Subgroups — #16018969

Order 3328.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125495	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117108	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117091	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117077	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117067	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116895	C₂	1	2	yes	maximal, complemented
7	#13116873	C₂	1	2	yes	maximal, complemented
8	#13116776	C₂	1	2	yes	maximal, complemented
9	#13116765	C₂	1	2	yes	maximal, complemented
10	#13116643	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3155	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
15	C₁₃×#535	—	2	1	no	—
16	#11719223	—	2	1	no	—
17	C₁₃×#496	—	2	1	no	—
18	C₁₃×#496	—	2	1	no	—
19	C₁₃×#495	C₂²	1	1	yes	characteristic
20	C₁₃×#495	C₂²	1	1	yes	characteristic
21	C₁₃×#493	C₂²	1	1	yes	characteristic
22	C₁₃×#493	C₂²	1	1	yes	characteristic
23	C₁₃×#405	C₂²	1	1	yes	characteristic
24	C₂×#88069	C₂²	1	2	yes	complemented
25	C₂×#88067	C₂²	1	2	yes	—
26	C₂×#88067	C₂²	1	2	yes	—
27	C₂×#88067	C₂²	1	2	yes	complemented
28	#11718798	—	2	1	no	—
29	#11718777	C₂²	1	2	yes	—
30	#11718377	—	2	1	no	—
31	#11718364	—	2	1	no	—
32	#11718360	C₂²	1	1	yes	characteristic, complemented
33	C₄×Dic₅₂	C₂²	1	2	yes	complemented
34	#11718344	C₂²	1	1	yes	characteristic, complemented
35	#11718156	C₂²	1	2	yes	complemented
36	#11718154	C₂²	1	2	yes	—
37	#11718154	C₂²	1	2	yes	—
38	#11718154	C₂²	1	2	yes	complemented
39	C₂²⋊#6090	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	2	yes	complemented
41	#11718053	C₂²	1	1	yes	characteristic, complemented
42	C₄×#6090	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₁₃×#41	—	2	1	no	—
44	C₂²×D₄×C₁₃	—	2	1	no	—
45	C₁₃×AGL(1,ℤ/8)	—	4	4	no	—
46	C₁₃×#135	—	4	2	no	—
47	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₁₃×SD₁₆	—	2	1	no	—
49	C₂×C₁₃×SD₁₆	—	2	1	no	—
50	C₂×C₁₃×SD₁₆	—	2	1	no	—
51	C₂×C₁₃×D₈	—	2	1	no	—
52	C₂×C₁₃×D₈	—	2	1	no	—
53	C₂×C₁₃×D₈	—	2	1	no	—
54	C₂×C₁₃×D₈	—	4	2	no	—
55	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
56	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×#127	—	2	1	no	—
61	C₁₃×#127	—	2	1	no	—
62	C₁₃×#124	—	2	1	no	—
63	C₁₃×#124	—	2	1	no	—
64	C₁₃×#121	—	4	2	no	—
65	C₁₃×#120	—	4	2	no	—
66	C₁₃×#117	D₄	1	1	yes	characteristic
67	#88178	—	2	2	no	—
68	C₂×Dic₅₂	—	2	1	no	—
69	C₂×Dic₅₂	C₂³	1	2	yes	—
70	#88143	—	4	2	no	—
71	#88133	—	4	2	no	—
72	#88124	—	2	1	no	—
73	#88123	—	2	1	no	—
74	C₄×Dic₂₆	—	2	2	no	—
75	C₁₃×#97	D₄	1	1	yes	characteristic
76	C₁₃×#97	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#88069	—	4	2	no	—
80	#88067	—	4	2	no	—
81	#88067	—	4	2	no	—
82	#88067	—	4	2	no	—
83	#88058	C₂³	1	2	yes	—
84	#88057	—	2	1	no	—
85	#88056	—	2	2	no	—
86	#88056	—	2	2	no	—
87	C₂×#6090	C₂³	1	2	yes	—
88	C₂×#6090	C₂³	1	2	yes	—
89	#33788	—	13	1	no	maximal, Sylow
90	C₂⁴×C₁₃	—	4	1	no	—
91	C₁₃×#41	—	4	2	no	—
92	C₁₃×#41	—	4	2	no	—
93	C₂×Q₈×C₁₃	—	2	1	no	—
