Subgroups — #16017493

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718562	C₂	1	1	yes	maximal, characteristic, complemented
2	#13124552	C₂	1	2	yes	maximal, complemented
3	#13116963	C₂	1	4	yes	maximal, complemented
4	#13116958	C₂	1	2	yes	maximal, complemented
5	#13116953	C₂	1	2	yes	maximal, complemented
6	C₂×#11718098	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116585	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	#11719527	—	2	2	no	—
11	C₂×#88155	C₂²	1	1	yes	characteristic
12	C₁₃×#508	—	2	1	no	—
13	C₁₃×#508	—	2	1	no	—
14	C₁₃×#508	—	2	1	no	—
15	C₁₃×#508	—	2	1	no	—
16	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
20	#11718898	—	2	2	no	—
21	#11718853	C₂²	1	4	yes	—
22	#11718838	—	2	4	no	—
23	#11718821	C₂²	1	4	yes	—
24	C₂×#88059	C₂²	1	2	yes	—
25	C₂×#88052	C₂²	1	1	yes	characteristic
26	#11718562	—	2	1	no	—
27	#11718562	—	2	1	no	—
28	#11718562	C₂²	1	2	yes	—
29	#11718562	C₂²	1	2	yes	—
30	C₂×#88038	—	2	1	no	—
31	C₂×C₁₆×D₁₃	—	2	1	no	—
32	#11718157	C₂²	1	4	yes	complemented
33	#11718155	C₂²	1	4	yes	complemented
34	#11718098	C₂²	1	2	yes	complemented
35	#11718098	C₂²	1	2	yes	complemented
36	C₂×C₁₃×#41	—	2	2	no	—
37	C₂×#6127	D₄	1	1	yes	characteristic
38	C₁₃×#135	—	2	2	no	—
39	C₁₃×#135	—	2	2	no	—
40	C₁₃×#135	—	2	2	no	—
41	C₁₃×#135	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	#88205	—	4	2	no	—
47	#88204	—	4	4	no	—
48	#88203	—	4	2	no	—
49	#88196	—	2	4	no	—
50	#88186	—	2	4	no	—
51	C₂×#6099	—	2	2	no	—
52	C₂×#6099	D₄	1	1	yes	characteristic
53	#88155	—	2	1	no	—
54	#88155	—	2	1	no	—
55	#88155	D₄	1	2	yes	—
56	#88155	C₂³	1	2	yes	—
57	C₂×#6094	—	2	1	no	—
58	C₂×C₈×D₁₃	—	2	1	no	—
59	C₁₃×Q₃₂	—	2	1	no	—
60	C₁₃×Q₃₂	—	2	1	no	—
61	C₁₃×SD₃₂	—	2	2	no	—
62	C₁₃×SD₃₂	—	2	2	no	—
63	C₁₃×D₁₆	—	2	1	no	—
64	C₁₃×D₁₆	—	2	1	no	—
65	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
66	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
67	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
68	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
71	#88076	—	2	4	no	—
72	#88073	—	2	4	no	—
73	#88059	—	2	2	no	—
74	#88059	C₂³	1	4	yes	—
75	#88052	—	2	1	no	—
76	#88052	C₂³	1	2	yes	—
77	C₂×#6090	—	2	1	no	—
78	#88038	—	2	2	no	—
79	#88038	—	2	2	no	—
80	C₁₆×D₁₃	—	2	2	no	—
81	C₁₆×D₁₃	—	2	2	no	—
82	#33617	—	13	1	no	maximal, Sylow
83	C₁₃×#41	—	4	2	no	—
84	C₁₃×#41	—	4	2	no	—
85	C₁₃×#41	—	4	2	no	—
86	C₁₃×#41	—	4	2	no	—
87	C₂×Q₈×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	—	4	2	no	—
90	C₂²×C₄×C₁₃	—	4	2	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	—	2	1	no	—
93	#6127	—	2	1	no	—
94	#6127	—	2	1	no	—
95	#6127	D₈	1	2	yes	—
96	#6127	C₂×D₄	1	2	yes	—
97	C₂×D₅₂	D₈	1	1	yes	characteristic
98	C₂×C₄×D₁₃	—	2	1	no	—
99	C₂×Dic₂₆	D₈	1	1	yes	characteristic
100	C₁₃×Q₁₆	—	2	2	no	—
101	C₁₃×Q₁₆	—	2	2	no	—
102	C₁₃×SD₁₆	—	4	2	no	—
103	C₁₃×SD₁₆	—	4	2	no	—
104	C₁₃×D₈	—	2	2	no	—
105	C₁₃×D₈	—	2	2	no	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
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	#6107	—	4	4	no	—
113	#6106	—	4	4	no	—
114	#6105	—	4	4	no	—
115	#6104	—	4	4	no	—
116	#6099	—	2	1	no	—
117	#6099	—	2	4	no	—
118	#6099	—	4	2	no	—
119	#6099	C₂×D₄	1	2	yes	—
120	C₂×#1055	—	2	1	no	—
121	C₂×#1055	—	4	2	no	—
122	#6094	—	2	2	no	—
123	#6094	—	2	2	no	—
124	C₈×D₁₃	—	2	2	no	—
125	C₈×D₁₃	—	2	2	no	—
126	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
127	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
128	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
129	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
130	#6090	—	2	2	no	—
