Subgroups — #16016394

Order 3328.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718854	C₂	1	1	yes	maximal, characteristic
2	#13123689	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116734	C₂	1	8	yes	maximal, complemented
4	#13116610	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718087	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116572	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116441	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2957	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₁₃×#568	—	2	1	no	—
11	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
12	#11719226	—	2	1	no	—
13	C₁₃×#444	C₂²	1	4	yes	complemented
14	C₁₃×#443	—	2	2	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₁₃×#421	—	2	1	no	—
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	#11718854	C₄	1	4	yes	complemented
19	#11718854	C₂²	1	4	yes	—
20	C₂×#88068	C₄	1	2	yes	complemented
21	C₂×#88066	C₄	1	2	yes	complemented
22	#11718792	—	2	1	no	—
23	#11718767	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂×#88052	C₂²	1	1	yes	characteristic
26	#11718372	C₂²	1	4	yes	complemented
27	C₄×D₁₀₄	—	2	2	no	—
28	#11718158	C₂²	1	4	yes	complemented
29	#11718087	C₂²	1	4	yes	complemented
30	#11718086	C₂²	1	4	yes	complemented
31	#11718058	C₂²	1	4	yes	complemented
32	C₂×C₁₃×#41	—	2	1	no	—
33	C₂²×D₅₂	—	2	1	no	—
34	C₁₃×#135	—	2	1	no	—
35	C₁₃×#135	—	2	1	no	—
36	C₁₃×#135	C₂×C₄	1	2	yes	complemented
37	C₁₃×#135	C₂³	1	2	yes	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×C₁₃×SD₁₆	—	2	1	no	—
40	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
41	C₁₃×#131	—	2	4	no	—
42	C₁₃×#131	—	4	2	no	—
43	C₂×C₁₃×M₁₆	—	2	1	no	—
44	C₂×C₁₃×M₁₆	—	4	2	no	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#117	D₄	1	1	yes	characteristic
48	#88180	—	2	1	no	—
49	C₂×D₁₀₄	—	2	1	no	—
50	C₂×D₁₀₄	—	2	1	no	—
51	C₂×D₁₀₄	—	2	1	no	—
52	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
53	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
54	C₂×D₁₀₄	C₂³	1	2	yes	—
55	#88147	—	4	2	no	—
56	C₂²⋊C₄×D₁₃	—	4	2	no	—
57	C₄×D₅₂	—	2	4	no	—
58	C₁₃×#108	—	2	1	no	—
59	C₁₃×#108	C₂³	1	2	yes	—
60	C₁₃×#104	—	2	4	no	—
61	C₁₃×#97	C₂³	1	2	yes	—
62	C₄×C₈×C₁₃	D₄	1	2	yes	—
63	#88068	C₂×C₄	1	8	yes	complemented
64	#88066	C₂×C₄	1	8	yes	complemented
65	#88061	—	2	4	no	—
66	#88058	—	2	1	no	—
67	#88057	C₂³	1	2	yes	—
68	#88052	—	2	1	no	—
69	#88052	C₂×C₄	1	2	yes	complemented
70	#88052	C₂³	1	2	yes	—
71	#88052	C₂³	1	2	yes	—
72	C₂×#6090	—	2	1	no	—
73	C₂×#6090	C₂×C₄	1	2	yes	complemented
74	#33471	—	13	1	no	maximal, Sylow
75	C₂³×D₁₃	—	4	1	no	—
76	C₁₃×#41	—	2	2	no	—
77	C₁₃×#41	—	2	2	no	—
78	C₁₃×#41	—	4	1	no	—
79	C₁₃×#41	—	4	1	no	—
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×D₅₂	—	2	2	no	—
86	C₂×D₅₂	—	2	2	no	—
87	C₂×D₅₂	—	2	2	no	—
88	C₂×D₅₂	—	4	1	no	—
89	C₂×D₅₂	—	4	1	no	—
90	C₂×D₅₂	—	4	1	no	—
91	C₂×C₄×D₁₃	—	4	4	no	—
92	C₁₃×Q₁₆	—	2	1	no	—
93	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
94	C₁₃×SD₁₆	—	2	2	no	—
95	C₁₃×SD₁₆	—	2	2	no	—
96	C₁₃×D₈	—	2	1	no	—
97	C₁₃×D₈	C₂²×C₄	1	2	yes	—
98	C₁₃×M₁₆	—	2	1	no	—
99	C₁₃×M₁₆	—	2	1	no	—
100	C₁₃×M₁₆	—	2	2	no	—
101	C₁₃×M₁₆	—	4	4	no	—
102	C₁₃×M₁₆	—	4	1	no	—
103	C₂×C₈×C₁₃	—	4	4	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	—	2	1	no	—
113	C₁₃×C₂²⋊C₄	—	2	1	no	—
114	C₄²×C₁₃	C₂×D₄	1	2	yes	—
115	C₂²⋊Dic₁₃	—	4	1	no	—
116	#6103	—	4	4	no	—
117	C₄⋊Dic₁₃	—	2	2	no	—
118	C₄×Dic₁₃	—	4	1	no	—
119	D₁₀₄	—	2	2	no	—
120	D₁₀₄	—	2	2	no	—
121	D₁₀₄	—	2	2	no	—
122	D₁₀₄	C₂²×C₄	1	4	yes	—
123	#6090	—	2	2	no	—
124	#6090	C₂²×C₄	1	4	yes	—
125	C₂×#509	—	13	1	no	—
