Subgroups — #16018896

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718357	C₂	1	1	yes	maximal, characteristic
2	#13117105	C₂	1	2	yes	maximal, complemented
3	#13117075	C₂	1	2	yes	maximal, complemented
4	#13116898	C₂	1	4	yes	maximal, complemented
5	#13116769	C₂	1	2	yes	maximal, complemented
6	#13116764	C₂	1	2	yes	maximal
7	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#533	—	2	1	no	—
10	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
11	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
12	C₂²×#6095	—	2	1	no	—
13	C₂×#88129	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
16	C₁₃×#487	—	2	1	no	—
17	C₁₃×#467	—	2	1	no	—
18	C₁₃×#467	C₂²	1	2	yes	—
19	C₄×C₁₃×Q₁₆	—	2	1	no	—
20	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×#88069	C₂²	1	4	yes	complemented
23	C₂×#88068	C₂²	1	4	yes	—
24	#11718357	—	2	1	no	—
25	#11718357	—	2	1	no	—
26	#11718357	C₂²	1	2	yes	—
27	#11718357	C₂²	1	2	yes	—
28	C₂²⋊#6090	C₂²	1	2	yes	—
29	#11718084	C₂²	1	4	yes	complemented
30	#11718083	C₂²	1	4	yes	—
31	C₄⋊#6090	C₂²	1	2	yes	complemented
32	#11718053	C₂²	1	2	yes	complemented
33	C₂²×D₅₂	—	2	1	no	—
34	C₂²×Dic₂₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×#125	—	4	1	no	—
42	C₁₃×C₂²⋊Q₈	—	2	2	no	—
43	C₁₃×C₂²⋊Q₈	—	4	1	no	—
44	C₄×Q₈×C₁₃	—	2	2	no	—
45	C₄×Q₈×C₁₃	—	4	1	no	—
46	C₁₃×#117	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₂×#6103	—	4	1	no	—
49	C₂×#6103	—	4	1	no	—
50	C₂×Dic₅₂	—	2	1	no	—
51	C₂×Dic₅₂	—	4	2	no	—
52	C₂×Dic₅₂	C₂³	1	2	yes	—
53	C₂×D₁₀₄	—	2	1	no	—
54	C₂×D₁₀₄	—	4	2	no	—
55	C₂×D₁₀₄	C₂³	1	2	yes	—
56	C₂×#6095	—	2	2	no	—
57	C₂×#6095	—	2	2	no	—
58	C₂×#6095	—	4	4	no	—
59	#88129	—	2	2	no	—
60	#88129	—	2	2	no	—
61	#88129	—	4	1	no	—
62	#88129	—	4	1	no	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#103	—	2	2	no	—
66	C₁₃×#103	—	2	2	no	—
67	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#88069	—	4	4	no	—
72	#88068	—	4	4	no	—
73	C₂×#6090	C₂³	1	4	yes	—
74	#33781	—	13	1	no	maximal, Sylow
75	C₂³×D₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₁₃	—	4	1	no	—
82	C₂×D₅₂	—	2	2	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×D₅₂	—	4	2	no	—
85	C₂×D₅₂	—	4	1	no	—
86	C₂×Dic₂₆	—	2	2	no	—
87	C₂×Dic₂₆	—	4	2	no	—
88	C₂×Dic₂₆	—	4	2	no	—
89	C₂×Dic₂₆	—	4	1	no	—
90	C₁₃×Q₁₆	—	2	4	no	—
91	C₁₃×Q₁₆	—	4	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₄²×C₁₃	—	4	2	no	—
106	C₄²×C₁₃	D₈	1	1	yes	characteristic
107	C₄²×C₁₃	D₈	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	#6103	—	4	2	no	—
111	#6103	—	4	2	no	—
112	#6103	—	4	2	no	—
113	#6103	—	4	2	no	—
114	Dic₅₂	—	2	4	no	—
115	Dic₅₂	—	4	2	no	—
116	D₁₀₄	—	2	4	no	—
117	D₁₀₄	—	4	2	no	—
118	#6095	—	4	4	no	—
119	#6095	—	4	4	no	—
120	#6090	—	2	4	no	—
121	#3170	D₁₃	1	1	yes	characteristic, complemented
122	C₂×#495	—	13	1	no	—
123	#2251	—	13	2	no	—
124	#2243	—	13	2	no	—
125	#2211	—	13	4	no	—
