Subgroups — #16018016

Order 3328.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125406	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117039	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117022	C₂	1	1	yes	maximal, characteristic
5	#13117001	C₂	1	2	yes	maximal, complemented
6	#13116899	C₂	1	2	yes	maximal, complemented
7	#13116772	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116755	C₂	1	1	yes	maximal, characteristic
9	#13116708	C₂	1	2	yes	maximal, complemented
10	#13116702	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
13	#11719184	—	2	1	no	—
14	C₂×C₄×Dic₂₆	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718777	C₂²	1	1	yes	characteristic
17	#11718776	C₂²	1	1	yes	characteristic
18	#11718775	—	2	1	no	—
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88056	—	2	1	no	—
21	C₂×Dic₁₀₄	C₂²	1	2	yes	—
22	C₂×#88040	C₂²	1	2	yes	—
23	C₄×Dic₅₂	—	2	1	no	—
24	C₄×Dic₅₂	—	2	1	no	—
25	C₄×Dic₅₂	C₂²	1	2	yes	—
26	C₄×Dic₅₂	C₂²	1	2	yes	—
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	#11718358	—	2	1	no	—
29	#11718357	C₂²	1	1	yes	characteristic
30	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
31	C₄×#6095	—	2	1	no	—
32	#11718344	C₂²	1	1	yes	characteristic
33	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₁₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₁₃×#346	C₂²	1	2	yes	—
37	#11718101	C₂²	1	2	yes	—
38	#11718097	C₂²	1	2	yes	—
39	#11718096	C₂²	1	2	yes	—
40	#11718095	C₂²	1	2	yes	—
41	#11718095	C₂²	1	2	yes	—
42	#11718095	C₂²	1	2	yes	—
43	C₂²×Dic₂₆	—	2	1	no	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
47	#88184	—	2	1	no	—
48	#88183	—	4	1	no	—
49	C₄×C₂²⋊D₁₃	—	4	1	no	—
50	C₂×C₄⋊Dic₁₃	—	2	1	no	—
51	#88178	—	2	1	no	—
52	C₂×#6101	—	4	1	no	—
53	C₂×C₄×Dic₁₃	—	4	1	no	—
54	C₂×Dic₅₂	—	2	1	no	—
55	C₂×Dic₅₂	—	2	4	no	—
56	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
57	C₂×Dic₅₂	C₂³	1	2	yes	—
58	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
59	C₂×#6095	—	2	1	no	—
60	#88130	—	4	1	no	—
61	#88129	—	2	1	no	—
62	C₄×D₅₂	—	2	1	no	—
63	#88126	—	4	1	no	—
64	#88124	—	2	1	no	—
65	C₄×Dic₂₆	—	2	2	no	—
66	C₄×Dic₂₆	—	2	2	no	—
67	C₄×Dic₂₆	—	2	1	no	—
68	C₄×Dic₂₆	—	4	1	no	—
69	C₄×Dic₂₆	—	4	1	no	—
70	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
71	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
72	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#88061	—	2	1	no	—
77	#88061	—	2	1	no	—
78	#88058	—	2	1	no	—
79	#88058	C₂³	1	2	yes	—
80	#88058	C₂³	1	1	yes	characteristic
81	#88058	C₂³	1	1	yes	characteristic
82	#88057	—	2	1	no	—
83	#88056	—	2	2	no	—
84	#88056	—	2	2	no	—
85	#88056	—	2	1	no	—
86	#88056	—	2	1	no	—
87	Dic₁₀₄	—	2	4	no	—
88	#88040	—	2	4	no	—
