Subgroups — #16019755

Order 3328.

622 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	#13127230	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₅₂	C₂	1	1	yes	maximal, characteristic
3	#13117060	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117023	C₂	1	2	yes	maximal, complemented
5	#13116983	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116935	C₂	1	2	yes	maximal, complemented
7	#13116876	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116763	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₃×#357	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₁₃	C₂²	1	2	yes	—
13	#11719284	—	2	1	no	—
14	D₄×D₅₂	—	2	1	no	—
15	C₂×C₁₃×SD₃₂	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
17	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
18	#11718849	C₂²	1	2	yes	complemented
19	C₂×#88068	C₂²	1	2	yes	complemented
20	#11718682	—	2	1	no	—
21	#11718676	—	2	1	no	—
22	C₂×C₁₆×D₁₃	C₂²	1	2	yes	complemented
23	#11718544	C₂²	1	2	yes	—
24	D₁₃×#107	C₂²	1	2	yes	complemented
25	#11718470	—	2	2	no	—
26	#11718434	—	2	2	no	—
27	#11718356	C₂²	1	1	yes	characteristic
28	C₈×D₅₂	C₂²	1	1	yes	characteristic
29	#11718343	C₂²	1	1	yes	characteristic, complemented
30	C₁₃×#367	C₂²	1	1	yes	characteristic, complemented
31	C₁₃×#358	C₂²	1	1	yes	characteristic, complemented
32	C₁₃×#357	C₂²	1	1	yes	characteristic
33	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
34	#11718154	C₂²	1	2	yes	—
35	#11718101	C₂²	1	2	yes	complemented
36	#11718101	C₂²	1	2	yes	complemented
37	#11718099	C₂²	1	2	yes	complemented
38	#11718083	C₂²	1	2	yes	complemented
39	#11718082	C₂²	1	2	yes	complemented
40	C₄⋊#6090	C₂²	1	1	yes	characteristic, complemented
41	C₂×D₄×D₁₃	—	2	2	no	—
42	C₂²×D₅₂	—	4	1	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₁₃	—	2	1	no	—
46	C₄×D₄×C₁₃	—	2	1	no	—
47	#88200	—	2	2	no	—
48	C₂×#6104	—	2	2	no	—
49	#88184	—	4	1	no	—
50	D₈×D₁₃	—	2	4	no	—
51	D₈×D₁₃	—	4	2	no	—
52	C₂×D₁₀₄	—	2	1	no	—
53	C₂×D₁₀₄	C₂³	1	2	yes	—
54	C₂×C₈×D₁₃	C₂³	1	2	yes	—
55	#88150	—	4	2	no	—
56	#88149	—	2	2	no	—
57	C₄⋊C₄×D₁₃	—	2	2	no	—
58	#88136	—	4	2	no	—
59	#88128	—	2	1	no	—
60	C₄×D₅₂	—	4	1	no	—
61	C₄×D₅₂	D₄	1	1	yes	characteristic
62	#88124	—	4	1	no	—
63	#88123	—	2	1	no	—
64	C₁₃×SD₃₂	—	2	2	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#103	—	2	1	no	—
70	C₁₃×#102	—	2	1	no	—
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	#88075	—	2	2	no	—
73	#88068	—	4	2	no	—
74	#88060	D₄	1	2	yes	—
75	#88058	—	2	1	no	—
76	#88058	C₂³	1	2	yes	—
77	C₂×#6090	C₂³	1	2	yes	—
78	#88049	—	2	2	no	—
79	#88047	—	2	2	no	—
80	C₄⋊#1055	D₄	1	1	yes	characteristic
81	C₁₆×D₁₃	—	4	2	no	—
82	#33862	—	13	1	no	maximal, Sylow
83	C₂³×D₁₃	—	4	2	no	—
84	C₂×Q₈×C₁₃	—	2	1	no	—
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₂×C₂²⋊D₁₃	—	4	2	no	—
88	D₄×D₁₃	—	4	4	no	—
89	D₄×D₁₃	—	8	2	no	—
90	C₂×D₅₂	—	2	2	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×D₅₂	—	4	1	no	—