126	#2188	—	13	2	no	—
127	#2187	—	13	2	no	—
128	#2163	—	13	1	no	—
129	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₁₃	—	4	2	no	—
131	C₂²×D₁₃	—	8	2	no	—
132	C₂²×D₁₃	—	8	2	no	—
133	C₂²×D₁₃	—	8	2	no	—
134	Q₈×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	4	2	no	—
136	Q₈×C₁₃	—	8	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	8	2	no	—
151	D₅₂	—	4	2	no	—
152	D₅₂	—	4	2	no	—
153	D₅₂	—	8	2	no	—
154	Dic₂₆	—	4	2	no	—
155	Dic₂₆	—	4	2	no	—
156	Dic₂₆	—	8	2	no	—
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	—	2	1	no	—
159	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
160	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
161	C₂²×Q₁₆	—	13	1	no	—
162	C₂²×SD₁₆	—	26	1	no	—
163	C₂²×D₈	—	13	1	no	—
164	#533	—	2	1	no	—
165	C₂×#124	—	26	1	no	—
166	C₂×Q₃₂	—	13	4	no	—
167	C₂×SD₃₂	—	13	4	no	—
168	#500	C₂×D₁₃	1	1	yes	characteristic, complemented
169	#495	—	13	2	no	—
170	#495	—	13	2	no	—
171	#495	—	26	1	no	—
172	#495	—	26	1	no	—
173	#494	C₂×D₁₃	1	1	yes	characteristic, complemented
174	#487	—	2	1	no	—
175	#467	—	2	1	no	—
176	#467	C₂×D₁₃	1	2	yes	—
177	C₄×Q₁₆	—	2	1	no	—
178	C₄×Q₁₆	C₂×D₁₃	1	2	yes	complemented
179	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
180	C₄⋊C₁₆	—	13	2	no	—
181	#358	—	13	4	no	—
182	#357	—	13	4	no	—
183	C₂²⋊C₁₆	—	13	2	no	—
184	#346	—	13	2	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	#510	1	2	yes	—
188	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
189	C₂×D₁₃	—	8	2	no	—
190	C₂×D₁₃	—	8	2	no	—
191	C₂×D₁₃	—	8	2	no	—
192	C₂×D₁₃	—	16	4	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	#535	1	1	yes	characteristic
201	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	Dic₁₃	—	8	2	no	—
203	C₂²×Q₈	—	26	1	no	—
204	C₂²×D₄	—	26	1	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	13	2	no	—
207	C₂×Q₁₆	—	26	1	no	—
208	C₂×Q₁₆	—	52	2	no	—
209	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
210	C₂×SD₁₆	—	26	2	no	—
211	C₂×SD₁₆	—	26	2	no	—
212	C₂×SD₁₆	—	52	4	no	—
213	C₂×D₈	—	13	2	no	—
214	C₂×D₈	—	26	1	no	—
215	C₂×D₈	—	52	2	no	—
216	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#125	—	4	1	no	—
221	#124	—	26	2	no	—
222	#124	—	26	2	no	—
223	#124	—	52	1	no	—
224	#124	—	52	1	no	—
225	C₂²⋊Q₈	—	2	2	no	—
226	C₂²⋊Q₈	—	4	1	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×Q₈	—	4	1	no	—
229	#117	—	4	1	no	—
230	C₂×C₂²⋊C₄	—	52	1	no	—
231	C₂×C₂²⋊C₄	—	52	1	no	—
232	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
233	Q₃₂	—	52	4	no	—
234	SD₃₂	—	52	4	no	—
235	C₂×C₁₆	—	13	4	no	—
236	#107	—	2	1	no	—
237	#107	C₂²×D₁₃	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
241	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×C₁₃	—	2	2	no	—
245	C₂×C₁₃	#3205	1	1	yes	characteristic
246	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
247	C₂×C₁₃	#3414	1	1	yes	characteristic