131	C₂×#508	D₁₃	1	1	yes	characteristic, complemented
132	C₂×#504	—	13	1	no	—
133	#3067	—	13	2	no	—
134	#2223	—	13	4	no	—
135	#2222	—	13	2	no	—
136	#2221	—	13	2	no	—
137	C₂×#368	—	13	1	no	—
138	#2148	—	13	2	no	—
139	C₂³×C₁₃	—	4	2	no	—
140	C₂³×C₁₃	#135	1	1	yes	characteristic
141	C₂²×D₁₃	—	2	1	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	4	2	no	—
145	D₄×C₁₃	—	4	2	no	—
146	D₄×C₁₃	—	8	2	no	—
147	D₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	8	2	no	—
150	C₂×C₄×C₁₃	—	8	2	no	—
151	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₁₃	—	2	2	no	—
158	C₂²⋊D₁₃	—	2	2	no	—
159	C₂×Dic₁₃	—	2	1	no	—
160	D₅₂	—	2	1	no	—
161	D₅₂	C₂×D₈	1	2	yes	—
162	C₄×D₁₃	—	2	2	no	—
163	C₄×D₁₃	—	2	2	no	—
164	Dic₂₆	—	2	1	no	—
165	Dic₂₆	C₂×D₈	1	2	yes	—
166	C₈×C₁₃	#121	1	1	yes	characteristic
167	C₈×C₁₃	#121	1	1	yes	characteristic
168	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
169	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
170	#1055	—	2	2	no	—
171	#1055	—	4	4	no	—
172	#575	—	26	2	no	—
173	C₂×#135	C₂×D₁₃	1	2	yes	—
174	C₂×#131	—	13	1	no	—
175	#508	—	2	1	no	—
176	#508	—	2	1	no	—
177	#508	—	2	1	no	—
178	#508	—	2	1	no	—
179	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic, complemented
180	C₂×SD₃₂	C₂×D₁₃	1	2	yes	complemented
181	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
182	#504	—	13	2	no	—
183	#504	—	13	2	no	—
184	#504	—	26	1	no	—
185	#504	—	26	1	no	—
186	C₂×M₃₂	—	13	1	no	—
187	C₂×M₃₂	—	26	1	no	—
188	C₂²×C₁₆	—	26	1	no	—
189	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
190	#473	—	13	4	no	—
191	#472	—	13	4	no	—
192	#471	—	26	4	no	—
193	C₂×#108	—	13	2	no	—
194	#413	—	26	2	no	—
195	#368	—	13	2	no	—
196	#368	—	13	2	no	—
197	#362	—	13	4	no	—
198	#361	—	13	4	no	—
199	C₂²×C₁₃	—	4	1	no	—
200	C₂²×C₁₃	—	8	2	no	—
201	C₂²×C₁₃	—	16	2	no	—
202	C₂²×C₁₃	#466	1	1	yes	characteristic
203	C₂²×C₁₃	#466	1	1	yes	characteristic
204	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
205	C₂×D₁₃	—	2	2	no	—
206	C₂×D₁₃	—	4	2	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	#466	1	1	yes	characteristic
209	C₄×C₁₃	#466	1	1	yes	characteristic
210	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
211	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
212	Dic₁₃	—	2	2	no	—
213	C₂×#41	—	2	2	no	—
214	C₂×#41	—	13	1	no	—
215	#137	—	26	4	no	—
216	#137	—	52	2	no	—
217	AGL(1,ℤ/8)	—	26	4	no	—
218	AGL(1,ℤ/8)	—	52	2	no	—
219	#135	—	2	2	no	—
220	#135	—	2	2	no	—
221	#135	—	2	2	no	—
222	#135	—	2	2	no	—
223	#135	—	52	4	no	—
224	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	C₂²×D₁₃	1	2	yes	—
227	#131	—	13	2	no	—
228	#131	—	13	2	no	—
229	#131	—	26	1	no	—
230	#131	—	26	1	no	—
231	C₂×M₁₆	—	13	1	no	—
232	C₂×M₁₆	—	26	1	no	—
233	C₂×M₁₆	—	26	2	no	—
234	C₂²×C₈	—	26	1	no	—
235	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	Q₃₂	—	2	1	no	—
237	Q₃₂	—	2	1	no	—
238	SD₃₂	—	2	2	no	—
239	SD₃₂	—	2	2	no	—
240	D₁₆	—	2	1	no	—
241	D₁₆	—	2	1	no	—
242	M₃₂	—	13	2	no	—
243	M₃₂	—	26	1	no	—
244	M₃₂	—	26	2	no	—
245	M₃₂	—	26	2	no	—
246	C₂×C₁₆	—	26	1	no	—
247	C₂×C₁₆	—	26	2	no	—
248	C₂×C₁₆	—	26	2	no	—
249	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
252	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
254	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
255	#108	—	13	4	no	—
256	#108	—	26	2	no	—
257	#101	—	26	4	no	—
258	#100	—	26	4	no	—
259	C₂×C₁₃	—	2	1	no	—
260	C₂×C₁₃	—	2	1	no	—
261	C₂×C₁₃	—	2	1	no	—
262	C₂×C₁₃	—	16	2	no	—
263	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
264	D₁₃	—	4	2	no	—
265	#41	—	4	2	no	—
266	#41	—	4	2	no	—
267	#41	—	4	2	no	—