93	C₂×D₅₂	—	4	1	no	—
94	C₂×D₅₂	—	8	4	no	—
95	C₂×D₅₂	D₈	1	1	yes	characteristic
96	C₂×C₄×D₁₃	—	4	2	no	—
97	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
98	C₂×Dic₂₆	—	4	1	no	—
99	C₁₃×Q₁₆	—	2	1	no	—
100	C₁₃×Q₁₆	—	2	1	no	—
101	C₁₃×D₈	—	2	1	no	—
102	C₁₃×D₈	#41	1	2	yes	—
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₄²×C₁₃	—	4	1	no	—
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	#6104	—	4	4	no	—
112	#6103	—	4	2	no	—
113	#6103	—	4	2	no	—
114	#6103	D₈	1	2	yes	—
115	C₄⋊Dic₁₃	—	2	2	no	—
116	C₄⋊Dic₁₃	—	4	1	no	—
117	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
118	#6101	—	4	2	no	—
119	#6101	—	4	2	no	—
120	C₂×#1055	C₂×D₄	1	2	yes	—
121	D₁₀₄	—	2	2	no	—
122	D₁₀₄	—	2	2	no	—
123	D₁₀₄	—	8	1	no	—
124	C₈×D₁₃	—	2	2	no	—
125	C₈×D₁₃	—	2	2	no	—
126	C₁₃×C₁₆	—	2	1	no	—
127	C₁₃×C₁₆	—	2	1	no	—
128	#6090	—	2	2	no	—
129	#3338	—	13	1	no	—
130	D₄×D₈	—	13	1	no	—
131	#2240	—	13	1	no	—
132	#2233	—	13	2	no	—
133	#2226	—	13	1	no	—
134	#2217	—	13	2	no	—
135	#2207	—	13	1	no	—
136	#2187	—	13	2	no	—
137	C₄×SD₃₂	D₁₃	1	1	yes	characteristic, complemented
138	D₄×C₁₆	—	13	1	no	—
139	C₂×#357	—	13	2	no	—
140	C₂³×C₁₃	—	4	1	no	—
141	C₂²×D₁₃	—	4	2	no	—
142	C₂²×D₁₃	—	8	4	no	—
143	C₂²×D₁₃	—	8	4	no	—
144	C₂²×D₁₃	—	16	2	no	—
145	C₂²×D₁₃	C₂×D₈	1	2	yes	—
146	Q₈×C₁₃	—	4	1	no	—
147	Q₈×C₁₃	—	4	1	no	—
148	D₄×C₁₃	—	2	2	no	—
149	D₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	4	1	no	—
151	C₂×C₄×C₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	8	2	no	—
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₁₃	—	8	4	no	—
158	C₂×Dic₁₃	—	4	2	no	—
159	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
160	D₅₂	—	4	2	no	—
161	D₅₂	—	4	2	no	—
162	D₅₂	—	8	4	no	—
163	D₅₂	—	8	2	no	—
164	D₅₂	—	16	1	no	—
165	D₅₂	SD₃₂	1	4	yes	complemented
166	C₄×D₁₃	—	2	2	no	—
167	C₄×D₁₃	—	2	2	no	—
168	C₄×D₁₃	—	16	2	no	—
169	Dic₂₆	—	16	1	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
172	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
173	#1055	—	2	2	no	—
174	C₂²×D₈	—	13	2	no	—
175	#554	—	26	1	no	—
176	D₄²	—	26	1	no	—
177	C₂×SD₃₂	—	13	2	no	—
178	C₂×SD₃₂	C₂×D₁₃	1	1	yes	characteristic
179	C₂²×C₁₆	—	13	2	no	—
180	#500	—	13	1	no	—
181	#493	—	13	1	no	—
182	#484	—	26	2	no	—
183	#477	—	26	1	no	—
184	#467	—	13	2	no	—
185	#466	—	13	2	no	—
186	#459	—	26	1	no	—
187	#447	—	26	2	no	—
188	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
189	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
190	C₈×D₄	—	13	1	no	—
191	C₂×#107	—	13	2	no	—
192	#367	—	13	2	no	—
193	#367	C₂×D₁₃	1	1	yes	characteristic, complemented
194	C₄⋊C₁₆	—	13	1	no	—
195	#358	C₂×D₁₃	1	1	yes	characteristic, complemented
196	#357	—	13	2	no	—
197	#357	—	13	2	no	—
198	#357	—	13	2	no	—
199	#357	—	13	2	no	—
200	#357	C₂×D₁₃	1	1	yes	characteristic
201	C₂²⋊C₁₆	—	13	2	no	—
202	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
