Subgroups — #16018974

Order 3328.

562 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125330	C₂	1	1	yes	maximal, characteristic
2	#13117105	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117088	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117083	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117070	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116898	C₂	1	2	yes	maximal, complemented
7	#13116886	C₂	1	2	yes	maximal, complemented
8	#13116774	C₂	1	2	yes	maximal
9	#13116769	C₂	1	2	yes	maximal, complemented
10	#13116643	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#541	—	2	1	no	—
13	C₂×#88163	—	2	1	no	—
14	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
15	C₂×#88158	C₂²	1	1	yes	characteristic
16	#11719231	—	2	1	no	—
17	C₁₃×#499	C₂²	1	1	yes	characteristic
18	C₁₃×#494	C₂²	1	1	yes	characteristic
19	C₁₃×#491	—	2	1	no	—
20	C₁₃×#470	—	2	1	no	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₁₃×#441	—	2	1	no	—
23	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
24	C₁₃×#405	C₂²	1	1	yes	characteristic
25	C₂×#88069	C₂²	1	2	yes	—
26	C₂×#88069	C₂²	1	2	yes	—
27	C₂×#88069	C₂²	1	2	yes	—
28	C₂×#88068	C₂²	1	2	yes	—
29	#11718376	—	2	1	no	—
30	#11718376	—	2	1	no	—
31	#11718360	C₂²	1	1	yes	characteristic
32	#11718360	C₂²	1	1	yes	characteristic
33	#11718357	C₂²	1	1	yes	characteristic
34	#11718357	C₂²	1	1	yes	characteristic
35	C₂²⋊#6090	C₂²	1	2	yes	—
36	#11718084	C₂²	1	2	yes	—
37	#11718084	C₂²	1	2	yes	—
38	#11718084	C₂²	1	2	yes	—
39	#11718083	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	2	yes	—
41	#11718053	C₂²	1	1	yes	characteristic
42	C₄×#6090	C₂²	1	1	yes	characteristic
43	C₂×#6127	—	2	1	no	—
44	C₂²×Dic₂₆	—	2	1	no	—
45	C₂×C₁₃×Q₁₆	—	2	1	no	—
46	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#126	—	4	2	no	—
51	C₁₃×#125	—	2	1	no	—
52	C₁₃×#125	—	4	2	no	—
53	C₁₃×C₂²⋊Q₈	—	2	2	no	—
54	C₄×Q₈×C₁₃	—	2	2	no	—
55	C₁₃×#117	D₄	1	1	yes	characteristic
56	#88163	—	4	4	no	—
57	C₂×Dic₅₂	—	2	1	no	—
58	C₂×Dic₅₂	—	2	1	no	—
59	C₂×Dic₅₂	—	2	1	no	—
60	C₂×Dic₅₂	—	4	2	no	—
61	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
63	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
64	#88158	—	4	2	no	—
65	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
66	C₂×#6095	—	2	1	no	—
67	C₂×#6095	—	2	1	no	—
68	C₂×#6095	—	2	1	no	—
69	#88151	—	4	2	no	—
70	#88140	—	4	2	no	—
71	#88129	—	2	1	no	—
72	#88129	—	2	1	no	—
73	#88123	—	2	1	no	—
74	#88123	—	2	1	no	—
75	C₁₃×#107	C₂³	1	2	yes	—
76	C₁₃×#106	—	2	1	no	—
77	C₁₃×#103	—	2	2	no	—
78	C₁₃×#103	—	2	2	no	—
79	C₁₃×#97	D₄	1	1	yes	characteristic
80	C₁₃×#97	D₄	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
