Subgroups — #16019794

Order 3328.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127184	C₂	1	1	yes	maximal, characteristic
2	#13117063	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117031	C₂	1	2	yes	maximal, complemented
4	#13116978	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116934	C₂	1	2	yes	maximal, complemented
6	#13116878	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116753	C₂	1	2	yes	maximal, complemented
8	C₄×#88068	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11718156	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	D₄×Q₈×C₁₃	—	2	1	no	—
14	#11719260	—	2	1	no	—
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	C₁₃×#467	C₂²	1	2	yes	—
17	C₁₃×#462	—	2	1	no	—
18	C₁₃×#451	—	2	2	no	—
19	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
20	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×#88068	C₂²	1	1	yes	characteristic
22	C₂×#88068	C₂²	1	2	yes	—
23	#11718776	C₂²	1	2	yes	—
24	C₂×#88058	C₂²	1	2	yes	—
25	C₂²×#6090	C₂²	1	2	yes	—
26	#11718507	—	2	1	no	—
27	#11718397	—	2	2	no	—
28	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
29	#11718343	C₂²	1	1	yes	characteristic
30	#11718156	C₂²	1	2	yes	—
31	#11718156	C₂²	1	1	yes	characteristic
32	#11718156	C₂²	1	2	yes	—
33	#11718156	C₂²	1	2	yes	—
34	#11718156	C₂²	1	2	yes	—
35	C₂²⋊#6090	C₂²	1	2	yes	—
36	#11718083	C₂²	1	1	yes	characteristic
37	#11718082	C₂²	1	2	yes	—
38	#11718082	C₂²	1	1	yes	characteristic
39	C₄⋊#6090	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₃	—	2	2	no	—
42	C₂×#6129	—	4	1	no	—
43	C₂×C₁₃×Q₁₆	—	2	4	no	—
44	C₂×C₁₃×Q₁₆	—	2	2	no	—
45	C₂×C₁₃×Q₁₆	—	2	1	no	—
46	C₂×C₁₃×Q₁₆	—	4	2	no	—
47	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
48	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
50	C₁₃×C₄⋊Q₈	—	2	1	no	—
51	C₁₃×C₄⋊Q₈	—	4	1	no	—
52	C₁₃×C₂²⋊Q₈	—	2	2	no	—
53	C₁₃×C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈×C₁₃	—	2	1	no	—
55	C₄×D₄×C₁₃	—	4	1	no	—
56	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
57	#88184	—	2	2	no	—
58	C₂×C₄⋊Dic₁₃	—	2	2	no	—
59	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
60	#88151	—	4	1	no	—
61	#88140	—	4	2	no	—
62	C₄×D₅₂	—	2	1	no	—
63	#88123	—	2	1	no	—
64	C₁₃×#107	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₃×#103	—	2	1	no	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88068	—	2	2	no	—
71	#88068	—	4	2	no	—
72	#88061	—	2	2	no	—
73	#88061	—	2	1	no	—
74	#88058	—	2	2	no	—
75	#88058	—	2	1	no	—
76	#88058	C₂³	1	1	yes	characteristic
77	#88058	C₂³	1	2	yes	—
78	C₂×#6090	—	4	2	no	—
79	C₂×#6090	C₂³	1	1	yes	characteristic
80	C₂×#6090	C₂³	1	2	yes	—
81	C₂×#6090	C₂³	1	1	yes	characteristic
82	#33865	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₃	—	2	2	no	—
85	C₂×Q₈×C₁₃	—	4	4	no	—
86	C₂×Q₈×C₁₃	—	4	2	no	—
87	C₂×Q₈×C₁₃	—	8	2	no	—
88	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	—	4	2	no	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₂²⋊D₁₃	—	4	2	no	—
92	C₂²×Dic₁₃	—	4	2	no	—
93	#6129	—	8	4	no	—
94	C₂×D₅₂	—	2	1	no	—
95	C₂×C₄×D₁₃	—	4	1	no	—
96	C₂×Dic₂₆	—	4	1	no	—
97	C₁₃×Q₁₆	—	2	2	no	—
98	C₁₃×Q₁₆	—	2	1	no	—
99	C₁₃×Q₁₆	—	2	2	no	—
100	C₁₃×Q₁₆	—	4	4	no	—
101	C₁₃×Q₁₆	—	8	1	no	—
102	C₁₃×Q₁₆	#41	1	2	yes	complemented
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₁₃×C₄⋊C₄	—	2	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₄⋊C₄	—	4	2	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
116	C₄²×C₁₃	—	4	1	no	—