203	C₂²×C₁₃	—	4	2	no	—
204	C₂²×C₁₃	—	8	1	no	—
205	C₂²×C₁₃	—	8	1	no	—
206	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
207	C₂×D₁₃	—	2	2	no	—
208	C₂×D₁₃	—	2	2	no	—
209	C₂×D₁₃	—	8	2	no	—
210	C₂×D₁₃	—	8	2	no	—
211	C₂×D₁₃	—	8	2	no	—
212	C₂×D₁₃	—	16	8	no	—
213	C₂×D₁₃	C₂×SD₃₂	1	4	yes	—
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	—	8	1	no	—
216	C₄×C₁₃	—	8	1	no	—
217	C₄×C₁₃	—	8	1	no	—
218	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
219	C₄×C₁₃	#546	1	1	yes	characteristic
220	C₄×C₁₃	#577	1	1	yes	characteristic
221	Dic₁₃	—	2	2	no	—
222	Dic₁₃	—	8	2	no	—
223	C₂²×D₄	—	26	2	no	—
224	C₂²×D₄	—	52	1	no	—
225	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×D₈	—	13	2	no	—
227	C₂×D₈	—	26	2	no	—
228	C₂×D₈	—	26	4	no	—
229	C₂×D₈	—	26	1	no	—
230	C₂×D₈	—	52	2	no	—
231	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₂²×C₈	—	13	2	no	—
233	C₄⋊Q₈	—	26	1	no	—
234	#127	—	26	1	no	—
235	#125	—	52	1	no	—
236	C₂²⋊Q₈	—	26	2	no	—
237	C₂²⋊Q₈	—	52	2	no	—
238	#121	—	26	2	no	—
239	#121	—	52	1	no	—
240	#120	—	52	2	no	—
241	C₄×Q₈	—	2	1	no	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	—	13	1	no	—
244	C₄×D₄	—	52	1	no	—
245	C₂×C₄⋊C₄	—	26	2	no	—
246	SD₃₂	—	2	2	no	—
247	SD₃₂	—	52	2	no	—
248	C₂×C₁₆	—	13	2	no	—
249	C₂×C₁₆	—	52	2	no	—
250	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
251	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
252	#107	—	13	2	no	—
253	#107	—	26	2	no	—
254	#107	—	26	1	no	—
255	#107	C₂²×D₁₃	1	1	yes	characteristic
256	#107	C₂²×D₁₃	1	1	yes	characteristic
257	C₄⋊C₈	—	13	1	no	—
258	#103	—	2	1	no	—
259	#103	—	26	2	no	—
260	#102	—	2	1	no	—
261	#102	—	26	2	no	—
262	C₂²⋊C₈	—	13	2	no	—
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₂×C₁₃	—	8	2	no	—
265	C₂×C₁₃	#3294	1	1	yes	characteristic
266	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
267	C₂×C₁₃	#3415	1	1	yes	characteristic
268	D₁₃	—	2	4	no	—
269	D₁₃	—	16	2	no	—
270	C₂⁴	—	52	2	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	52	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	13	1	no	—
275	C₂×D₄	—	26	2	no	—
276	C₂×D₄	—	52	2	no	—
277	C₂×D₄	—	52	2	no	—
278	C₂×D₄	—	52	4	no	—
279	C₂×D₄	—	52	1	no	—
280	C₂×D₄	—	52	1	no	—
281	C₂×D₄	—	104	4	no	—
282	C₂×D₄	—	104	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	13	2	no	—
285	C₂²×C₄	—	52	2	no	—
286	Q₁₆	—	2	1	no	—
287	Q₁₆	—	2	1	no	—
288	D₈	—	2	1	no	—
289	D₈	—	26	2	no	—
290	D₈	—	26	2	no	—
291	D₈	—	52	4	no	—
292	D₈	—	104	1	no	—
293	D₈	#6127	1	2	yes	—
294	C₂×C₈	—	13	2	no	—
295	C₂×C₈	—	26	2	no	—
296	C₂×C₈	—	26	2	no	—
297	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
298	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	13	1	no	—
303	C₄⋊C₄	—	26	2	no	—
304	C₄⋊C₄	—	52	2	no	—
305	C₄⋊C₄	—	52	2	no	—
306	C₄⋊C₄	—	52	1	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	13	2	no	—
309	C₂²⋊C₄	—	52	2	no	—