89	#33677	—	13	1	no	maximal, Sylow
90	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₃	—	4	1	no	—
94	C₂²×Dic₁₃	—	4	1	no	—
95	C₂²×Dic₁₃	—	4	1	no	—
96	C₂×D₅₂	—	2	1	no	—
97	C₂×C₄×D₁₃	—	4	1	no	—
98	C₂×Dic₂₆	—	2	4	no	—
99	C₂×Dic₂₆	—	2	2	no	—
100	C₂×Dic₂₆	—	2	1	no	—
101	C₂×Dic₂₆	—	4	2	no	—
102	C₂×Dic₂₆	—	4	1	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₄²×C₁₃	D₈	1	1	yes	characteristic
110	C₄²×C₁₃	D₈	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂²⋊Dic₁₃	—	4	1	no	—
114	#6103	—	4	1	no	—
115	#6103	—	4	1	no	—
116	#6103	—	4	1	no	—
117	#6103	—	4	1	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	—	2	1	no	—
120	C₄⋊Dic₁₃	—	2	1	no	—
121	C₄⋊Dic₁₃	—	4	1	no	—
122	#6101	—	4	2	no	—
123	#6101	—	4	2	no	—
124	#6101	—	4	1	no	—
125	#6101	—	4	1	no	—
126	#6101	—	4	1	no	—
127	#6101	—	4	1	no	—
128	C₄×Dic₁₃	—	4	2	no	—
129	C₄×Dic₁₃	—	4	2	no	—
130	C₄×Dic₁₃	—	4	1	no	—
131	Dic₅₂	—	2	4	no	—
132	Dic₅₂	—	2	2	no	—
133	Dic₅₂	—	2	2	no	—
134	Dic₅₂	#41	1	4	yes	—
135	D₁₀₄	—	2	2	no	—
136	#6095	—	4	2	no	—
137	C₁₃×C₁₆	—	2	2	no	—
138	C₁₃×C₁₆	—	2	2	no	—
139	#3168	—	13	1	no	—
140	C₂×C₄×Q₁₆	—	13	1	no	—
141	#2236	—	13	1	no	—
142	#2233	—	13	1	no	—
143	#2229	—	13	2	no	—
144	#2211	—	13	2	no	—
145	#2188	—	13	1	no	—
146	#2185	—	13	1	no	—
147	#2176	—	13	2	no	—
148	#2175	—	13	2	no	—
149	#2163	D₁₃	1	1	yes	characteristic, complemented
150	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂²×D₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	1	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂²⋊D₁₃	—	8	2	no	—
163	C₂×Dic₁₃	—	4	4	no	—
164	C₂×Dic₁₃	—	4	2	no	—
165	C₂×Dic₁₃	—	4	2	no	—
166	C₂×Dic₁₃	—	4	1	no	—
167	C₂×Dic₁₃	—	4	1	no	—
168	C₂×Dic₁₃	—	4	1	no	—
169	C₂×Dic₁₃	—	8	2	no	—
170	D₅₂	—	4	1	no	—
171	D₅₂	—	4	1	no	—
172	C₄×D₁₃	—	8	2	no	—
173	Dic₂₆	—	2	4	no	—
174	Dic₂₆	—	4	4	no	—
175	Dic₂₆	—	4	2	no	—
176	Dic₂₆	—	4	1	no	—
177	Dic₂₆	—	4	1	no	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	—	2	1	no	—
180	C₈×C₁₃	—	2	1	no	—
181	C₈×C₁₃	C₂×#41	1	2	yes	—
182	C₂²×Q₁₆	—	13	1	no	—
183	#529	—	26	1	no	—
184	C₂×C₄×Q₈	—	26	1	no	—
185	C₂×Q₃₂	—	13	2	no	—
186	C₂×SD₃₂	—	13	2	no	—
187	#499	—	13	1	no	—
188	#495	—	13	1	no	—
189	#488	—	26	1	no	—
190	#467	—	13	1	no	—
191	#466	—	13	1	no	—
192	#465	—	26	1	no	—
193	C₄×Q₁₆	—	13	1	no	—
194	C₄×Q₁₆	—	13	2	no	—
195	C₄×Q₁₆	—	13	2	no	—
196	C₄×Q₁₆	—	26	1	no	—
197	C₄×Q₁₆	—	26	1	no	—
198	C₄×SD₁₆	—	26	1	no	—
199	C₄×D₈	—	13	1	no	—
200	C₂×#107	—	13	1	no	—
201	C₂×#103	—	26	1	no	—
202	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
203	#367	—	13	2	no	—
204	#366	—	13	2	no	—
205	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
206	#358	—	13	2	no	—
207	#358	—	13	2	no	—
208	#358	—	13	2	no	—
209	#357	—	13	2	no	—