94	C₂×D₄×C₁₃	—	2	1	no	—
95	C₂×D₄×C₁₃	—	2	1	no	—
96	C₂×D₄×C₁₃	—	2	1	no	—
97	C₂×D₄×C₁₃	—	4	2	no	—
98	C₂×D₄×C₁₃	—	4	2	no	—
99	C₂×D₄×C₁₃	—	4	1	no	—
100	C₂×D₄×C₁₃	—	4	1	no	—
101	C₂×D₄×C₁₃	—	4	2	no	—
102	C₂×D₄×C₁₃	—	4	2	no	—
103	C₂²×C₄×C₁₃	—	4	1	no	—
104	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₂₆	—	2	2	no	—
106	C₁₃×Q₁₆	—	2	2	no	—
107	C₁₃×SD₁₆	—	4	2	no	—
108	C₁₃×SD₁₆	—	4	2	no	—
109	C₁₃×SD₁₆	—	4	2	no	—
110	C₁₃×D₈	—	2	2	no	—
111	C₁₃×D₈	—	2	2	no	—
112	C₁₃×D₈	—	2	2	no	—
113	C₁₃×D₈	—	4	2	no	—
114	C₁₃×D₈	—	4	2	no	—
115	C₁₃×D₈	—	4	2	no	—
116	C₁₃×M₁₆	—	2	2	no	—
117	C₂×C₈×C₁₃	—	2	2	no	—
118	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
122	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
123	C₁₃×C₂²⋊C₄	—	4	2	no	—
124	C₁₃×C₂²⋊C₄	—	4	2	no	—
125	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
126	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
128	C₂²⋊Dic₁₃	—	4	2	no	—
129	C₄⋊Dic₁₃	—	2	2	no	—
130	C₄⋊Dic₁₃	—	4	2	no	—
131	#6101	—	4	2	no	—
132	#6101	—	4	2	no	—
133	#6101	—	4	2	no	—
134	C₄×Dic₁₃	—	4	2	no	—
135	Dic₅₂	—	2	2	no	—
136	Dic₅₂	—	2	2	no	—
137	Dic₅₂	—	4	2	no	—
138	#6090	—	2	2	no	—
139	#6090	—	2	2	no	—
140	#3177	—	13	1	no	—
141	#3155	D₁₃	1	1	yes	characteristic
142	#2251	—	13	1	no	—
143	#2248	—	13	1	no	—
144	#2244	—	13	1	no	—
145	#2242	—	13	1	no	—
146	#2211	—	13	2	no	—
147	#2207	—	13	2	no	—
148	#2189	—	13	2	no	—
149	#2187	—	13	2	no	—
150	#2164	—	13	1	no	—
151	C₂³×C₁₃	—	4	1	no	—
152	C₂³×C₁₃	—	4	1	no	—
153	C₂³×C₁₃	—	4	1	no	—
154	C₂³×C₁₃	—	8	2	no	—
155	C₂³×C₁₃	—	16	2	no	—
156	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
157	Q₈×C₁₃	—	4	1	no	—
158	Q₈×C₁₃	—	4	1	no	—
159	D₄×C₁₃	—	4	1	no	—
160	D₄×C₁₃	—	4	1	no	—
161	D₄×C₁₃	—	4	1	no	—
162	D₄×C₁₃	—	4	1	no	—
163	D₄×C₁₃	—	4	1	no	—
164	D₄×C₁₃	—	4	1	no	—
165	D₄×C₁₃	—	8	2	no	—
166	D₄×C₁₃	—	8	2	no	—
167	D₄×C₁₃	—	16	2	no	—
168	D₄×C₁₃	—	16	2	no	—
169	C₂×C₄×C₁₃	—	2	1	no	—
170	C₂×C₄×C₁₃	—	2	1	no	—
171	C₂×C₄×C₁₃	—	4	1	no	—
172	C₂×C₄×C₁₃	—	8	2	no	—
173	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
174	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
176	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×Dic₁₃	—	4	2	no	—
178	C₂×Dic₁₃	—	4	2	no	—
179	Dic₂₆	—	4	2	no	—
180	Dic₂₆	—	4	2	no	—
181	C₈×C₁₃	—	2	1	no	—
182	C₈×C₁₃	—	2	1	no	—
183	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
184	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
185	C₂×AGL(1,ℤ/8)	—	2	1	no	—
186	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
187	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
188	#541	—	26	1	no	—
189	#535	—	2	1	no	—
190	C₂×Q₃₂	—	13	2	no	—
191	C₂×SD₃₂	—	13	2	no	—
192	C₂×SD₃₂	—	13	2	no	—
193	C₂×SD₃₂	—	13	2	no	—
194	#499	—	13	1	no	—
195	#496	—	2	1	no	—
196	#496	—	2	1	no	—
197	#495	C₂×D₁₃	1	1	yes	characteristic
198	#495	C₂×D₁₃	1	1	yes	characteristic
199	#494	—	13	1	no	—
200	#493	C₂×D₁₃	1	1	yes	characteristic
201	#493	C₂×D₁₃	1	1	yes	characteristic
202	#491	—	26	1	no	—
203	#470	—	26	1	no	—
