Subgroups — #16794966

Order 3328.

666 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131755	C₂	1	2	yes	maximal
2	#13130400	C₂	1	4	yes	maximal, complemented
3	#13130398	C₂	1	2	yes	maximal, complemented
4	#13130390	C₂	1	2	yes	maximal, complemented
5	#13130387	C₂	1	2	yes	maximal, complemented
6	#13130385	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718255	C₂	1	1	yes	maximal, characteristic, complemented
8	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
10	#41×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88202	C₄	1	1	yes	characteristic
12	#11719158	—	2	2	no	—
13	#11719145	—	2	2	no	—
14	#11719144	—	2	4	no	—
15	#11719130	C₂²	1	4	yes	—
16	#11719129	C₂²	1	4	yes	—
17	#11719119	C₂²	1	2	yes	—
18	C₂×C₄⋊#1057	C₂²	1	2	yes	—
19	#11718783	C₄	1	2	yes	—
20	#11718783	C₄	1	1	yes	characteristic
21	M₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
22	M₁₆×Dic₁₃	C₂²	1	2	yes	—
23	#11718762	C₄	1	1	yes	characteristic
24	#11718762	C₄	1	2	yes	—
25	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
27	#11718256	C₂²	1	2	yes	—
28	#11718255	C₂²	1	2	yes	—
29	#11718255	C₂²	1	2	yes	—
30	#11718255	C₂²	1	1	yes	characteristic
31	#11718255	C₂²	1	1	yes	characteristic
32	#11718254	C₂²	1	4	yes	—
33	#11718253	C₂²	1	4	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₂×#6122	—	2	4	no	—
36	C₂²×#1057	—	2	2	no	—
37	C₁₃×#131	—	2	4	no	—
38	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
42	#88202	—	2	4	no	—
43	Q₈×Dic₁₃	D₄	1	1	yes	characteristic
44	Q₈×Dic₁₃	Q₈	1	1	yes	characteristic
45	D₄×Dic₁₃	D₄	1	2	yes	—
46	D₄×Dic₁₃	D₄	1	1	yes	characteristic
47	D₄×Dic₁₃	Q₈	1	2	yes	—
48	D₄×Dic₁₃	Q₈	1	1	yes	characteristic
49	#88180	C₂×C₄	1	2	yes	—
50	#88180	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₂×#6099	C₂×C₄	1	1	yes	characteristic
54	C₂×#6099	C₂×C₄	1	2	yes	—
55	C₂²×#1055	C₂×C₄	1	2	yes	—
56	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
57	C₂²⋊#1057	—	2	2	no	—
58	C₂²⋊#1057	—	2	4	no	—
59	#88112	—	2	4	no	—
60	#88110	—	2	4	no	—
61	C₄⋊#1057	—	2	2	no	—
62	C₄⋊#1057	C₂³	1	2	yes	—
63	C₄⋊#1057	C₂³	1	4	yes	—
64	C₄⋊#1057	C₂³	1	2	yes	—
65	C₄×#1057	—	2	2	no	—
66	#88055	C₂×C₄	1	2	yes	—
67	#88055	C₂×C₄	1	2	yes	—
68	#88055	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
70	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
73	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
74	C₈×Dic₁₃	C₂³	1	2	yes	—
75	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
76	#17798	—	13	1	no	maximal, Sylow
77	C₁₃×#41	—	2	2	no	—
78	C₁₃×#41	—	2	2	no	—
79	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₃	C₄⋊C₄	1	2	yes	—
81	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
86	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
87	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
88	#6122	—	8	4	no	—
89	C₂×#1057	—	2	2	no	—
90	C₂×#1057	—	2	2	no	—
91	C₂×#1057	—	2	4	no	—
92	C₂×#1057	—	8	2	no	—
93	C₁₃×M₁₆	—	2	4	no	—
94	C₁₃×M₁₆	M₁₆	1	4	yes	complemented
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	4	no	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂²⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
101	C₂²⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
102	C₂²⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
103	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