83	#88069	—	4	2	no	—
84	#88069	—	4	2	no	—
85	#88069	—	4	2	no	—
86	#88068	—	4	2	no	—
87	C₂×#6090	C₂³	1	2	yes	—
88	C₂×#6090	C₂³	1	2	yes	—
89	#33789	—	13	1	no	maximal, Sylow
90	C₂×Q₈×C₁₃	—	2	2	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	—	4	1	no	—
93	C₂²×Dic₁₃	—	4	1	no	—
94	#6127	—	4	2	no	—
95	#6127	—	4	2	no	—
96	C₂×D₅₂	—	2	1	no	—
97	C₂×C₄×D₁₃	—	4	1	no	—
98	C₂×Dic₂₆	—	2	1	no	—
99	C₂×Dic₂₆	—	2	1	no	—
100	C₂×Dic₂₆	—	2	1	no	—
101	C₂×Dic₂₆	—	4	2	no	—
102	C₂×Dic₂₆	—	4	2	no	—
103	C₂×Dic₂₆	—	4	1	no	—
104	C₁₃×Q₁₆	—	2	2	no	—
105	C₁₃×Q₁₆	—	2	2	no	—
106	C₁₃×Q₁₆	—	4	2	no	—
107	C₁₃×M₁₆	—	2	2	no	—
108	C₂×C₈×C₁₃	—	2	2	no	—
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₁₃×C₄⋊C₄	—	2	2	no	—
114	C₁₃×C₄⋊C₄	—	4	2	no	—
115	C₁₃×C₄⋊C₄	—	4	2	no	—
116	C₁₃×C₄⋊C₄	—	4	2	no	—
117	C₁₃×C₄⋊C₄	—	4	2	no	—
118	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
119	C₁₃×C₂²⋊C₄	—	4	2	no	—
120	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
121	C₄²×C₁₃	—	4	2	no	—
122	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
124	#6103	—	4	2	no	—
125	#6103	—	4	2	no	—
126	C₄⋊Dic₁₃	—	4	2	no	—
127	C₄⋊Dic₁₃	—	4	2	no	—
128	Dic₅₂	—	2	2	no	—
129	Dic₅₂	—	2	2	no	—
130	Dic₅₂	—	2	2	no	—
131	Dic₅₂	—	4	2	no	—
132	Dic₅₂	—	4	2	no	—
133	Dic₅₂	—	4	2	no	—
134	D₁₀₄	—	2	2	no	—
135	#6095	—	4	2	no	—
136	#6095	—	4	2	no	—
137	#6095	—	4	2	no	—
138	#6090	—	2	2	no	—
139	#6090	—	2	2	no	—
140	#3177	D₁₃	1	1	yes	characteristic, complemented
141	#3156	—	13	1	no	—
142	#2251	—	13	1	no	—
143	#2247	—	13	1	no	—
144	#2245	—	13	1	no	—
145	#2242	—	13	1	no	—
146	#2211	—	13	2	no	—
147	#2209	—	13	2	no	—
148	#2189	—	13	2	no	—
149	#2188	—	13	2	no	—
150	#2164	—	13	1	no	—
151	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂²×D₁₃	—	4	1	no	—
153	Q₈×C₁₃	—	4	2	no	—
154	Q₈×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	2	1	no	—
156	C₂×C₄×C₁₃	—	2	1	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	—	4	2	no	—
159	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂²⋊D₁₃	—	8	2	no	—
164	C₂×Dic₁₃	—	4	1	no	—
165	C₂×Dic₁₃	—	4	1	no	—
166	C₂×Dic₁₃	—	4	1	no	—
167	C₂×Dic₁₃	—	8	2	no	—
168	D₅₂	—	4	1	no	—
169	D₅₂	—	4	1	no	—
170	C₄×D₁₃	—	8	2	no	—
171	Dic₂₆	—	4	1	no	—
172	Dic₂₆	—	4	1	no	—
173	Dic₂₆	—	4	1	no	—
174	Dic₂₆	—	4	1	no	—
175	Dic₂₆	—	4	1	no	—
176	Dic₂₆	—	4	1	no	—
177	Dic₂₆	—	8	2	no	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	—	2	1	no	—
180	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
181	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
182	C₂×#137	—	26	1	no	—
183	C₂×#135	—	13	1	no	—
184	C₂²×Q₁₆	—	13	1	no	—
185	#541	—	2	1	no	—
186	#536	—	26	1	no	—
