	52	1	no	—
287	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
288	Q₁₆	—	2	1	no	—
289	Q₁₆	—	2	1	no	—
290	SD₁₆	—	4	2	no	—
291	D₈	—	2	1	no	—
292	D₈	—	2	1	no	—
293	M₁₆	—	26	2	no	—
294	M₁₆	—	26	4	no	—
295	C₂×C₈	—	13	2	no	—
296	C₂×C₈	—	26	2	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	#6127	1	2	yes	—
299	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
300	C₂×C₈	#6129	1	2	yes	—
301	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	26	2	no	—
305	C₄⋊C₄	—	26	2	no	—
306	C₄⋊C₄	—	52	2	no	—
307	C₄⋊C₄	—	52	2	no	—
308	C₄⋊C₄	—	52	1	no	—
309	C₄⋊C₄	—	52	1	no	—
310	C₂²⋊C₄	—	4	1	no	—
311	C₂²⋊C₄	—	52	2	no	—
312	C₂²⋊C₄	—	52	2	no	—
313	C₂²⋊C₄	—	52	2	no	—
314	C₄²	—	4	1	no	—
315	C₄²	—	52	2	no	—
316	C₁₆	—	2	1	no	—
317	C₁₆	—	2	1	no	—
318	C₁₆	—	26	2	no	—
319	C₁₃	#33653	1	1	yes	characteristic, complemented, Sylow
320	C₂³	—	4	1	no	—
321	C₂³	—	13	2	no	—
322	C₂³	D₈×D₁₃	1	1	yes	characteristic
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	26	2	no	—
326	D₄	—	4	1	no	—
327	D₄	—	4	1	no	—
328	D₄	—	8	2	no	—
329	D₄	—	26	4	no	—
330	D₄	—	26	2	no	—
331	D₄	—	208	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	13	2	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	26	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	—	52	2	no	—
341	C₂×C₄	—	208	2	no	—
342	C₂×C₄	—	208	1	no	—
343	C₂×C₄	—	208	1	no	—
344	C₂×C₄	#88137	1	2	yes	—
345	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
346	C₂×C₄	#88169	1	2	yes	—
347	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
348	C₈	—	26	2	no	—
349	C₈	#88137	1	2	yes	—
350	C₈	C₂×#6127	1	1	yes	characteristic
351	C₈	C₂×#6129	1	1	yes	characteristic
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	26	2	no	—
358	C₂²	—	26	2	no	—
359	C₂²	—	26	2	no	—
360	C₂²	#11718435	1	2	yes	—
361	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	26	2	no	—
366	C₄	—	104	2	no	—
367	C₄	—	104	2	no	—
368	C₄	#11718435	1	2	yes	—
369	C₄	C₂×#88137	1	1	yes	characteristic
370	C₄	C₂×#88169	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	16	1	no	—
373	C₂	—	52	2	no	—
374	C₂	C₂×#11718435	1	1	yes	characteristic
375	C₂	#13127979	1	1	yes	characteristic
376	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	#13124502	C₂
hypercenter	#2138	D₁₃
hypocenter	C₁₃	#33653
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
2. #13123433	C₂	semidirect
3. #13117042	C₂	semidirect
4. #13117032	C₂	semidirect
5. #13117026	C₂	semidirect
6. #13117020	C₂	semidirect
7. #13116625	C₂	semidirect
8. C₂×#11718097	C₂	semidirect
9. #13116548	C₂	semidirect
10. #13116449	C₂	semidirect
11. C₁₃×#2138	C₂	semidirect
141. #2138	D₁₃	semidirect
319. C₁₃	#33653	semidirect

Maximal subgroups

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

subgroup	index
#13124502	2
#13123433	2
#13117042	2
#13117032	2
#13117026	2
#13117020	2
#13116625	2
C₂×#11718097	2
#13116548	2
#13116449	2
C₁₃×#2138	2
#33653	13