Subgroups — #16780465

Order 3328.

610 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130622	C₂	1	1	yes	maximal, characteristic
2	C₄×#88110	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718244	C₂	1	2	yes	maximal
4	#13129672	C₂	1	2	yes	maximal
5	C₄×#88141	C₂	1	1	yes	maximal, characteristic
6	C₂×#88141	C₄	1	1	yes	characteristic
7	C₂×#88141	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
9	C₂×#88110	—	2	1	no	—
10	C₂×#88110	—	2	1	no	—
11	C₂×#88110	—	2	1	no	—
12	C₂×C₄⋊#1057	C₂²	1	2	yes	—
13	C₂×C₄×#1057	C₂²	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
15	C₄⋊C₄×Dic₁₃	C₄	1	1	yes	characteristic
16	C₄⋊C₄×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₄×#6101	—	2	1	no	—
18	C₄²×Dic₁₃	—	2	1	no	—
19	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
20	#11718310	—	2	1	no	—
21	#11718310	—	2	1	no	—
22	#11718290	—	2	1	no	—
23	#11718244	C₄	1	8	yes	complemented
24	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1057	—	2	2	no	—
26	C₂²×#1057	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	—	2	1	no	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#6101	—	2	1	no	—
33	C₂×#6101	—	2	1	no	—
34	C₂×#6101	—	2	1	no	—
35	C₂×C₄×Dic₁₃	—	2	1	no	—
36	C₂×C₄×Dic₁₃	—	2	1	no	—
37	C₂×C₄×Dic₁₃	—	2	1	no	—
38	C₂×C₄×Dic₁₃	—	2	1	no	—
39	C₂×C₄×Dic₁₃	—	2	1	no	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
46	#88141	—	2	2	no	—
47	#88141	—	2	2	no	—
48	#88141	—	2	2	no	—
49	#88141	C₂×C₄	1	4	yes	—
50	C₄×Dic₂₆	—	2	2	no	—
51	C₄×Dic₂₆	—	2	2	no	—
52	#88110	—	2	4	no	—
53	#88110	—	2	4	no	—
54	#88110	—	4	2	no	—
55	C₄⋊#1057	—	2	2	no	—
56	C₄⋊#1057	C₂×C₄	1	4	yes	—
57	C₄×#1057	—	2	2	no	—
58	C₄×#1057	C₂×C₄	1	4	yes	—
59	C₁₃×#95	—	2	1	no	—
60	#88071	—	2	1	no	—
61	#88071	—	2	1	no	—
62	#8173	—	13	1	no	maximal, Sylow
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	1	no	—
69	C₂²×Dic₁₃	—	2	1	no	—
70	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×Dic₂₆	—	2	1	no	—
75	C₂×Dic₂₆	—	2	1	no	—
76	C₂×Dic₂₆	—	2	1	no	—
77	C₂×Dic₂₆	C₄²	1	2	yes	—
78	C₂×Dic₂₆	M₁₆	1	2	yes	—
79	C₂×Dic₂₆	M₁₆	1	2	yes	—
80	C₂×#1057	—	2	4	no	—
81	C₂×#1057	—	2	4	no	—
82	C₂×#1057	—	2	4	no	—
83	C₂×#1057	—	2	4	no	—
84	C₂×#1057	—	4	2	no	—
85	C₂×#1057	—	4	2	no	—
86	C₁₃×C₄⋊C₄	—	2	1	no	—
87	C₁₃×C₄⋊C₄	—	2	1	no	—
88	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
89	C₁₃×C₄⋊C₄	M₁₆	1	2	yes	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	—	4	2	no	—
93	C₄⋊Dic₁₃	—	2	1	no	—
94	C₄⋊Dic₁₃	M₁₆	1	2	yes	—
95	#6101	—	2	4	no	—
96	#6101	—	2	4	no	—
97	#6101	—	2	4	no	—
98	C₄×Dic₁₃	—	2	4	no	—
99	C₄×Dic₁₃	—	2	4	no	—
100	C₄×Dic₁₃	—	2	2	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	C₄×Dic₁₃	—	2	1	no	—
103	C₄×Dic₁₃	—	2	2	no	—
104	C₄×Dic₁₃	—	2	4	no	—
105	C₄×Dic₁₃	—	2	1	no	—
106	C₄×Dic₁₃	—	2	2	no	—
107	C₄×Dic₁₃	—	2	1	no	—
108	C₄×Dic₁₃	—	4	2	no	—
109	C₄×Dic₁₃	—	4	1	no	—
110	C₄×Dic₁₃	—	4	1	no	—
111	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
112	C₄×Dic₁₃	#41	1	2	yes	—
113	C₄²×Q₈	—	13	1	no	—
114	#1767	—	13	1	no	—
115	C₄×#97	—	13	1	no	—
116	C₂×Q₈⋊C₈	—	13	2	no	—
117	#1279	—	13	2	no	—
118	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	—	4	1	no	—
130	C₂×C₄×C₁₃	#104	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	1	no	—