126	#2957	D₁₃	1	1	yes	characteristic, complemented
127	#2948	—	13	1	no	—
128	AGL(1,ℤ/16)	—	13	8	no	—
129	#2155	—	13	1	no	—
130	C₂×#361	—	13	1	no	—
131	#2145	—	13	1	no	—
132	#2110	—	13	1	no	—
133	C₂³×C₁₃	—	4	1	no	—
134	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂²×D₁₃	—	4	2	no	—
136	C₂²×D₁₃	—	4	2	no	—
137	C₂²×D₁₃	—	4	2	no	—
138	C₂²×D₁₃	—	16	2	no	—
139	Q₈×C₁₃	—	2	2	no	—
140	Q₈×C₁₃	—	4	1	no	—
141	D₄×C₁₃	—	2	2	no	—
142	D₄×C₁₃	—	4	2	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×Dic₁₃	—	4	2	no	—
156	D₅₂	—	2	4	no	—
157	D₅₂	—	4	2	no	—
158	D₅₂	—	4	2	no	—
159	D₅₂	—	4	2	no	—
160	C₄×D₁₃	—	16	4	no	—
161	C₈×C₁₃	—	2	2	no	—
162	C₈×C₁₃	—	4	2	no	—
163	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
164	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
165	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
166	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
167	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
168	C₂²×D₈	—	13	1	no	—
169	#568	—	2	1	no	—
170	#518	—	26	1	no	—
171	#509	—	13	4	no	—
172	#509	—	13	4	no	—
173	C₂×SD₃₂	—	13	2	no	—
174	C₂×D₁₆	—	13	2	no	—
175	C₂×M₃₂	—	13	1	no	—
176	C₂×M₃₂	—	13	1	no	—
177	#444	C₂×D₁₃	1	4	yes	complemented
178	#443	—	2	2	no	—
179	#442	—	13	4	no	—
180	C₄×D₈	—	26	2	no	—
181	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
182	#427	—	13	1	no	—
183	#421	—	2	1	no	—
184	#418	—	26	1	no	—
185	#405	C₂×D₁₃	1	1	yes	characteristic
186	#365	—	13	4	no	—
187	#361	—	13	4	no	—
188	#360	—	13	4	no	—
189	#347	—	13	4	no	—
190	C₂²×C₁₃	—	4	1	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	16	1	no	—
193	C₂²×C₁₃	#442	1	1	yes	characteristic
194	C₂²×C₁₃	#442	1	1	yes	characteristic
195	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₂×D₁₃	—	4	4	no	—
197	C₂×D₁₃	—	8	4	no	—
198	C₂×D₁₃	—	16	2	no	—
199	C₂×D₁₃	—	16	2	no	—
200	C₄×C₁₃	—	4	2	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	#442	1	1	yes	characteristic
203	C₄×C₁₃	#442	1	1	yes	characteristic
204	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
205	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	Dic₁₃	—	8	2	no	—
207	C₂×#41	—	2	1	no	—
208	C₂²×D₄	—	26	1	no	—
209	#135	—	2	1	no	—
210	#135	—	2	1	no	—
211	#135	C₄×D₁₃	1	2	yes	complemented
212	#135	C₂²×D₁₃	1	2	yes	—
213	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic, complemented
214	C₂×SD₁₆	—	2	1	no	—
215	C₂×D₈	—	13	2	no	—
216	C₂×D₈	—	13	2	no	—
217	C₂×D₈	—	13	2	no	—
218	C₂×D₈	—	26	1	no	—
219	C₂×D₈	—	26	1	no	—
220	C₂×D₈	—	26	1	no	—
221	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic, complemented
222	#131	—	2	4	no	—
223	#131	—	4	2	no	—
224	C₂×M₁₆	—	2	1	no	—
225	C₂×M₁₆	—	4	2	no	—
226	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₄×D₄	—	26	4	no	—
229	C₄×D₄	—	52	2	no	—
230	#117	—	26	1	no	—
231	#117	C₂²⋊D₁₃	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	—	52	2	no	—
233	SD₃₂	—	13	8	no	—
234	D₁₆	—	13	8	no	—
235	M₃₂	—	13	2	no	—
236	M₃₂	—	13	2	no	—
237	M₃₂	—	13	2	no	—
238	M₃₂	—	26	1	no	—
239	C₂×C₁₆	—	13	2	no	—
240	C₂×C₁₆	—	26	1	no	—
241	#108	—	2	1	no	—
242	#108	C₂²×D₁₃	1	2	yes	—
243	#107	—	26	1	no	—
244	#106	—	13	2	no	—
245	#104	—	2	4	no	—
246	#102	—	26	4	no	—
247	#97	C₂²×D₁₃	1	2	yes	—
248	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
249	C₂×C₁₃	—	2	1	no	—
250	C₂×C₁₃	—	2	1	no	—
251	C₂×C₁₃	—	2	1	no	—
252	C₂×C₁₃	—	8	2	no	—
253	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
254	D₁₃	—	8	4	no	—
255	C₂⁴	—	52	1	no	—
256	#41	—	2	2	no	—
257	#41	—	2	2	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	26	2	no	—
264	C₂×D₄	—	26	2	no	—
265	C₂×D₄	—	26	2	no	—
266	C₂×D₄	—	52	1	no	—
267	C₂×D₄	—	52	1	no	—
