Subgroups — #16016380

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127972	C₂	1	1	yes	maximal, characteristic
2	#13123754	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123721	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116736	C₂	1	4	yes	maximal, complemented
5	#13116729	C₂	1	4	yes	maximal, complemented
6	C₂×#11718105	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116586	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11718110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719450	—	2	1	no	—
11	C₂×#88158	C₂²	1	1	yes	characteristic
12	C₂×#88158	C₂²	1	1	yes	characteristic
13	#11719228	—	2	1	no	—
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₁₃×#405	C₂²	1	1	yes	characteristic
17	#11718794	—	2	1	no	—
18	C₂×#88059	C₂²	1	1	yes	characteristic
19	C₂×#88057	C₂²	1	1	yes	characteristic
20	#11718666	—	2	1	no	—
21	#11718570	C₄	1	2	yes	complemented
22	#11718570	C₂²	1	2	yes	—
23	#11718569	C₄	1	2	yes	complemented
24	#11718569	C₂²	1	2	yes	—
25	#11718565	C₄	1	4	yes	complemented
26	#11718378	C₂²	1	4	yes	—
27	#11718377	C₂²	1	2	yes	complemented
28	#11718372	C₂²	1	2	yes	complemented
29	#11718361	—	2	2	no	—
30	C₄×#6095	—	2	2	no	—
31	C₁₃×#347	C₂²	1	4	yes	complemented
32	#11718110	C₂²	1	4	yes	—
33	#11718109	C₂²	1	2	yes	complemented
34	#11718108	C₂²	1	2	yes	complemented
35	#11718105	C₂²	1	4	yes	complemented
36	C₂×#6127	—	2	1	no	—
37	C₂×#6127	—	2	1	no	—
38	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×#117	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₃	—	2	1	no	—
42	C₂×#6099	—	2	1	no	—
43	#88161	—	4	2	no	—
44	M₁₆×D₁₃	—	4	2	no	—
45	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
47	#88158	—	2	1	no	—
48	#88158	—	2	1	no	—
49	#88158	—	2	1	no	—
50	#88158	—	2	1	no	—
51	#88158	C₂×C₄	1	2	yes	—
52	#88158	C₂×C₄	1	2	yes	—
53	#88158	C₂×C₄	1	2	yes	complemented
54	#88158	C₂³	1	2	yes	—
55	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic, complemented
56	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
57	C₂×#6095	—	2	1	no	—
58	C₂×#6095	—	2	1	no	—
59	#88155	—	2	4	no	—
60	#88146	—	4	1	no	—
61	C₄⋊C₄×D₁₃	—	4	1	no	—
62	#88135	—	4	2	no	—
63	C₄×D₅₂	—	2	2	no	—
64	C₄×Dic₂₆	—	2	2	no	—
65	C₁₃×M₃₂	—	2	1	no	—
66	C₁₃×M₃₂	C₂×C₄	1	2	yes	complemented
67	C₁₃×M₃₂	C₂³	1	2	yes	—
68	C₁₃×M₃₂	C₂³	1	2	yes	—
69	C₂×C₁₃×C₁₆	—	2	1	no	—
70	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
71	C₁₃×#97	C₂³	1	2	yes	—
72	C₄×C₈×C₁₃	D₄	1	2	yes	—
73	#88061	—	2	2	no	—
74	#88059	—	2	1	no	—
75	#88059	C₂³	1	2	yes	—
76	#88057	—	2	1	no	—
77	#88057	C₂³	1	2	yes	—
78	#88056	—	2	2	no	—
79	#88045	—	2	4	no	—
80	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
81	#88040	C₂×C₄	1	4	yes	complemented
82	#88040	C₂×C₄	1	4	yes	complemented
83	D₂₀₈	C₂×C₄	1	4	yes	complemented
84	#33469	—	13	1	no	maximal, Sylow
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₂²⋊D₁₃	—	4	1	no	—
87	C₂×C₂²⋊D₁₃	—	4	1	no	—
88	C₂²×Dic₁₃	—	4	1	no	—
89	#6127	—	2	2	no	—
90	#6127	—	2	2	no	—
91	#6127	—	2	2	no	—
92	#6127	—	2	2	no	—
93	#6127	—	4	1	no	—
94	#6127	—	4	1	no	—