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	2	1	no	—
142	C₂×Dic₁₃	—	2	1	no	—
143	C₂×Dic₁₃	—	2	1	no	—
144	C₂×Dic₁₃	—	2	1	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	1	no	—
147	C₂×Dic₁₃	—	2	1	no	—
148	C₂×Dic₁₃	—	4	1	no	—
149	C₂×Dic₁₃	—	4	1	no	—
150	C₂×Dic₁₃	#104	1	2	yes	—
151	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
152	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
153	C₂×Dic₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×Dic₁₃	#137	1	2	yes	—
155	C₂×Dic₁₃	#137	1	1	yes	characteristic
156	Dic₂₆	—	2	2	no	—
157	Dic₂₆	—	2	2	no	—
158	Dic₂₆	—	2	2	no	—
159	Dic₂₆	#97	1	4	yes	complemented
160	#1057	—	2	8	no	—
161	#1057	—	4	4	no	—
162	C₂×C₄×Q₈	—	13	1	no	—
163	C₂×C₄×Q₈	—	13	1	no	—
164	C₂×C₄×Q₈	—	13	1	no	—
165	C₂×C₄⋊C₈	—	13	2	no	—
166	C₂×#97	—	26	1	no	—
167	C₂×#97	—	26	1	no	—
168	C₂×#97	—	26	1	no	—
169	C₂×C₄×C₈	—	13	2	no	—
170	C₄×C₄⋊C₄	—	13	1	no	—
171	C₄×C₄⋊C₄	—	13	1	no	—
172	C₄×C₄⋊C₄	—	26	1	no	—
173	C₄×C₄⋊C₄	—	26	1	no	—
174	C₄×C₄⋊C₄	—	26	1	no	—
175	C₄×C₄⋊C₄	#261	1	1	yes	characteristic
176	C₄³	—	13	1	no	—
177	C₄³	—	26	1	no	—
178	#336	—	26	1	no	—
179	Q₈⋊C₈	—	13	8	no	—
180	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₁₃	#407	1	1	yes	characteristic
182	C₂²×C₁₃	#407	1	1	yes	characteristic
183	C₂²×C₁₃	#417	1	1	yes	characteristic
184	C₂²×C₁₃	#419	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	4	1	no	—
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	4	2	no	—
191	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
192	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
193	C₄×C₁₃	#432	1	1	yes	characteristic
194	C₄×C₁₃	#432	1	1	yes	characteristic
195	Dic₁₃	—	2	2	no	—
196	Dic₁₃	—	2	4	no	—
197	Dic₁₃	—	2	2	no	—
198	Dic₁₃	—	4	1	no	—
199	Dic₁₃	—	4	2	no	—
200	Dic₁₃	#440	1	2	yes	—
201	Dic₁₃	#441	1	2	yes	—
202	C₂²×Q₈	—	13	1	no	—
203	C₂²×C₈	—	26	2	no	—
204	C₂²×C₈	—	26	2	no	—
205	C₄×Q₈	—	13	4	no	—
206	C₄×Q₈	—	26	2	no	—
207	C₄×Q₈	—	26	2	no	—
208	C₄×Q₈	—	26	2	no	—
209	C₄×Q₈	—	26	2	no	—
210	C₄×Q₈	—	26	2	no	—
211	C₂×C₄⋊C₄	—	13	1	no	—
212	C₂×C₄⋊C₄	—	26	1	no	—
213	C₂×C₄⋊C₄	—	26	1	no	—
214	C₂×C₄⋊C₄	—	26	1	no	—
215	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	—	13	1	no	—
219	C₂×C₄²	—	13	1	no	—
220	C₂×C₄²	—	13	1	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	1	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	26	1	no	—
225	C₂×C₄²	—	26	1	no	—
226	C₂×C₄²	—	26	1	no	—
227	C₂×C₄²	—	26	1	no	—
228	C₂×C₄²	—	26	1	no	—
229	C₂×C₄²	C₂×#261	1	1	yes	characteristic
230	C₄⋊C₈	—	13	4	no	—
231	C₄⋊C₈	—	26	2	no	—
232	#97	—	26	4	no	—
233	#97	—	26	4	no	—
234	#97	—	52	2	no	—
235	C₄×C₈	—	13	4	no	—
236	C₄×C₈	—	26	2	no	—
237	#95	—	2	1	no	—
238	#95	—	26	1	no	—
239	#95	—	26	1	no	—
240	C₂×C₁₃	#1489	1	1	yes	characteristic
241	C₂×C₁₃	#1489	1	1	yes	characteristic
242	C₂×C₁₃	#1491	1	1	yes	characteristic
243	C₂×C₁₃	#1491	1	1	yes	characteristic
244	C₂×C₁₃	#1749	1	1	yes	characteristic
245	C₂×C₁₃	C₄×#104	1	1	yes	characteristic
246	C₂×C₁₃	#1763	1	1	yes	characteristic
247	C₂×Q₈	—	13	2	no	—
248	C₂×Q₈	—	13	2	no	—
249	C₂×Q₈	—	13	2	no	—
250	C₂×Q₈	—	26	1	no	—
251	C₂×Q₈	—	26	1	no	—
252	C₂×Q₈	—	26	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	13	1	no	—
256	C₂²×C₄	—	13	1	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	26	1	no	—
