Subgroups — #16796371

Order 3328.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6122	C₂	1	2	yes	maximal
2	#13130483	C₂	1	2	yes	maximal, complemented
3	#13130461	C₂	1	4	yes	maximal, complemented
4	#13130429	C₂	1	2	yes	maximal, complemented
5	#13130390	C₂	1	1	yes	maximal, characteristic
6	#13130164	C₂	1	2	yes	maximal
7	#13130045	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₃	C₂²	1	2	yes	—
11	#11719158	—	2	4	no	—
12	#11719158	C₂²	1	4	yes	—
13	Q₈×#1057	C₂²	1	4	yes	—
14	#11719129	—	2	2	no	—
15	#11719129	C₂²	1	2	yes	—
16	#11719125	C₂²	1	1	yes	characteristic
17	C₄×#6122	—	2	2	no	—
18	C₄×#6122	C₂²	1	1	yes	characteristic
19	C₂×#88075	C₄	1	2	yes	—
20	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic
21	Q₁₆×Dic₁₃	C₄	1	2	yes	—
22	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic
23	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
25	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×#88058	C₄	1	1	yes	characteristic
27	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
28	Q₈⋊#1057	C₂²	1	4	yes	—
29	Q₈⋊#1057	C₂²	1	4	yes	—
30	#11718255	C₂²	1	2	yes	—
31	#11718253	C₂²	1	2	yes	—
32	#1057⋊C₈	C₂²	1	2	yes	—
33	#11718251	C₂²	1	2	yes	—
34	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×#6122	—	2	2	no	—
36	C₂×#6122	D₄	1	2	yes	—
37	C₂×C₁₃×Q₁₆	—	2	4	no	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	Q₈×Dic₁₃	—	2	2	no	—
43	Q₈×Dic₁₃	—	2	2	no	—
44	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
45	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
46	Q₈×Dic₁₃	C₂³	1	2	yes	—
47	Q₈×Dic₁₃	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₁₃	—	2	2	no	—
49	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₁₃	—	2	2	no	—
51	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
53	C₂²⋊#1057	—	2	1	no	—
54	#88112	—	2	4	no	—
55	#88112	—	2	2	no	—
56	#88112	—	2	1	no	—
57	#88110	—	2	4	no	—
58	#88110	D₄	1	2	yes	—
59	C₄⋊#1057	—	2	4	no	—
60	C₄⋊#1057	C₂³	1	4	yes	—
61	C₄×#1057	—	2	4	no	—
62	C₄×#1057	C₂³	1	2	yes	—
63	C₄×#1057	C₂³	1	2	yes	—
64	#88075	C₂×C₄	1	2	yes	—
65	#88075	C₂×C₄	1	2	yes	—
66	#88075	C₂×C₄	1	2	yes	—
67	#88075	C₂×C₄	1	2	yes	—
68	#88058	C₂×C₄	1	2	yes	—
69	#88058	C₂×C₄	1	1	yes	characteristic
70	#88058	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
74	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
75	#18042	—	13	1	no	maximal, Sylow
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₃	—	4	2	no	—
79	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×Dic₁₃	—	2	2	no	—
84	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₁₃	#41	1	1	yes	characteristic
86	#6122	—	2	4	no	—
87	#6122	—	4	4	no	—
88	C₂×#1057	—	2	4	no	—
89	C₂×#1057	—	2	2	no	—
90	C₂×#1057	C₂×D₄	1	4	yes	—
91	C₁₃×Q₁₆	—	2	4	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	2	2	no	—
102	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
104	C₄×Dic₁₃	—	2	2	no	—
105	C₄×Dic₁₃	—	2	2	no	—
106	C₄×Dic₁₃	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₁₃	#41	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
112	C₂×#1055	—	4	1	no	—
113	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
114	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
115	Q₈×M₁₆	—	13	2	no	—
116	C₂×C₄×Q₁₆	—	13	1	no	—
117	#1593	—	13	2	no	—
118	#1587	—	13	4	no	—
119	#1579	—	13	2	no	—
120	#1566	—	13	1	no	—
121	#1487	—	13	2	no	—
122	#1451	—	13	1	no	—
123	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
124	Q₈×C₁₃	—	2	2	no	—
125	Q₈×C₁₃	—	2	2	no	—
126	Q₈×C₁₃	—	2	2	no	—
127	Q₈×C₁₃	—	4	4	no	—
128	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	4	4	no	—
134	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	4	1	no	—
139	C₂×Dic₁₃	—	8	2	no	—
140	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₁₃	#137	1	4	yes	—
143	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
145	#1057	—	2	4	no	—
146	#1057	—	4	2	no	—
147	#1057	—	4	4	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
150	#1055	—	2	1	no	—
151	#1055	—	2	1	no	—
152	C₂²×Q₁₆	#261	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	13	2	no	—
154	#446	—	13	4	no	—
155	#446	—	26	4	no	—
156	C₈×Q₈	—	13	4	no	—
157	C₄×Q₁₆	—	13	1	no	—
158	C₄×Q₁₆	—	13	2	no	—
159	C₄×Q₁₆	—	13	1	no	—
160	C₄×Q₁₆	—	13	2	no	—
161	C₄×Q₁₆	—	13	1	no	—
162	C₄×Q₁₆	—	13	1	no	—
163	#431	—	13	1	no	—
164	C₂×#107	—	13	1	no	—
165	#423	—	13	2	no	—
166	#423	—	26	2	no	—
167	C₂×#103	—	13	2	no	—