95	#6127	—	4	1	no	—
96	#6127	—	4	1	no	—
97	C₂×D₅₂	—	2	1	no	—
98	C₂×D₅₂	—	2	1	no	—
99	C₂×C₄×D₁₃	—	4	1	no	—
100	C₂×C₄×D₁₃	—	4	1	no	—
101	C₂×C₄×D₁₃	—	4	2	no	—
102	C₂×Dic₂₆	—	2	1	no	—
103	C₂×Dic₂₆	—	2	1	no	—
104	C₁₃×M₁₆	—	2	1	no	—
105	C₁₃×M₁₆	—	2	1	no	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
111	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₁₃×C₄⋊C₄	—	2	1	no	—
114	C₁₃×C₂²⋊C₄	—	2	1	no	—
115	C₄²×C₁₃	C₂×D₄	1	2	yes	—
116	#6103	—	4	2	no	—
117	C₄⋊Dic₁₃	—	2	2	no	—
118	C₄⋊Dic₁₃	—	4	1	no	—
119	#6101	—	4	2	no	—
120	C₄×Dic₁₃	—	4	2	no	—
121	#6099	—	2	2	no	—
122	#6099	—	4	1	no	—
123	C₂×#1055	—	4	1	no	—
124	Dic₅₂	—	2	1	no	—
125	Dic₅₂	—	2	1	no	—
126	Dic₅₂	C₂²×C₄	1	2	yes	—
127	Dic₅₂	C₂²×C₄	1	2	yes	—
128	D₁₀₄	—	2	1	no	—
129	D₁₀₄	—	2	1	no	—
130	D₁₀₄	C₂²×C₄	1	2	yes	—
131	D₁₀₄	C₂²×C₄	1	2	yes	—
132	#6095	—	2	2	no	—
133	#6095	—	2	2	no	—
134	#6095	—	2	2	no	—
135	#6095	—	2	2	no	—
136	#6094	—	4	4	no	—
137	C₈×D₁₃	—	4	4	no	—
138	C₁₃×C₁₆	—	2	2	no	—
139	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
140	#3414	—	13	1	no	—
141	#2957	—	13	1	no	—
142	#2952	—	13	1	no	—
143	AGL(1,ℤ/16)	—	13	4	no	—
144	#2180	—	13	4	no	—
145	C₂×#365	—	13	1	no	—
146	#2148	—	13	1	no	—
147	C₂×#360	—	13	1	no	—
148	#2110	D₁₃	1	1	yes	characteristic, complemented
149	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂²×D₁₃	—	4	1	no	—
151	C₂²×D₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²⋊D₁₃	—	4	2	no	—
160	C₂²⋊D₁₃	—	4	2	no	—
161	C₂²⋊D₁₃	—	4	2	no	—
162	C₂²⋊D₁₃	—	4	2	no	—
163	C₂×Dic₁₃	—	4	1	no	—
164	C₂×Dic₁₃	—	4	1	no	—
165	C₂×Dic₁₃	—	4	2	no	—
166	C₂×Dic₁₃	—	16	1	no	—
167	D₅₂	—	2	2	no	—
168	D₅₂	—	2	2	no	—
169	D₅₂	—	4	1	no	—
170	D₅₂	—	4	1	no	—
171	C₄×D₁₃	—	4	2	no	—
172	C₄×D₁₃	—	4	2	no	—
173	C₄×D₁₃	—	4	2	no	—
174	C₄×D₁₃	—	4	2	no	—
175	C₄×D₁₃	—	16	2	no	—
176	Dic₂₆	—	2	2	no	—
177	Dic₂₆	—	2	2	no	—
178	Dic₂₆	—	4	1	no	—
179	Dic₂₆	—	4	1	no	—
180	C₈×C₁₃	—	2	2	no	—
181	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
182	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
183	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
184	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
185	#1055	—	4	2	no	—
186	C₂×#135	—	13	1	no	—
187	C₂×#135	—	13	1	no	—
188	#568	—	26	1	no	—
189	#520	—	26	1	no	—
190	#510	—	13	2	no	—
191	#510	—	13	2	no	—
192	#509	—	13	2	no	—
193	#509	—	13	2	no	—
194	#508	—	13	4	no	—
195	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
196	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
197	#444	—	13	4	no	—
198	#443	—	26	2	no	—
199	#442	—	13	2	no	—
200	#441	—	13	2	no	—
201	C₄×SD₁₆	—	26	2	no	—
202	C₂×#108	—	13	1	no	—
203	C₂×#106	—	13	1	no	—
204	#421	—	26	1	no	—
205	#417	—	26	1	no	—
206	#405	C₂×D₁₃	1	1	yes	characteristic
207	#365	—	13	4	no	—
208	#362	—	13	2	no	—
209	#361	—	13	2	no	—
210	#360	—	13	4	no	—
211	#347	C₂×D₁₃	1	4	yes	complemented
212	C₂²×C₁₃	—	4	1	no	—
213	C₂²×C₁₃	#442	1	1	yes	characteristic
214	C₂²×C₁₃	#442	1	1	yes	characteristic
215	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂×D₁₃	—	4	2	no	—