268	C₂×D₄	—	52	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	52	4	no	—
271	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
272	Q₁₆	—	2	1	no	—
273	Q₁₆	C₂×C₄×D₁₃	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	D₈	—	2	1	no	—
277	D₈	—	13	4	no	—
278	D₈	—	26	2	no	—
279	D₈	—	26	2	no	—
280	D₈	—	26	2	no	—
281	D₈	C₂×C₄×D₁₃	1	2	yes	—
282	M₁₆	—	2	1	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	4	4	no	—
286	M₁₆	—	4	1	no	—
287	C₂×C₈	—	4	4	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
291	C₂×C₈	#6127	1	1	yes	characteristic
292	C₂×C₈	#6127	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
294	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	52	4	no	—
300	C₂²⋊C₄	—	52	1	no	—
301	C₄²	—	52	1	no	—
302	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
303	C₁₆	—	13	4	no	—
304	C₁₆	—	26	2	no	—
305	C₁₃	#33471	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	4	1	no	—
307	C₂³	—	52	2	no	—
308	C₂³	—	52	2	no	—
309	C₂³	—	52	2	no	—
310	C₂³	—	208	2	no	—
311	C₂³	#88203	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	26	4	no	—
319	D₄	—	52	2	no	—
320	D₄	—	52	2	no	—
321	D₄	—	52	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	208	4	no	—
328	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
329	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
330	C₂×C₄	#88186	1	1	yes	characteristic
331	C₂×C₄	#88186	1	1	yes	characteristic
332	C₂×C₄	#88203	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	4	2	no	—
336	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
337	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
339	C₈	C₂×#6127	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	16	1	no	—
343	C₂²	—	52	4	no	—
344	C₂²	—	104	4	no	—
345	C₂²	—	208	2	no	—
346	C₂²	—	208	2	no	—
347	C₂²	#11718675	1	1	yes	characteristic
348	C₂²	#11718675	1	1	yes	characteristic
349	C₂²	C₂×#88203	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	104	2	no	—
353	C₄	#11718675	1	1	yes	characteristic
354	C₄	#11718675	1	1	yes	characteristic
355	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
356	C₄	C₂×#88186	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	2	no	—
361	C₂	—	104	4	no	—
362	C₂	C₂×#11718675	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718675
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2957	C₂
socle	C₂×C₁₃	C₂×#442
prime radical	C₂×#11718854	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2957	D₁₃
hypocenter	C₁₃	#33471
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2957

Complements

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

N	H ≅ G/N	type
2. #13123689	C₂	semidirect
3. #13116734	C₂	semidirect
4. #13116610	C₂	semidirect
5. C₂×#11718087	C₂	semidirect
6. #13116572	C₂	semidirect
7. #13116441	C₂	semidirect
8. C₁₃×#2957	C₂	semidirect
13. C₁₃×#444	C₂²	semidirect
18. #11718854	C₄	semidirect
20. C₂×#88068	C₄	semidirect
21. C₂×#88066	C₄	semidirect
26. #11718372	C₂²	semidirect
28. #11718158	C₂²	semidirect
29. #11718087	C₂²	semidirect
30. #11718086	C₂²	semidirect
31. #11718058	C₂²	semidirect
36. C₁₃×#135	C₂×C₄	semidirect
38. C₂×C₁₃×Q₁₆	C₂×C₄	semidirect
40. C₂×C₁₃×D₈	C₂×C₄	semidirect
63. #88068	C₂×C₄	semidirect
64. #88066	C₂×C₄	semidirect
69. #88052	C₂×C₄	semidirect
73. C₂×#6090	C₂×C₄	semidirect
126. #2957	D₁₃	semidirect
177. #444	C₂×D₁₃	semidirect
211. #135	C₄×D₁₃	semidirect
213. C₂×Q₁₆	C₄×D₁₃	semidirect
221. C₂×D₈	C₄×D₁₃	semidirect
305. C₁₃	#33471	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718854	2
#13123689	2
#13116734	2
#13116610	2
C₂×#11718087	2
#13116572	2
#13116441	2
C₁₃×#2957	2
#33471	13