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	#6103	—	4	1	no	—
119	#6103	—	4	2	no	—
120	C₄⋊Dic₁₃	—	2	2	no	—
121	C₄⋊Dic₁₃	—	2	1	no	—
122	C₄⋊Dic₁₃	—	4	1	no	—
123	C₄⋊Dic₁₃	—	4	2	no	—
124	D₁₀₄	—	2	1	no	—
125	D₁₀₄	—	2	1	no	—
126	#6090	—	2	1	no	—
127	#6090	—	2	2	no	—
128	#6090	—	2	1	no	—
129	#3334	—	13	1	no	—
130	D₄×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
131	#2241	—	13	1	no	—
132	#2235	—	13	2	no	—
133	#2225	—	13	1	no	—
134	#2217	—	13	2	no	—
135	#2208	—	13	1	no	—
136	#2185	—	13	2	no	—
137	C₄×SD₃₂	—	13	1	no	—
138	D₄×C₁₆	—	13	1	no	—
139	C₂×#358	—	13	2	no	—
140	C₂³×C₁₃	C₂×D₈	1	2	yes	—
141	C₂²×D₁₃	—	4	1	no	—
142	Q₈×C₁₃	—	2	2	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	4	2	no	—
145	Q₈×C₁₃	—	4	2	no	—
146	Q₈×C₁₃	—	8	4	no	—
147	D₄×C₁₃	SD₃₂	1	4	yes	complemented
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	4	1	no	—
151	C₂×C₄×C₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	8	4	no	—
154	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂²⋊D₁₃	—	8	4	no	—
159	C₂×Dic₁₃	—	4	2	no	—
160	C₂×Dic₁₃	—	4	1	no	—
161	C₂×Dic₁₃	—	8	4	no	—
162	D₅₂	—	4	1	no	—
163	D₅₂	—	4	1	no	—
164	C₄×D₁₃	—	8	2	no	—
165	Dic₂₆	—	8	2	no	—
166	C₈×C₁₃	—	2	2	no	—
167	C₈×C₁₃	—	2	1	no	—
168	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
169	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
171	D₄×Q₈	—	2	1	no	—
172	#547	—	26	1	no	—
173	C₂×SD₃₂	—	13	1	no	—
174	C₂×SD₃₂	—	13	2	no	—
175	C₂²×C₁₆	—	13	2	no	—
176	#500	—	13	1	no	—
177	#494	C₂×D₁₃	1	1	yes	characteristic
178	#480	—	26	2	no	—
179	#474	—	26	1	no	—
180	#467	C₂×D₁₃	1	2	yes	—
181	#466	—	13	2	no	—
182	#462	—	2	1	no	—
183	#451	—	2	2	no	—
184	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
185	C₄×D₈	—	13	1	no	—
186	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
187	C₂×#107	—	13	2	no	—
188	#367	—	13	1	no	—
189	#367	—	13	2	no	—
190	C₄⋊C₁₆	—	13	1	no	—
191	#358	—	13	1	no	—
192	#358	—	13	2	no	—
193	#358	—	13	2	no	—
194	#358	—	13	2	no	—
195	#358	—	13	2	no	—
196	#357	—	13	1	no	—
197	C₂²⋊C₁₆	—	13	2	no	—
198	C₄×C₁₆	—	13	1	no	—
199	C₂²×C₁₃	—	2	2	no	—
200	C₂²×C₁₃	—	2	2	no	—
201	C₂²×C₁₃	C₂×SD₃₂	1	4	yes	—
202	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
203	C₂×D₁₃	—	8	1	no	—
204	C₂×D₁₃	—	8	1	no	—
205	C₂×D₁₃	—	8	1	no	—
206	C₄×C₁₃	—	2	1	no	—
207	C₄×C₁₃	—	2	2	no	—
208	C₄×C₁₃	—	4	2	no	—
209	C₄×C₁₃	—	8	1	no	—
210	C₄×C₁₃	—	8	2	no	—
211	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
212	C₄×C₁₃	#546	1	1	yes	characteristic
213	C₄×C₁₃	#577	1	1	yes	characteristic
214	Dic₁₃	—	8	2	no	—
215	Dic₁₃	—	8	1	no	—
216	C₂×#41	—	52	1	no	—
217	C₂²×Q₈	—	2	2	no	—
218	C₂×Q₁₆	—	2	4	no	—
219	C₂×Q₁₆	—	2	2	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	4	2	no	—
222	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
223	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×D₈	—	13	1	no	—
225	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	4	1	no	—
228	C₄⋊Q₈	—	26	1	no	—
229	#123	—	52	2	no	—
230	C₂²⋊Q₈	—	2	2	no	—
231	C₂²⋊Q₈	—	4	2	no	—
232	C₂²⋊Q₈	—	52	1	no	—
233	#121	—	26	2	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×D₄	—	4	1	no	—
236	C₄×D₄	—	26	1	no	—
237	C₄×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
238	C₂×C₄⋊C₄	—	26	2	no	—
239	SD₃₂	—	26	2	no	—
240	SD₃₂	—	52	2	no	—
241	C₂×C₁₆	—	13	1	no	—
242	C₂×C₁₆	—	13	1	no	—
243	C₂×C₁₆	—	13	2	no	—
244	C₂×C₁₆	—	52	2	no	—
245	#107	—	13	1	no	—
246	#107	—	13	2	no	—
247	#107	—	26	2	no	—
248	#107	—	26	1	no	—
249	#107	C₂²×D₁₃	1	1	yes	characteristic