217	C₂×D₁₃	—	4	2	no	—
218	C₂×D₁₃	—	8	1	no	—
219	C₂×D₁₃	—	8	1	no	—
220	C₂×D₁₃	—	16	1	no	—
221	C₄×C₁₃	—	4	2	no	—
222	C₄×C₁₃	#442	1	1	yes	characteristic
223	C₄×C₁₃	#442	1	1	yes	characteristic
224	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	Dic₁₃	—	4	2	no	—
227	Dic₁₃	—	4	2	no	—
228	Dic₁₃	—	8	2	no	—
229	C₂×#41	—	26	1	no	—
230	C₂×#41	—	26	1	no	—
231	#135	—	13	2	no	—
232	#135	—	13	2	no	—
233	#135	—	13	2	no	—
234	#135	—	13	2	no	—
235	#135	—	26	1	no	—
236	#135	—	26	1	no	—
237	#135	—	26	1	no	—
238	#135	—	26	1	no	—
239	C₂×Q₁₆	—	13	1	no	—
240	C₂×Q₁₆	—	13	1	no	—
241	C₂×SD₁₆	—	26	1	no	—
242	C₂×SD₁₆	—	26	1	no	—
243	C₂×D₈	—	13	1	no	—
244	C₂×D₈	—	13	1	no	—
245	#131	—	26	4	no	—
246	#131	—	52	2	no	—
247	C₂×M₁₆	—	26	1	no	—
248	C₂×M₁₆	—	52	2	no	—
249	C₂×M₁₆	D₅₂	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
251	C₄×Q₈	—	26	2	no	—
252	C₄×D₄	—	26	2	no	—
253	C₄×D₄	—	52	2	no	—
254	#117	—	52	1	no	—
255	#117	D₅₂	1	1	yes	characteristic
256	C₂×C₄⋊C₄	—	26	1	no	—
257	C₂×C₄⋊C₄	—	52	1	no	—
258	Q₃₂	—	13	4	no	—
259	SD₃₂	—	13	4	no	—
260	SD₃₂	—	13	4	no	—
261	D₁₆	—	13	4	no	—
262	M₃₂	—	2	1	no	—
263	M₃₂	C₄×D₁₃	1	2	yes	complemented
264	M₃₂	C₂²×D₁₃	1	2	yes	—
265	M₃₂	C₂²×D₁₃	1	2	yes	—
266	C₂×C₁₆	—	2	1	no	—
267	C₂×C₁₆	C₄×D₁₃	1	2	yes	complemented
268	#108	—	13	2	no	—
269	#108	—	26	1	no	—
270	#106	—	13	2	no	—
271	#106	—	26	1	no	—
272	#104	—	26	4	no	—
273	#103	—	26	2	no	—
274	#102	—	26	2	no	—
275	#97	C₂²×D₁₃	1	2	yes	—
276	C₄×C₈	D₅₂	1	2	yes	—
277	C₂×C₁₃	—	2	1	no	—
278	C₂×C₁₃	—	2	1	no	—
279	C₂×C₁₃	—	2	1	no	—
280	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
281	D₁₃	—	8	2	no	—
282	D₁₃	—	8	2	no	—
283	#41	—	26	2	no	—
284	#41	—	26	2	no	—
285	#41	—	26	2	no	—
286	#41	—	26	2	no	—
287	#41	—	52	1	no	—
288	#41	—	52	1	no	—
289	#41	—	52	1	no	—
290	#41	—	52	1	no	—
291	C₂×Q₈	—	26	1	no	—
292	C₂×Q₈	—	26	1	no	—
293	C₂×D₄	—	26	1	no	—
294	C₂×D₄	—	26	1	no	—
295	C₂×D₄	—	52	1	no	—
296	C₂×D₄	—	52	1	no	—
297	C₂²×C₄	—	52	2	no	—
298	C₂²×C₄	—	52	1	no	—
299	C₂²×C₄	—	52	1	no	—
300	C₂²×C₄	—	52	1	no	—
301	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
302	Q₁₆	—	13	2	no	—
303	Q₁₆	—	13	2	no	—
304	Q₁₆	—	26	1	no	—
305	Q₁₆	—	26	1	no	—
306	SD₁₆	—	26	2	no	—
307	SD₁₆	—	26	2	no	—
308	SD₁₆	—	26	2	no	—
309	SD₁₆	—	26	2	no	—
310	D₈	—	13	2	no	—
311	D₈	—	13	2	no	—
312	D₈	—	26	1	no	—
313	D₈	—	26	1	no	—
314	M₁₆	—	2	1	no	—
315	M₁₆	—	2	1	no	—
316	M₁₆	—	26	2	no	—
317	M₁₆	—	52	4	no	—
318	M₁₆	—	52	1	no	—
319	C₂×C₈	—	52	4	no	—
320	C₂×C₈	—	52	1	no	—
321	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
323	C₂×C₈	C₂×D₅₂	1	2	yes	—
324	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
325	C₂×C₈	#6127	1	1	yes	characteristic
326	C₂×C₈	#6127	1	1	yes	characteristic
327	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	26	2	no	—
330	C₄⋊C₄	—	52	2	no	—
331	C₄⋊C₄	—	52	1	no	—
332	C₂²⋊C₄	—	2	1	no	—
333	C₂²⋊C₄	—	52	2	no	—
334	C₄²	—	52	2	no	—
335	C₄²	C₂×D₅₂	1	2	yes	—
336	C₁₆	—	2	2	no	—
337	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
338	C₁₃	#33469	1	1	yes	characteristic, complemented, Sylow