260	C₂²×C₄	—	26	1	no	—
261	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
263	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
264	C₂×C₈	—	26	4	no	—
265	C₂×C₈	—	26	4	no	—
266	C₂×C₈	—	26	4	no	—
267	C₂×C₈	—	26	4	no	—
268	C₂×C₈	—	52	2	no	—
269	C₂×C₈	—	52	2	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	13	2	no	—
273	C₄⋊C₄	—	26	4	no	—
274	C₄⋊C₄	—	26	4	no	—
275	C₄⋊C₄	—	26	4	no	—
276	C₄⋊C₄	—	26	1	no	—
277	C₄⋊C₄	#6118	1	2	yes	—
278	C₄⋊C₄	C₄×#261	1	2	yes	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	4	2	no	—
282	C₄²	—	13	2	no	—
283	C₄²	—	13	2	no	—
284	C₄²	—	26	2	no	—
285	C₄²	—	26	1	no	—
286	C₄²	—	26	2	no	—
287	C₄²	—	26	1	no	—
288	C₄²	—	26	4	no	—
289	C₄²	—	26	4	no	—
290	C₄²	—	26	4	no	—
291	C₄²	—	26	2	no	—
292	C₄²	—	26	1	no	—
293	C₄²	—	26	1	no	—
294	C₄²	—	52	2	no	—
295	C₄²	—	52	1	no	—
296	C₄²	—	52	1	no	—
297	C₁₃	#8173	1	1	yes	characteristic, complemented, Sylow
298	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
299	Q₈	—	13	4	no	—
300	Q₈	—	26	2	no	—
301	Q₈	—	26	2	no	—
302	Q₈	—	26	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	13	2	no	—
315	C₂×C₄	—	13	2	no	—
316	C₂×C₄	—	13	1	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	13	2	no	—
320	C₂×C₄	—	26	2	no	—
321	C₂×C₄	—	26	1	no	—
322	C₂×C₄	—	26	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	26	1	no	—
325	C₂×C₄	—	26	1	no	—
326	C₂×C₄	—	26	1	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	1	no	—
329	C₂×C₄	—	26	1	no	—
330	C₂×C₄	—	26	1	no	—
331	C₂×C₄	—	52	1	no	—
332	C₂×C₄	—	52	1	no	—
333	C₂×C₄	#88116	1	2	yes	—
334	C₂×C₄	C₂×#6118	1	1	yes	characteristic
335	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
336	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
337	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
339	C₂×C₄	#88237	1	1	yes	characteristic
340	C₈	—	26	8	no	—
341	C₈	—	52	4	no	—
342	C₂²	#11719126	1	1	yes	characteristic
343	C₂²	#11719127	1	1	yes	characteristic
344	C₂²	C₄×C₂²⋊#261	1	1	yes	characteristic
345	C₂²	C₂×#88116	1	1	yes	characteristic
346	C₂²	#11719153	1	1	yes	characteristic
347	C₂²	D₁₃⋊#104	1	1	yes	characteristic
348	C₂²	#11719164	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	13	2	no	—
354	C₄	—	13	2	no	—
355	C₄	—	26	4	no	—
356	C₄	—	26	2	no	—
357	C₄	—	26	2	no	—
358	C₄	—	52	1	no	—
359	C₄	—	52	2	no	—
360	C₄	#11719047	1	1	yes	characteristic
361	C₄	C₄×#6118	1	1	yes	characteristic
362	C₄	#11719054	1	1	yes	characteristic
363	C₄	#11719055	1	1	yes	characteristic
364	C₂	#13130172	1	1	yes	characteristic
365	C₂	#13130173	1	1	yes	characteristic
366	C₂	#13130179	1	1	yes	characteristic
367	C₂	#13130180	1	1	yes	characteristic
368	C₂	#13130563	1	1	yes	characteristic
369	C₂	C₄×#88116	1	1	yes	characteristic
370	C₂	#13130608	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₄×C₁₃×C₄⋊C₄	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₄×C₄⋊C₄	#261
hypocenter	C₁₃	#8173
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. #11718244	C₄	semidirect
159. Dic₂₆	#97	semidirect
297. C₁₃	#8173	semidirect

Maximal subgroups

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

subgroup	index
#13130622	2
C₄×#88110	2
C₂×#11718244	2
#13129672	2
C₄×#88141	2
#8173	13