248	D₁₃	—	16	2	no	—
249	C₂⁴	—	52	1	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	26	2	no	—
253	C₂×Q₈	—	52	2	no	—
254	C₂×Q₈	—	52	2	no	—
255	C₂×Q₈	—	52	1	no	—
256	C₂×D₄	—	26	2	no	—
257	C₂×D₄	—	52	2	no	—
258	C₂×D₄	—	52	2	no	—
259	C₂×D₄	—	52	1	no	—
260	C₂²×C₄	—	52	1	no	—
261	C₂²×C₄	#6104	1	1	yes	characteristic
262	C₂²×C₄	#6104	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	Q₁₆	—	2	4	no	—
265	Q₁₆	—	4	2	no	—
266	Q₁₆	—	26	4	no	—
267	Q₁₆	—	52	2	no	—
268	SD₁₆	—	52	4	no	—
269	SD₁₆	—	52	4	no	—
270	D₈	—	26	4	no	—
271	D₈	—	52	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
276	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	52	2	no	—
286	C₂²⋊C₄	—	52	2	no	—
287	C₂²⋊C₄	—	52	2	no	—
288	C₂²⋊C₄	—	52	2	no	—
289	C₄²	—	4	2	no	—
290	C₄²	#6104	1	1	yes	characteristic
291	C₄²	#6104	1	1	yes	characteristic
292	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
294	C₁₆	—	26	4	no	—
295	C₁₃	#33781	1	1	yes	characteristic, complemented, Sylow
296	C₂³	—	52	2	no	—
297	C₂³	—	104	2	no	—
298	C₂³	—	104	2	no	—
299	C₂³	—	104	2	no	—
300	C₂³	C₂×#6104	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	Q₈	—	52	2	no	—
305	Q₈	—	52	2	no	—
306	Q₈	—	104	2	no	—
307	D₄	—	52	2	no	—
308	D₄	—	52	2	no	—
309	D₄	—	104	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	52	2	no	—
317	C₂×C₄	—	104	2	no	—
318	C₂×C₄	C₂×#6104	1	1	yes	characteristic
319	C₂×C₄	C₂×#6104	1	1	yes	characteristic
320	C₂×C₄	C₂×#6104	1	1	yes	characteristic
321	C₂×C₄	C₂×#6104	1	1	yes	characteristic
322	C₂×C₄	C₂×#6104	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	#88203	1	2	yes	—
327	C₈	#88256	1	2	yes	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	104	2	no	—
331	C₂²	—	104	2	no	—
332	C₂²	—	104	2	no	—
333	C₂²	—	208	4	no	—
334	C₂²	#11718840	1	2	yes	—
335	C₂²	C₂²×#6104	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	104	2	no	—
344	C₄	C₂×#88203	1	1	yes	characteristic
345	C₄	#11719535	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	208	2	no	—
348	C₂	#13125751	1	1	yes	characteristic
349	C₂	C₂×#11718840	1	1	yes	characteristic
350	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#33781
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13117105	C₂	semidirect
3. #13117075	C₂	semidirect
4. #13116898	C₂	semidirect
5. #13116769	C₂	semidirect
7. #13116639	C₂	semidirect
8. C₁₃×#3170	C₂	semidirect
14. C₁₃×#500	C₂²	semidirect
15. C₁₃×#494	C₂²	semidirect
20. C₄×C₁₃×Q₁₆	C₂²	semidirect
22. C₂×#88069	C₂²	semidirect
29. #11718084	C₂²	semidirect
31. C₄⋊#6090	C₂²	semidirect
32. #11718053	C₂²	semidirect
121. #3170	D₁₃	semidirect
168. #500	C₂×D₁₃	semidirect
173. #494	C₂×D₁₃	semidirect
178. C₄×Q₁₆	C₂×D₁₃	semidirect
295. C₁₃	#33781	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718357	2
#13117105	2
#13117075	2
#13116898	2
#13116769	2
#13116764	2
#13116639	2
C₁₃×#3170	2
#33781	13