204	#467	—	13	2	no	—
205	#441	—	26	1	no	—
206	C₄×Q₁₆	—	13	2	no	—
207	#405	C₂×D₁₃	1	1	yes	characteristic
208	C₄⋊C₁₆	—	13	2	no	—
209	#358	—	13	2	no	—
210	#357	—	13	2	no	—
211	#357	—	13	2	no	—
212	#357	—	13	2	no	—
213	C₂²⋊C₁₆	—	13	2	no	—
214	#346	—	13	1	no	—
215	C₄×C₁₆	—	13	1	no	—
216	C₂²×C₁₃	—	2	2	no	—
217	C₂²×C₁₃	—	2	1	no	—
218	C₂²×C₁₃	—	8	2	no	—
219	C₂²×C₁₃	—	8	2	no	—
220	C₂²×C₁₃	—	8	2	no	—
221	C₂²×C₁₃	—	8	1	no	—
222	C₂²×C₁₃	—	8	1	no	—
223	C₂²×C₁₃	—	8	1	no	—
224	C₂²×C₁₃	—	16	4	no	—
225	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
226	C₄×C₁₃	—	2	2	no	—
227	C₄×C₁₃	—	2	2	no	—
228	C₄×C₁₃	—	2	1	no	—
229	C₄×C₁₃	—	8	1	no	—
230	C₄×C₁₃	#535	1	1	yes	characteristic
231	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	Dic₁₃	—	8	2	no	—
233	Dic₁₃	—	8	2	no	—
234	C₂×#41	—	2	1	no	—
235	C₂²×D₄	—	2	1	no	—
236	AGL(1,ℤ/8)	—	4	4	no	—
237	#135	—	4	2	no	—
238	C₂×Q₁₆	—	13	2	no	—
239	C₂×Q₁₆	—	26	1	no	—
240	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×SD₁₆	—	2	1	no	—
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×SD₁₆	—	2	1	no	—
244	C₂×D₈	—	2	1	no	—
245	C₂×D₈	—	2	1	no	—
246	C₂×D₈	—	2	1	no	—
247	C₂×D₈	—	4	2	no	—
248	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
249	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
250	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
251	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
252	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
253	C₄⋊Q₈	—	26	1	no	—
254	#127	—	2	1	no	—
255	#127	—	2	1	no	—
256	#126	—	52	2	no	—
257	#125	—	26	1	no	—
258	#125	—	52	2	no	—
259	#124	—	2	1	no	—
260	#124	—	2	1	no	—
261	C₂²⋊Q₈	—	26	2	no	—
262	#121	—	4	2	no	—
263	#120	—	4	2	no	—
264	C₄×Q₈	—	26	2	no	—
265	#117	C₂²⋊D₁₃	1	1	yes	characteristic
266	Q₃₂	—	52	2	no	—
267	SD₃₂	—	52	2	no	—
268	SD₃₂	—	52	2	no	—
269	SD₃₂	—	52	2	no	—
270	C₂×C₁₆	—	13	2	no	—
271	C₂×C₁₆	—	13	2	no	—
272	#107	—	13	2	no	—
273	#106	—	26	1	no	—
274	#103	—	26	2	no	—
275	#103	—	26	2	no	—
276	#97	C₂²⋊D₁₃	1	1	yes	characteristic
277	#97	C₂²⋊D₁₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
280	C₂×C₁₃	—	4	1	no	—
281	C₂×C₁₃	—	16	1	no	—
282	C₂×C₁₃	—	16	1	no	—
283	C₂×C₁₃	—	16	1	no	—
284	C₂×C₁₃	#3205	1	1	yes	characteristic
285	C₂×C₁₃	#3416	1	2	yes	—
286	C₂⁴	—	4	1	no	—
287	#41	—	4	2	no	—
288	#41	—	4	2	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	—	26	2	no	—
291	C₂×D₄	—	2	1	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	4	2	no	—
295	C₂×D₄	—	4	2	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	4	2	no	—
300	C₂²×C₄	—	4	1	no	—
301	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
302	Q₁₆	—	2	2	no	—
303	Q₁₆	—	26	2	no	—
304	Q₁₆	—	26	2	no	—
305	Q₁₆	—	52	2	no	—
306	SD₁₆	—	4	2	no	—
307	SD₁₆	—	4	2	no	—
308	SD₁₆	—	4	2	no	—
309	D₈	—	2	2	no	—
310	D₈	—	2	2	no	—
311	D₈	—	2	2	no	—
312	D₈	—	4	2	no	—
313	D₈	—	4	2	no	—
314	D₈	—	4	2	no	—
315	M₁₆	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
318	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