168	C₄×M₁₆	—	13	1	no	—
169	C₄×M₁₆	—	26	2	no	—
170	C₂×C₄×C₈	—	13	1	no	—
171	#334	—	13	2	no	—
172	#334	—	13	2	no	—
173	Q₈⋊C₈	—	13	4	no	—
174	Q₈⋊C₈	—	13	4	no	—
175	#322	—	13	2	no	—
176	#322	—	13	2	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	#441	1	2	yes	—
180	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	4	no	—
183	C₄×C₁₃	—	4	2	no	—
184	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
185	C₄×C₁₃	#568	1	1	yes	characteristic
186	Dic₁₃	—	2	1	no	—
187	Dic₁₃	—	2	1	no	—
188	Dic₁₃	—	4	2	no	—
189	Dic₁₃	—	4	2	no	—
190	Dic₁₃	#441	1	2	yes	—
191	Dic₁₃	C₂×#137	1	2	yes	—
192	C₂²×Q₈	C₂×#261	1	2	yes	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	C₂×#261	1	2	yes	—
195	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
196	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
197	C₂×M₁₆	—	13	2	no	—
198	C₂×M₁₆	—	26	2	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	C₂×#261	1	1	yes	characteristic
201	C₄×Q₈	—	13	2	no	—
202	C₄×Q₈	—	13	2	no	—
203	C₄×Q₈	—	13	2	no	—
204	C₄×Q₈	—	13	2	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×Q₈	—	26	2	no	—
207	C₂×C₄⋊C₄	—	13	2	no	—
208	C₂×C₄⋊C₄	—	26	2	no	—
209	C₂×C₄²	—	13	1	no	—
210	C₂×C₄²	—	26	2	no	—
211	#107	—	13	2	no	—
212	#107	—	13	1	no	—
213	#107	—	13	1	no	—
214	C₄⋊C₈	—	13	4	no	—
215	C₄⋊C₈	—	26	2	no	—
216	C₄⋊C₈	—	26	1	no	—
217	C₄⋊C₈	—	26	4	no	—
218	C₄⋊C₈	—	26	4	no	—
219	#103	—	13	2	no	—
220	#103	—	13	2	no	—
221	#103	—	13	2	no	—
222	#103	—	13	2	no	—
223	C₂²⋊C₈	—	26	1	no	—
224	#97	—	13	2	no	—
225	#97	—	26	4	no	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	—	13	1	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	2	no	—
232	C₄×C₈	—	26	4	no	—
233	C₂×C₁₃	—	2	2	no	—
234	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
235	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
236	C₂×C₁₃	#2957	1	1	yes	characteristic
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	C₂²×#261	1	2	yes	—
241	C₂×Q₈	C₂²×#261	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	26	2	no	—
246	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	#6122	1	4	yes	complemented
250	M₁₆	—	26	4	no	—
251	M₁₆	—	52	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	13	4	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	13	1	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	4	no	—
258	C₂×C₈	—	52	1	no	—
259	C₂×C₈	C₂²×#261	1	1	yes	characteristic
260	C₂×C₈	C₂²×#261	1	1	yes	characteristic
261	C₄⋊C₄	—	13	2	no	—
262	C₄⋊C₄	—	13	2	no	—
263	C₄⋊C₄	—	26	2	no	—
264	C₄⋊C₄	—	26	2	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	26	2	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	13	1	no	—
270	C₄²	—	13	1	no	—
271	C₄²	—	13	1	no	—
272	C₄²	—	26	2	no	—
273	C₄²	—	26	2	no	—
274	C₄²	—	26	2	no	—
275	C₄²	—	26	2	no	—
276	C₁₃	#18042	1	1	yes	characteristic, complemented, Sylow
277	C₂³	D₄×#261	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	4	4	no	—
282	Q₈	C₂×#6122	1	4	yes	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	13	4	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	13	1	no	—
292	C₂×C₄	—	13	1	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	52	1	no	—
296	C₂×C₄	—	104	2	no	—
297	C₂×C₄	D₄×#261	1	1	yes	characteristic
298	C₂×C₄	C₂³×#261	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	26	4	no	—
301	C₈	—	26	1	no	—
302	C₈	—	26	1	no	—
303	C₈	—	52	4	no	—
304	C₈	—	52	2	no	—
305	C₈	C₂×#6122	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#11719110	1	2	yes	—
309	C₂²	C₂×D₄×#261	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	2	no	—
313	C₄	—	13	2	no	—
314	C₄	—	13	2	no	—
315	C₄	—	26	1	no	—
316	C₄	—	26	1	no	—
317	C₄	—	52	2	no	—
318	C₄	—	52	2	no	—
319	C₄	#11719621	1	1	yes	characteristic
320	C₄	C₂²×#6122	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	D₄×#6122	1	1	yes	characteristic
323	C₂	C₂×#11719110	1	1	yes	characteristic
324	C₂	#13131616	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×Q₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂²×Q₁₆	#261
hypocenter	C₁₃	#18042
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13130483	C₂	semidirect
3. #13130461	C₂	semidirect
4. #13130429	C₂	semidirect
93. C₁₃×Q₁₆	M₁₆	semidirect
249. Q₁₆	#6122	semidirect
276. C₁₃	#18042	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6122	2
#13130483	2
#13130461	2
#13130429	2
#13130390	2
#13130164	2
#13130045	2
C₂×Q₁₆×Dic₁₃	2
#18042	13