104	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
106	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
109	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
111	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
112	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
113	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
114	#6099	—	2	2	no	—
115	#6099	—	2	4	no	—
116	C₂×#1055	—	2	4	no	—
117	C₂×#1055	M₁₆	1	4	yes	complemented
118	C₂×#1055	C₂²×C₄	1	2	yes	—
119	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
120	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
121	#2989	—	13	2	no	—
122	C₄×#131	—	13	1	no	—
123	#1569	—	13	4	no	—
124	#1568	—	13	2	no	—
125	#1566	—	13	2	no	—
126	#1565	—	13	2	no	—
127	#1564	—	13	1	no	—
128	C₂×#334	—	13	1	no	—
129	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	Q₈×C₁₃	—	2	2	no	—
132	D₄×C₁₃	—	2	4	no	—
133	D₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	2	4	no	—
143	C₂×Dic₁₃	—	4	1	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
151	#1057	—	4	2	no	—
152	#1057	—	4	4	no	—
153	#1057	—	4	2	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	—	2	1	no	—
156	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
157	#1055	—	2	2	no	—
158	#1055	C₂×M₁₆	1	2	yes	—
159	#1055	C₂×M₁₆	1	2	yes	—
160	C₂×#131	—	13	1	no	—
161	C₂×#131	#261	1	1	yes	characteristic
162	C₄×#41	—	13	1	no	—
163	#446	—	26	2	no	—
164	#436	—	26	2	no	—
165	#435	—	26	4	no	—
166	#432	—	13	4	no	—
167	#431	—	13	2	no	—
168	#423	—	13	4	no	—
169	C₂×C₄⋊C₈	—	13	2	no	—
170	#405	—	13	1	no	—
171	#405	—	13	2	no	—
172	#405	—	13	2	no	—
173	#405	—	13	1	no	—
174	C₄×M₁₆	—	13	1	no	—
175	C₄×M₁₆	—	13	2	no	—
176	C₂×C₄×C₈	—	13	1	no	—
177	C₂×C₄×C₈	—	13	2	no	—
178	#335	—	13	2	no	—
179	#335	—	13	4	no	—
180	#334	—	13	2	no	—
181	#334	—	13	2	no	—
182	#334	—	13	1	no	—
183	#334	—	13	1	no	—
184	#334	—	13	4	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	2	2	no	—
188	C₂²×C₁₃	—	2	2	no	—
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	#423	1	2	yes	—
191	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	#423	1	2	yes	—
194	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
195	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
196	Dic₁₃	—	2	1	no	—
197	Dic₁₃	—	2	2	no	—
198	Dic₁₃	—	2	1	no	—
199	Dic₁₃	—	2	1	no	—
200	Dic₁₃	—	2	2	no	—
201	Dic₁₃	#577	1	2	yes	—
202	C₂×#41	C₂×#261	1	1	yes	characteristic
203	#131	—	2	4	no	—
204	#131	—	26	4	no	—
205	C₂×M₁₆	—	13	2	no	—
206	C₂×M₁₆	—	13	1	no	—
207	C₂×M₁₆	—	26	4	no	—
208	C₂×M₁₆	C₂×#261	1	2	yes	—
209	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
210	C₂²×C₈	—	13	1	no	—
211	C₂²×C₈	—	13	2	no	—
212	C₂²×C₈	—	26	2	no	—
213	C₂²×C₈	C₂×#261	1	1	yes	characteristic
214	C₂²×C₈	C₂×#261	1	2	yes	—
215	C₄×Q₈	—	13	1	no	—
216	C₄×Q₈	—	13	1	no	—
217	C₄×D₄	—	13	2	no	—
218	C₄×D₄	—	13	1	no	—
219	C₄×D₄	—	13	2	no	—
220	C₄×D₄	—	13	1	no	—
221	#117	—	13	2	no	—
222	#117	—	13	1	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	13	2	no	—
225	C₄⋊C₈	—	13	4	no	—
226	C₄⋊C₈	—	13	2	no	—
227	C₄⋊C₈	—	13	2	no	—
228	C₄⋊C₈	—	26	2	no	—
229	C₄⋊C₈	—	26	4	no	—
230	C₂²⋊C₈	—	26	2	no	—
231	C₂²⋊C₈	—	26	4	no	—
232	#97	—	13	2	no	—
233	#97	—	13	2	no	—
234	#97	—	13	2	no	—
235	#97	—	26	4	no	—
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	—	13	1	no	—