250	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
251	#103	—	2	1	no	—
252	#103	—	2	2	no	—
253	#102	—	26	1	no	—
254	#102	—	26	2	no	—
255	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
256	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
257	C₂×C₁₃	—	2	4	no	—
258	C₂×C₁₃	#3294	1	1	yes	characteristic
259	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
260	C₂×C₁₃	#3417	1	1	yes	characteristic
261	D₁₃	—	16	1	no	—
262	#41	—	104	4	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	4	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	—	8	2	no	—
268	C₂×Q₈	—	52	1	no	—
269	C₂×D₄	—	26	1	no	—
270	C₂×D₄	—	52	2	no	—
271	C₂×D₄	#6104	1	1	yes	characteristic
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	—	52	2	no	—
274	C₂²×C₄	—	52	1	no	—
275	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	4	4	no	—
280	Q₁₆	—	8	1	no	—
281	Q₁₆	#6129	1	2	yes	complemented
282	D₈	—	26	1	no	—
283	D₈	—	26	1	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
288	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	26	1	no	—
295	C₄⋊C₄	—	26	2	no	—
296	C₄⋊C₄	—	52	2	no	—
297	C₄⋊C₄	—	52	1	no	—
298	C₄⋊C₄	#6104	1	1	yes	characteristic
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	52	1	no	—
301	C₂²⋊C₄	—	52	2	no	—
302	C₂²⋊C₄	#6104	1	2	yes	—
303	C₄²	—	4	1	no	—
304	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₁₆	—	26	1	no	—
306	C₁₆	—	26	1	no	—
307	C₁₆	—	26	2	no	—
308	C₁₃	#33865	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	52	1	no	—
310	C₂³	C₂×#6104	1	2	yes	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	8	4	no	—
316	Q₈	—	104	2	no	—
317	D₄	—	52	1	no	—
318	D₄	—	52	1	no	—
319	D₄	—	104	4	no	—
320	D₄	#88068	1	4	yes	complemented
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	—	52	1	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	—	104	4	no	—
330	C₂×C₄	—	104	2	no	—
331	C₂×C₄	C₂×#6104	1	1	yes	characteristic
332	C₂×C₄	C₂×#6104	1	2	yes	—
333	C₂×C₄	C₂×#6104	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	C₂×#6129	1	1	yes	characteristic
338	C₈	#88251	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	104	1	no	—
342	C₂²	—	104	1	no	—
343	C₂²	—	104	1	no	—
344	C₂²	C₂×#88068	1	4	yes	—
345	C₂²	C₂²×#6104	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	2	no	—
351	C₄	—	104	1	no	—
352	C₄	—	104	2	no	—
353	C₄	C₂×#88068	1	2	yes	—
354	C₄	#11719511	1	1	yes	characteristic
355	C₄	#11719541	1	1	yes	characteristic
356	C₂	—	2	4	no	—
357	C₂	—	208	1	no	—
358	C₂	#13126649	1	1	yes	characteristic
359	C₂	C₂²×#88068	1	1	yes	characteristic
360	C₂	#13127987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	D₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13127184	C₂
hypercenter	D₄×Q₁₆	D₁₃
hypocenter	C₁₃	#33865
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13117063	C₂	semidirect
3. #13117031	C₂	semidirect
4. #13116978	C₂	semidirect
5. #13116934	C₂	semidirect
6. #13116878	C₂	semidirect
7. #13116753	C₂	semidirect
8. C₄×#88068	C₂	semidirect
9. D₄×#6090	C₂	semidirect
10. C₂×#11718156	C₂	semidirect
11. D₄×C₁₃×Q₁₆	C₂	semidirect
102. C₁₃×Q₁₆	#41	semidirect
130. D₄×Q₁₆	D₁₃	semidirect
147. D₄×C₁₃	SD₃₂	semidirect
281. Q₁₆	#6129	semidirect
308. C₁₃	#33865	semidirect
320. D₄	#88068	semidirect

Maximal subgroups

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

subgroup	index
#13127184	2
#13117063	2
#13117031	2
#13116978	2
#13116934	2
#13116878	2
#13116753	2
C₄×#88068	2
D₄×#6090	2
C₂×#11718156	2
D₄×C₁₃×Q₁₆	2
#33865	13