339	C₂³	—	52	1	no	—
340	C₂³	—	52	1	no	—
341	C₂³	#88162	1	1	yes	characteristic
342	Q₈	—	26	2	no	—
343	Q₈	—	26	2	no	—
344	Q₈	—	52	1	no	—
345	Q₈	—	52	1	no	—
346	D₄	—	26	2	no	—
347	D₄	—	26	2	no	—
348	D₄	—	52	2	no	—
349	D₄	—	52	2	no	—
350	D₄	—	52	2	no	—
351	D₄	—	52	2	no	—
352	D₄	—	52	1	no	—
353	D₄	—	52	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	52	1	no	—
356	C₂×C₄	—	52	1	no	—
357	C₂×C₄	—	52	2	no	—
358	C₂×C₄	—	52	2	no	—
359	C₂×C₄	—	52	2	no	—
360	C₂×C₄	—	52	2	no	—
361	C₂×C₄	—	52	2	no	—
362	C₂×C₄	—	208	2	no	—
363	C₂×C₄	—	208	1	no	—
364	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
365	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
366	C₂×C₄	#88162	1	1	yes	characteristic
367	C₂×C₄	#88162	1	1	yes	characteristic
368	C₂×C₄	#88162	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	52	2	no	—
372	C₈	C₄×D₅₂	1	1	yes	characteristic
373	C₈	C₄×D₅₂	1	1	yes	characteristic
374	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
375	C₈	C₂×#6127	1	1	yes	characteristic
376	C₂²	—	4	1	no	—
377	C₂²	—	52	2	no	—
378	C₂²	—	52	2	no	—
379	C₂²	—	104	1	no	—
380	C₂²	—	104	1	no	—
381	C₂²	—	208	1	no	—
382	C₂²	#11718372	1	1	yes	characteristic
383	C₂²	#11718372	1	1	yes	characteristic
384	C₂²	C₂×#88162	1	1	yes	characteristic
385	C₄	—	4	2	no	—
386	C₄	—	52	2	no	—
387	C₄	—	52	2	no	—
388	C₄	—	104	2	no	—
389	C₄	#11718372	1	1	yes	characteristic
390	C₄	#11718372	1	1	yes	characteristic
391	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
392	C₄	C₂×#88162	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	104	2	no	—
397	C₂	—	104	2	no	—
398	C₂	C₂×#11718372	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718372
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2110	C₂
socle	C₂×C₁₃	C₂×#442
prime radical	#13127972	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2110	D₁₃
hypocenter	C₁₃	#33469
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
2. #13123754	C₂	semidirect
3. #13123721	C₂	semidirect
4. #13116736	C₂	semidirect
5. #13116729	C₂	semidirect
6. C₂×#11718105	C₂	semidirect
7. #13116586	C₂	semidirect
8. C₂×#11718110	C₂	semidirect
9. C₁₃×#2110	C₂	semidirect
21. #11718570	C₄	semidirect
23. #11718569	C₄	semidirect
25. #11718565	C₄	semidirect
27. #11718377	C₂²	semidirect
28. #11718372	C₂²	semidirect
31. C₁₃×#347	C₂²	semidirect
33. #11718109	C₂²	semidirect
34. #11718108	C₂²	semidirect
35. #11718105	C₂²	semidirect
45. C₂×Dic₅₂	C₂×C₄	semidirect
53. #88158	C₂×C₄	semidirect
55. C₂×D₁₀₄	C₂×C₄	semidirect
66. C₁₃×M₃₂	C₂×C₄	semidirect
70. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
80. Dic₁₀₄	C₂×C₄	semidirect
81. #88040	C₂×C₄	semidirect
82. #88040	C₂×C₄	semidirect
83. D₂₀₈	C₂×C₄	semidirect
148. #2110	D₁₃	semidirect
211. #347	C₂×D₁₃	semidirect
263. M₃₂	C₄×D₁₃	semidirect
267. C₂×C₁₆	C₄×D₁₃	semidirect
338. C₁₃	#33469	semidirect

Maximal subgroups

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

subgroup	index
#13127972	2
#13123754	2
#13123721	2
#13116736	2
#13116729	2
C₂×#11718105	2
#13116586	2
C₂×#11718110	2
C₁₃×#2110	2
#33469	13