238	C₄×C₈	—	13	2	no	—
239	C₄×C₈	—	13	1	no	—
240	C₄×C₈	—	13	2	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	2	no	—
243	C₄×C₈	—	26	2	no	—
244	C₂×C₁₃	—	2	2	no	—
245	C₂×C₁₃	—	4	2	no	—
246	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
247	C₂×C₁₃	#2912	1	1	yes	characteristic
248	C₂×C₁₃	#2916	1	1	yes	characteristic
249	#41	—	2	2	no	—
250	#41	—	2	2	no	—
251	C₂×Q₈	C₄⋊#261	1	1	yes	characteristic
252	C₂×D₄	C₄⋊#261	1	2	yes	—
253	C₂×D₄	C₄⋊#261	1	1	yes	characteristic
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	C₂²×#261	1	2	yes	—
259	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
260	M₁₆	—	2	4	no	—
261	M₁₆	—	26	4	no	—
262	M₁₆	—	26	2	no	—
263	M₁₆	—	104	4	no	—
264	M₁₆	#6122	1	4	yes	complemented
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	13	4	no	—
268	C₂×C₈	—	13	1	no	—
269	C₂×C₈	—	13	1	no	—
270	C₂×C₈	—	13	2	no	—
271	C₂×C₈	—	26	4	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	—	26	2	no	—
274	C₂×C₈	—	26	4	no	—
275	C₂×C₈	—	104	2	no	—
276	C₂×C₈	C₂²×#261	1	1	yes	characteristic
277	C₂×C₈	C₂²×#261	1	1	yes	characteristic
278	C₂×C₈	C₂²×#261	1	2	yes	—
279	C₄⋊C₄	—	13	2	no	—
280	C₄⋊C₄	—	13	2	no	—
281	C₄⋊C₄	—	13	1	no	—
282	C₄⋊C₄	—	13	1	no	—
283	C₂²⋊C₄	—	13	2	no	—
284	C₂²⋊C₄	—	13	2	no	—
285	C₂²⋊C₄	—	13	2	no	—
286	C₄²	—	13	1	no	—
287	C₄²	—	13	2	no	—
288	C₄²	—	13	2	no	—
289	C₄²	—	13	1	no	—
290	C₄²	—	13	1	no	—
291	C₄²	—	13	1	no	—
292	C₄²	—	13	2	no	—
293	C₁₃	#17798	1	1	yes	characteristic, complemented, Sylow
294	C₂³	C₂×C₄⋊#261	1	2	yes	—
295	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
296	Q₈	—	2	2	no	—
297	D₄	—	2	4	no	—
298	D₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	13	1	no	—
304	C₂×C₄	—	13	2	no	—
305	C₂×C₄	—	13	1	no	—
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	52	1	no	—
310	C₂×C₄	—	52	2	no	—
311	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
314	C₂×C₄	C₂×#6122	1	2	yes	—
315	C₂×C₄	C₂³×#261	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	13	2	no	—
319	C₈	—	13	2	no	—
320	C₈	—	26	2	no	—
321	C₈	—	52	4	no	—
322	C₈	—	52	2	no	—
323	C₈	—	52	2	no	—
324	C₈	C₂×#6122	1	4	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	#11719129	1	2	yes	—
331	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	13	2	no	—
334	C₄	—	26	1	no	—
335	C₄	—	26	2	no	—
336	C₄	—	26	1	no	—
337	C₄	—	26	2	no	—
338	C₄	—	26	1	no	—
339	C₄	#11719129	1	2	yes	—
340	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
341	C₄	C₂²×#6122	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	4	2	no	—
344	C₂	C₂×#11719129	1	1	yes	characteristic
345	C₂	#13131490	1	1	yes	characteristic
346	C₂	#13131498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#131	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#17798
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #13130400	C₂	semidirect
3. #13130398	C₂	semidirect
4. #13130390	C₂	semidirect
5. #13130387	C₂	semidirect
6. #13130385	C₂	semidirect
7. C₂×#11718255	C₂	semidirect
94. C₁₃×M₁₆	M₁₆	semidirect
117. C₂×#1055	M₁₆	semidirect
264. M₁₆	#6122	semidirect
293. C₁₃	#17798	semidirect

Maximal subgroups

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

subgroup	index
#13131755	2
#13130400	2
#13130398	2
#13130390	2
#13130387	2
#13130385	2
C₂×#11718255	2
#131×Dic₁₃	2
#17798	13