210	C₂²⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
211	C₂²⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
212	#346	C₂×D₁₃	1	2	yes	—
213	C₂²×C₁₃	—	2	1	no	—
214	C₂²×C₁₃	—	2	1	no	—
215	C₂²×C₁₃	#510	1	2	yes	—
216	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
217	C₂×D₁₃	—	8	1	no	—
218	C₂×D₁₃	—	8	1	no	—
219	C₂×D₁₃	—	8	1	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	2	1	no	—
222	C₄×C₁₃	—	2	1	no	—
223	C₄×C₁₃	—	2	1	no	—
224	C₄×C₁₃	—	2	1	no	—
225	C₄×C₁₃	#525	1	1	yes	characteristic
226	C₄×C₁₃	#575	1	1	yes	characteristic
227	Dic₁₃	—	4	4	no	—
228	Dic₁₃	—	8	1	no	—
229	Dic₁₃	—	8	1	no	—
230	Dic₁₃	—	8	1	no	—
231	Dic₁₃	—	8	2	no	—
232	C₂²×Q₈	—	26	1	no	—
233	C₂×Q₁₆	—	13	2	no	—
234	C₂×Q₁₆	—	13	1	no	—
235	C₂×Q₁₆	—	26	4	no	—
236	C₂×Q₁₆	—	26	1	no	—
237	C₂×SD₁₆	—	26	1	no	—
238	C₂×D₈	—	13	1	no	—
239	C₂²×C₈	D₅₂	1	1	yes	characteristic
240	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	#126	—	52	1	no	—
242	#125	—	26	1	no	—
243	#124	—	26	1	no	—
244	#123	—	52	1	no	—
245	C₂²⋊Q₈	—	26	1	no	—
246	#121	—	26	1	no	—
247	C₄×Q₈	—	26	2	no	—
248	C₄×Q₈	—	26	2	no	—
249	C₄×Q₈	—	26	1	no	—
250	C₄×Q₈	—	52	1	no	—
251	C₄×Q₈	—	52	1	no	—
252	C₄×D₄	—	26	1	no	—
253	C₄×D₄	—	52	1	no	—
254	#117	—	52	1	no	—
255	C₂×C₄⋊C₄	—	26	1	no	—
256	C₂×C₄⋊C₄	—	52	1	no	—
257	C₂×C₄²	—	52	1	no	—
258	C₂×C₄²	D₅₂	1	1	yes	characteristic
259	Q₃₂	—	26	4	no	—
260	SD₃₂	—	26	4	no	—
261	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
262	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
263	#107	—	13	2	no	—
264	#107	—	13	1	no	—
265	#107	—	13	1	no	—
266	#107	—	26	1	no	—
267	#106	—	26	1	no	—
268	#103	—	26	2	no	—
269	#103	—	26	2	no	—
270	#103	—	26	1	no	—
271	#103	—	26	1	no	—
272	#102	—	26	1	no	—
273	#102	—	26	1	no	—
274	C₄×C₈	D₅₂	1	1	yes	characteristic
275	C₄×C₈	D₅₂	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
278	C₂×C₁₃	—	2	2	no	—
279	C₂×C₁₃	#3100	1	1	yes	characteristic
280	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
281	C₂×C₁₃	#3414	1	1	yes	characteristic
282	D₁₃	—	16	1	no	—
283	C₂×Q₈	—	26	4	no	—
284	C₂×Q₈	—	26	2	no	—
285	C₂×Q₈	—	26	1	no	—
286	C₂×Q₈	—	52	2	no	—
287	C₂×Q₈	—	52	1	no	—
288	C₂×D₄	—	26	1	no	—
289	C₂×D₄	—	52	1	no	—
290	C₂²×C₄	—	52	1	no	—
291	C₂²×C₄	—	52	1	no	—
292	C₂²×C₄	—	52	1	no	—
293	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
294	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
296	Q₁₆	—	13	4	no	—
297	Q₁₆	—	26	4	no	—
298	Q₁₆	—	26	2	no	—
299	Q₁₆	—	26	2	no	—
300	SD₁₆	—	52	2	no	—
301	D₈	—	26	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
305	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
306	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
308	C₄⋊C₄	—	26	1	no	—