187	C₂×Q₃₂	—	13	2	no	—
188	C₂×Q₃₂	—	13	2	no	—
189	C₂×Q₃₂	—	13	2	no	—
190	C₂×SD₃₂	—	13	2	no	—
191	#499	C₂×D₁₃	1	1	yes	characteristic
192	#497	—	26	1	no	—
193	#497	—	26	1	no	—
194	#495	—	13	1	no	—
195	#495	—	13	1	no	—
196	#494	—	13	1	no	—
197	#494	—	13	1	no	—
198	#494	C₂×D₁₃	1	1	yes	characteristic
199	#491	—	2	1	no	—
200	#470	—	2	1	no	—
201	#467	C₂×D₁₃	1	2	yes	—
202	#441	—	2	1	no	—
203	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
204	#405	C₂×D₁₃	1	1	yes	characteristic
205	C₄⋊C₁₆	—	13	2	no	—
206	#358	—	13	2	no	—
207	#358	—	13	2	no	—
208	#358	—	13	2	no	—
209	#357	—	13	2	no	—
210	C₂²⋊C₁₆	—	13	2	no	—
211	#346	—	13	1	no	—
212	C₄×C₁₆	—	13	1	no	—
213	C₂²×C₁₃	—	2	2	no	—
214	C₂²×C₁₃	—	2	1	no	—
215	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
216	C₂×D₁₃	—	8	2	no	—
217	C₂×D₁₃	—	8	1	no	—
218	C₄×C₁₃	—	2	2	no	—
219	C₄×C₁₃	—	2	2	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	8	2	no	—
222	C₄×C₁₃	—	8	2	no	—
223	C₄×C₁₃	#535	1	1	yes	characteristic
224	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	Dic₁₃	—	8	1	no	—
226	Dic₁₃	—	8	1	no	—
227	Dic₁₃	—	8	1	no	—
228	C₂×#41	—	26	1	no	—
229	C₂²×Q₈	—	26	1	no	—
230	#137	—	52	4	no	—
231	#135	—	52	2	no	—
232	C₂×Q₁₆	—	2	1	no	—
233	C₂×Q₁₆	—	13	1	no	—
234	C₂×Q₁₆	—	13	1	no	—
235	C₂×Q₁₆	—	13	1	no	—
236	C₂×Q₁₆	—	26	1	no	—
237	C₂×Q₁₆	—	26	1	no	—
238	C₂×Q₁₆	—	26	1	no	—
239	C₂×Q₁₆	—	52	2	no	—
240	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
241	C₂×SD₁₆	—	26	1	no	—
242	C₂×SD₁₆	—	26	1	no	—
243	C₂×SD₁₆	—	26	1	no	—
244	C₂×D₈	—	13	1	no	—
245	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	26	1	no	—
249	C₄⋊Q₈	—	26	1	no	—
250	#126	—	4	2	no	—
251	#125	—	2	1	no	—
252	#125	—	4	2	no	—
253	#124	—	26	1	no	—
254	#124	—	26	1	no	—
255	#123	—	52	2	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	52	2	no	—
258	C₄×Q₈	—	2	2	no	—
259	#117	C₂²⋊D₁₃	1	1	yes	characteristic
260	Q₃₂	—	52	2	no	—
261	Q₃₂	—	52	2	no	—
262	Q₃₂	—	52	2	no	—
263	SD₃₂	—	52	2	no	—
264	C₂×C₁₆	—	13	2	no	—
265	C₂×C₁₆	—	13	2	no	—
266	#107	C₂²×D₁₃	1	2	yes	—
267	#106	—	2	1	no	—
268	#103	—	2	2	no	—
269	#103	—	2	2	no	—
270	#97	C₂²⋊D₁₃	1	1	yes	characteristic
271	#97	C₂²⋊D₁₃	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
274	C₂×C₁₃	—	4	1	no	—
275	C₂×C₁₃	#3205	1	1	yes	characteristic
276	C₂×C₁₃	#3417	1	2	yes	—
277	D₁₃	—	16	1	no	—
278	#41	—	52	2	no	—
279	#41	—	52	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	26	1	no	—
282	C₂×Q₈	—	26	1	no	—
283	C₂×Q₈	—	26	1	no	—
284	C₂×Q₈	—	52	2	no	—
285	C₂×Q₈	—	52	2	no	—
286	C₂×Q₈	—	52	1	no	—
287	C₂×D₄	—	26	1	no	—
288	C₂×D₄	—	52	1	no	—
289	C₂²×C₄	—	52	1	no	—
290	C₂²×C₄	—	52	1	no	—