310	C₂²⋊C₄	—	52	2	no	—
311	C₄²	—	4	1	no	—
312	C₄²	D₄×D₁₃	1	1	yes	characteristic
313	C₁₆	—	2	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	26	2	no	—
316	C₁₃	#33862	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	4	1	no	—
318	C₂³	—	13	2	no	—
319	C₂³	—	52	2	no	—
320	C₂³	—	104	4	no	—
321	C₂³	—	104	4	no	—
322	C₂³	—	208	2	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	208	1	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	13	4	no	—
329	D₄	—	52	2	no	—
330	D₄	—	52	2	no	—
331	D₄	—	104	4	no	—
332	D₄	—	104	4	no	—
333	D₄	—	104	2	no	—
334	D₄	—	208	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	13	2	no	—
340	C₂×C₄	—	26	2	no	—
341	C₂×C₄	—	26	2	no	—
342	C₂×C₄	—	52	2	no	—
343	C₂×C₄	—	208	2	no	—
344	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
345	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	26	2	no	—
349	C₈	C₂×#6127	1	1	yes	characteristic
350	C₈	#88256	1	1	yes	characteristic
351	C₂²	—	4	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	13	4	no	—
355	C₂²	—	26	2	no	—
356	C₂²	—	26	2	no	—
357	C₂²	—	104	2	no	—
358	C₂²	—	104	2	no	—
359	C₂²	—	104	2	no	—
360	C₂²	—	208	8	no	—
361	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	26	2	no	—
367	C₄	—	104	2	no	—
368	C₄	D₁₃×SD₃₂	1	2	yes	—
369	C₄	#11719255	1	1	yes	characteristic
370	C₄	#11719475	1	1	yes	characteristic
371	C₂	—	8	2	no	—
372	C₂	—	26	4	no	—
373	C₂	—	208	2	no	—
374	C₂	#13126651	1	1	yes	characteristic
375	C₂	C₂×D₁₃×SD₃₂	1	1	yes	characteristic
376	C₂	#13127974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×SD₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	D₈×D₅₂	C₂
hypercenter	C₄×SD₃₂	D₁₃
hypocenter	C₁₃	#33862
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #13127230	C₂	semidirect
3. #13117060	C₂	semidirect
4. #13117023	C₂	semidirect
5. #13116983	C₂	semidirect
6. #13116935	C₂	semidirect
7. #13116876	C₂	semidirect
8. #13116763	C₂	semidirect
9. C₁₆×D₅₂	C₂	semidirect
10. D₁₃×#357	C₂	semidirect
11. C₄×C₁₃×SD₃₂	C₂	semidirect
16. C₄×C₁₃×Q₁₆	C₂²	semidirect
18. #11718849	C₂²	semidirect
19. C₂×#88068	C₂²	semidirect
22. C₂×C₁₆×D₁₃	C₂²	semidirect
24. D₁₃×#107	C₂²	semidirect
29. #11718343	C₂²	semidirect
30. C₁₃×#367	C₂²	semidirect
31. C₁₃×#358	C₂²	semidirect
33. C₄×C₁₃×C₁₆	C₂²	semidirect
35. #11718101	C₂²	semidirect
36. #11718101	C₂²	semidirect
37. #11718099	C₂²	semidirect
38. #11718083	C₂²	semidirect
39. #11718082	C₂²	semidirect
40. C₄⋊#6090	C₂²	semidirect
137. C₄×SD₃₂	D₁₃	semidirect
165. D₅₂	SD₃₂	semidirect
188. C₄×Q₁₆	C₂×D₁₃	semidirect
193. #367	C₂×D₁₃	semidirect
195. #358	C₂×D₁₃	semidirect
202. C₄×C₁₆	C₂×D₁₃	semidirect
316. C₁₃	#33862	semidirect

Maximal subgroups

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

subgroup	index
#13127230	2
D₈×D₅₂	2
#13117060	2
#13117023	2
#13116983	2
#13116935	2
#13116876	2
#13116763	2
C₁₆×D₅₂	2
D₁₃×#357	2
C₄×C₁₃×SD₃₂	2
#33862	13