268	#41	—	4	2	no	—
269	#41	—	13	2	no	—
270	#41	—	13	2	no	—
271	#41	—	26	1	no	—
272	#41	—	26	1	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	13	1	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	13	1	no	—
278	C₂×D₄	—	26	1	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	—	26	1	no	—
281	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
282	Q₁₆	—	2	2	no	—
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	52	4	no	—
285	SD₁₆	—	4	2	no	—
286	SD₁₆	—	4	2	no	—
287	SD₁₆	—	52	4	no	—
288	SD₁₆	—	52	4	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	52	4	no	—
292	M₁₆	—	13	2	no	—
293	M₁₆	—	26	1	no	—
294	M₁₆	—	26	2	no	—
295	M₁₆	—	26	2	no	—
296	M₁₆	—	26	4	no	—
297	M₁₆	—	52	2	no	—
298	C₂×C₈	—	26	1	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	—	52	2	no	—
302	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
303	C₂×C₈	#6129	1	1	yes	characteristic
304	C₂×C₈	#6129	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
308	C₁₆	—	26	2	no	—
309	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
311	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
312	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
313	C₁₃	#33617	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	4	2	no	—
315	C₂³	—	26	1	no	—
316	C₂³	#88166	1	1	yes	characteristic
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	13	2	no	—
320	Q₈	—	26	1	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	8	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	13	2	no	—
326	D₄	—	26	2	no	—
327	D₄	—	26	2	no	—
328	D₄	—	26	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	26	1	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
336	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
337	C₂×C₄	#88166	1	1	yes	characteristic
338	C₂×C₄	#88192	1	1	yes	characteristic
339	C₂×C₄	#88192	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
341	C₈	—	26	2	no	—
342	C₈	—	52	4	no	—
343	C₈	#88192	1	1	yes	characteristic
344	C₈	#88192	1	1	yes	characteristic
345	C₈	C₂×#6129	1	1	yes	characteristic
346	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	16	2	no	—
350	C₂²	—	26	2	no	—
351	C₂²	—	52	2	no	—
352	C₂²	#11718818	1	1	yes	characteristic
353	C₂²	#11718818	1	1	yes	characteristic
354	C₂²	C₂×#88166	1	1	yes	characteristic
355	C₄	—	8	2	no	—
356	C₄	—	26	2	no	—
357	C₄	#11718818	1	1	yes	characteristic
358	C₄	#11718818	1	1	yes	characteristic
359	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
360	C₄	C₂×#88192	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	16	2	no	—
365	C₂	—	52	2	no	—
366	C₂	C₂×#11718818	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718818
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#508	C₂
socle	C₂×C₁₃	C₂×#466
hypercenter	C₂×#508	D₁₃
hypocenter	C₁₃	#33617
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#11718562	C₂	semidirect
2. #13124552	C₂	semidirect
3. #13116963	C₂	semidirect
4. #13116958	C₂	semidirect
5. #13116953	C₂	semidirect
6. C₂×#11718098	C₂	semidirect
7. #13116585	C₂	semidirect
8. C₂×C₁₃×#508	C₂	semidirect
16. C₂×C₁₃×Q₃₂	C₂²	semidirect
17. C₂×C₁₃×SD₃₂	C₂²	semidirect
18. C₂×C₁₃×D₁₆	C₂²	semidirect
32. #11718157	C₂²	semidirect
33. #11718155	C₂²	semidirect
34. #11718098	C₂²	semidirect
35. #11718098	C₂²	semidirect
131. C₂×#508	D₁₃	semidirect
179. C₂×Q₃₂	C₂×D₁₃	semidirect
180. C₂×SD₃₂	C₂×D₁₃	semidirect
181. C₂×D₁₆	C₂×D₁₃	semidirect
313. C₁₃	#33617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718562	2
#13124552	2
#13116963	2
#13116958	2
#13116953	2
C₂×#11718098	2
#13116585	2
C₂×C₁₃×#508	2
#33617	13