309	C₄⋊C₄	—	26	1	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	52	1	no	—
312	C₄⋊C₄	—	52	2	no	—
313	C₄⋊C₄	—	52	2	no	—
314	C₄⋊C₄	—	52	1	no	—
315	C₄⋊C₄	—	52	1	no	—
316	C₄⋊C₄	—	52	1	no	—
317	C₄⋊C₄	—	52	1	no	—
318	C₂²⋊C₄	—	52	1	no	—
319	C₂²⋊C₄	—	52	1	no	—
320	C₂²⋊C₄	—	52	1	no	—
321	C₂²⋊C₄	—	52	1	no	—
322	C₂²⋊C₄	—	52	1	no	—
323	C₄²	—	52	2	no	—
324	C₄²	—	52	2	no	—
325	C₄²	—	52	1	no	—
326	C₄²	D₁₀₄	1	1	yes	characteristic
327	C₄²	D₁₀₄	1	1	yes	characteristic
328	C₄²	C₂×D₅₂	1	1	yes	characteristic
329	C₄²	C₂×D₅₂	1	1	yes	characteristic
330	C₁₆	—	2	2	no	—
331	C₁₆	—	2	2	no	—
332	C₁₃	#33677	1	1	yes	characteristic, complemented, Sylow
333	C₂³	—	52	1	no	—
334	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
335	Q₈	—	26	4	no	—
336	Q₈	—	52	4	no	—
337	Q₈	—	52	2	no	—
338	Q₈	—	52	1	no	—
339	Q₈	—	52	1	no	—
340	D₄	—	52	1	no	—
341	D₄	—	52	1	no	—
342	D₄	—	104	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	52	4	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	52	2	no	—
350	C₂×C₄	—	52	1	no	—
351	C₂×C₄	—	52	1	no	—
352	C₂×C₄	—	52	1	no	—
353	C₂×C₄	—	104	2	no	—
354	C₂×C₄	—	104	2	no	—
355	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
356	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
357	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
358	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
359	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
360	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	#41×D₁₃	1	2	yes	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	104	1	no	—
368	C₂²	—	104	1	no	—
369	C₂²	—	104	1	no	—
370	C₂²	#11718570	1	2	yes	—
371	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	52	4	no	—
378	C₄	—	104	1	no	—
379	C₄	—	104	1	no	—
380	C₄	—	104	2	no	—
381	C₄	—	104	1	no	—
382	C₄	#11719229	1	1	yes	characteristic
383	C₄	#11719453	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	208	1	no	—
386	C₂	#13124837	1	1	yes	characteristic
387	C₂	C₂×#11718570	1	1	yes	characteristic
388	C₂	#13127972	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2163	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#11718776	C₂²
hypercenter	#2163	D₁₃
hypocenter	C₁₃	#33677
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₅₂	C₂	semidirect
3. #13117039	C₂	semidirect
5. #13117001	C₂	semidirect
6. #13116899	C₂	semidirect
7. #13116772	C₂	semidirect
9. #13116708	C₂	semidirect
10. #13116702	C₂	semidirect
11. C₁₃×#2163	C₂	semidirect
149. #2163	D₁₃	semidirect
332. C₁₃	#33677	semidirect

Maximal subgroups

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

subgroup	index
#13125406	2
C₂×C₄×Dic₅₂	2
#13117039	2
#13117022	2
#13117001	2
#13116899	2
#13116772	2
#13116755	2
#13116708	2
#13116702	2
C₁₃×#2163	2
#33677	13