291	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	4	2	no	—
295	Q₁₆	—	26	2	no	—
296	Q₁₆	—	26	2	no	—
297	Q₁₆	—	26	2	no	—
298	Q₁₆	—	52	2	no	—
299	Q₁₆	—	52	2	no	—
300	Q₁₆	—	52	2	no	—
301	SD₁₆	—	52	2	no	—
302	SD₁₆	—	52	2	no	—
303	SD₁₆	—	52	2	no	—
304	D₈	—	26	2	no	—
305	M₁₆	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
308	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	52	2	no	—
317	C₄⋊C₄	—	52	2	no	—
318	C₄⋊C₄	#6104	1	2	yes	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	52	2	no	—
321	C₂²⋊C₄	—	52	2	no	—
322	C₂²⋊C₄	#6104	1	2	yes	—
323	C₄²	—	4	2	no	—
324	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
325	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
326	C₁₆	—	26	2	no	—
327	C₁₆	—	26	2	no	—
328	C₁₃	#33789	1	1	yes	characteristic, complemented, Sylow
329	C₂³	—	52	1	no	—
330	C₂³	C₂×#6104	1	1	yes	characteristic
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	52	1	no	—
334	Q₈	—	52	1	no	—
335	Q₈	—	52	1	no	—
336	Q₈	—	52	1	no	—
337	Q₈	—	52	1	no	—
338	Q₈	—	52	1	no	—
339	Q₈	—	104	2	no	—
340	D₄	—	52	1	no	—
341	D₄	—	52	1	no	—
342	D₄	—	104	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	52	1	no	—
348	C₂×C₄	—	52	1	no	—
349	C₂×C₄	—	52	1	no	—
350	C₂×C₄	—	104	2	no	—
351	C₂×C₄	—	104	2	no	—
352	C₂×C₄	C₂×#6104	1	2	yes	—
353	C₂×C₄	C₂×#6104	1	2	yes	—
354	C₂×C₄	C₂×#6104	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	#88203	1	2	yes	—
359	C₈	#88256	1	2	yes	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	104	2	no	—
363	C₂²	—	104	1	no	—
364	C₂²	C₂²×#6104	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	1	no	—
368	C₄	—	8	2	no	—
369	C₄	—	8	2	no	—
370	C₄	—	104	1	no	—
371	C₄	—	104	1	no	—
372	C₄	—	104	1	no	—
373	C₄	C₂×#88203	1	1	yes	characteristic
374	C₄	#11719535	1	1	yes	characteristic
375	C₂	—	4	1	no	—
376	C₂	—	208	1	no	—
377	C₂	#13125751	1	1	yes	characteristic
378	C₂	#13127987	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3177	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88158	C₂²
hypercenter	#3177	D₁₃
hypocenter	C₁₃	#33789
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #13117105	C₂	semidirect
3. #13117088	C₂	semidirect
4. #13117083	C₂	semidirect
5. #13117070	C₂	semidirect
6. #13116898	C₂	semidirect
7. #13116886	C₂	semidirect
9. #13116769	C₂	semidirect
10. #13116643	C₂	semidirect
11. C₁₃×#3177	C₂	semidirect
140. #3177	D₁₃	semidirect
328. C₁₃	#33789	semidirect

Maximal subgroups

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

subgroup	index
#13125330	2
#13117105	2
#13117088	2
#13117083	2
#13117070	2
#13116898	2
#13116886	2
#13116774	2
#13116769	2
#13116643	2
C₁₃×#3177	2
#33789	13