319	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
320	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
321	C₄⋊C₄	—	26	2	no	—
322	C₄⋊C₄	—	52	2	no	—
323	C₄⋊C₄	—	52	2	no	—
324	C₄⋊C₄	—	52	2	no	—
325	C₄⋊C₄	—	52	2	no	—
326	C₄⋊C₄	#6104	1	2	yes	—
327	C₂²⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	4	2	no	—
329	C₂²⋊C₄	—	52	2	no	—
330	C₂²⋊C₄	#6104	1	2	yes	—
331	C₄²	—	52	2	no	—
332	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
333	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
334	C₁₆	—	26	2	no	—
335	C₁₆	—	26	2	no	—
336	C₁₃	#33788	1	1	yes	characteristic, complemented, Sylow
337	C₂³	—	4	1	no	—
338	C₂³	—	4	1	no	—
339	C₂³	—	4	1	no	—
340	C₂³	—	8	2	no	—
341	C₂³	—	16	2	no	—
342	C₂³	C₂×#6104	1	1	yes	characteristic
343	Q₈	—	4	1	no	—
344	Q₈	—	4	1	no	—
345	Q₈	—	52	2	no	—
346	Q₈	—	52	2	no	—
347	D₄	—	4	1	no	—
348	D₄	—	4	1	no	—
349	D₄	—	4	1	no	—
350	D₄	—	4	1	no	—
351	D₄	—	4	1	no	—
352	D₄	—	4	1	no	—
353	D₄	—	8	2	no	—
354	D₄	—	8	2	no	—
355	D₄	—	16	2	no	—
356	D₄	—	16	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	8	2	no	—
361	C₂×C₄	—	52	2	no	—
362	C₂×C₄	—	52	2	no	—
363	C₂×C₄	C₂×#6104	1	2	yes	—
364	C₂×C₄	C₂×#6104	1	2	yes	—
365	C₂×C₄	C₂×#6104	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	#88186	1	2	yes	—
370	C₈	#88251	1	2	yes	—
371	C₂²	—	2	2	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	8	2	no	—
374	C₂²	—	8	2	no	—
375	C₂²	—	8	2	no	—
376	C₂²	—	8	1	no	—
377	C₂²	—	8	1	no	—
378	C₂²	—	8	1	no	—
379	C₂²	—	16	4	no	—
380	C₂²	C₂²×#6104	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	1	no	—
384	C₄	—	8	1	no	—
385	C₄	—	104	2	no	—
386	C₄	—	104	2	no	—
387	C₄	C₂×#88186	1	1	yes	characteristic
388	C₄	#11719510	1	1	yes	characteristic
389	C₂	—	4	1	no	—
390	C₂	—	16	1	no	—
391	C₂	—	16	1	no	—
392	C₂	—	16	1	no	—
393	C₂	#13125754	1	1	yes	characteristic
394	C₂	#13127983	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3155	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3155	D₁₃
hypocenter	C₁₃	#33788
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3155

Complements

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

N	H ≅ G/N	type
1. #13125495	C₂	semidirect
2. #13117108	C₂	semidirect
3. #13117091	C₂	semidirect
4. #13117077	C₂	semidirect
5. #13117067	C₂	semidirect
6. #13116895	C₂	semidirect
7. #13116873	C₂	semidirect
8. #13116776	C₂	semidirect
9. #13116765	C₂	semidirect
10. #13116643	C₂	semidirect
24. C₂×#88069	C₂²	semidirect
27. C₂×#88067	C₂²	semidirect
32. #11718360	C₂²	semidirect
33. C₄×Dic₅₂	C₂²	semidirect
34. #11718344	C₂²	semidirect
35. #11718156	C₂²	semidirect
38. #11718154	C₂²	semidirect
40. C₄⋊#6090	C₂²	semidirect
41. #11718053	C₂²	semidirect
42. C₄×#6090	C₂²	semidirect
336. C₁₃	#33788	semidirect

Maximal subgroups

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

subgroup	index
#13125495	2
#13117108	2
#13117091	2
#13117077	2
#13117067	2
#13116895	2
#13116873	2
#13116776	2
#13116765	2
#13116643	2
C₁₃×#3155	2
#33788	13