Subgroups — #16019753

Order 3328.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127229	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117059	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117019	C₂	1	2	yes	maximal, complemented
4	#13116983	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116930	C₂	1	2	yes	maximal, complemented
6	#13116873	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116765	C₂	1	2	yes	maximal, complemented
8	C₄×#88067	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11718154	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
13	D₄²×C₁₃	—	2	1	no	—
14	#11719248	—	2	1	no	—
15	C₁₃×#493	C₂²	1	1	yes	characteristic
16	C₁₃×#466	C₂²	1	2	yes	—
17	C₁₃×#459	—	2	1	no	—
18	C₁₃×#447	—	2	2	no	—
19	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
20	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×#88067	C₂²	1	1	yes	characteristic
22	C₂×#88067	C₂²	1	2	yes	complemented
23	#11718777	C₂²	1	2	yes	complemented
24	C₂×#88058	C₂²	1	2	yes	complemented
25	C₂²×#6090	C₂²	1	2	yes	complemented
26	#11718511	—	2	1	no	—
27	#11718383	—	2	2	no	—
28	C₄×Dic₅₂	C₂²	1	1	yes	characteristic, complemented
29	#11718343	C₂²	1	1	yes	characteristic, complemented
30	#11718154	C₂²	1	1	yes	characteristic
31	#11718154	C₂²	1	2	yes	—
32	#11718154	C₂²	1	2	yes	complemented
33	#11718154	C₂²	1	2	yes	complemented
34	#11718154	C₂²	1	2	yes	complemented
35	C₂²⋊#6090	C₂²	1	2	yes	complemented
36	#11718084	C₂²	1	1	yes	characteristic, complemented
37	#11718082	C₂²	1	2	yes	complemented
38	#11718082	C₂²	1	1	yes	characteristic, complemented
39	C₄⋊#6090	C₂²	1	1	yes	characteristic, complemented
40	C₄×#6090	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₄×C₁₃	—	2	2	no	—
42	C₂²×D₄×C₁₃	—	4	1	no	—
43	C₂×C₁₃×D₈	—	2	4	no	—
44	C₂×C₁₃×D₈	—	2	2	no	—
45	C₂×C₁₃×D₈	—	2	1	no	—
46	C₂×C₁₃×D₈	—	4	2	no	—
47	C₂×C₁₃×D₈	C₂³	1	2	yes	—
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
50	C₁₃×#127	—	2	1	no	—
51	C₁₃×#121	—	2	2	no	—
52	C₁₃×#121	—	4	1	no	—
53	C₁₃×#120	—	4	2	no	—
54	C₄×D₄×C₁₃	—	2	1	no	—
55	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
56	D₄×Dic₁₃	—	4	1	no	—
57	C₂×C₄⋊Dic₁₃	—	2	2	no	—
58	#88178	—	2	2	no	—
59	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
60	#88144	—	4	1	no	—
61	#88132	—	4	2	no	—
62	#88123	—	2	1	no	—
63	C₄×Dic₂₆	—	2	1	no	—
64	C₁₃×#107	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₃×#102	—	2	1	no	—
67	C₁₃×#102	—	2	2	no	—
68	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88067	—	2	2	no	—
71	#88067	—	4	2	no	—
72	#88058	—	2	2	no	—
73	#88058	—	2	1	no	—
74	#88058	C₂³	1	1	yes	characteristic
75	#88058	C₂³	1	2	yes	—
76	#88056	—	2	2	no	—
77	#88056	—	2	1	no	—
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	#33862	—	13	1	no	maximal, Sylow
83	C₂⁴×C₁₃	—	4	2	no	—
84	C₂×D₄×C₁₃	—	2	1	no	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	4	4	no	—
87	C₂×D₄×C₁₃	—	4	2	no	—
88	C₂×D₄×C₁₃	—	4	2	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	4	1	no	—
91	C₂×D₄×C₁₃	—	8	2	no	—
92	C₂×D₄×C₁₃	—	8	4	no	—
93	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	—	4	1	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂²×Dic₁₃	—	4	2	no	—
97	C₂×Dic₂₆	—	2	1	no	—
98	C₂×Dic₂₆	—	4	1	no	—
99	C₁₃×D₈	—	2	2	no	—
100	C₁₃×D₈	—	2	1	no	—
101	C₁₃×D₈	—	2	2	no	—
102	C₁₃×D₈	—	4	4	no	—
103	C₁₃×D₈	—	8	1	no	—
104	C₁₃×D₈	#41	1	2	yes	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
112	C₁₃×C₂²⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂²⋊Dic₁₃	—	4	2	no	—
117	C₄⋊Dic₁₃	—	2	1	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	—	4	1	no	—
120	C₄⋊Dic₁₃	—	4	2	no	—
121	#6101	—	4	1	no	—
122	#6101	—	4	2	no	—
123	C₄×Dic₁₃	—	4	1	no	—
124	Dic₅₂	—	2	1	no	—
125	Dic₅₂	—	2	1	no	—
126	#6090	—	2	1	no	—
127	#6090	—	2	2	no	—
128	#6090	—	2	1	no	—
129	#3338	—	13	1	no	—
130	D₄×D₈	D₁₃	1	1	yes	characteristic
131	#2240	—	13	1	no	—
132	#2233	—	13	2	no	—
133	#2226	—	13	1	no	—
134	#2217	—	13	2	no	—
135	#2207	—	13	1	no	—
136	#2187	—	13	2	no	—
137	C₄×SD₃₂	—	13	1	no	—
138	D₄×C₁₆	—	13	1	no	—
139	C₂×#357	—	13	2	no	—
140	C₂³×C₁₃	—	4	1	no	—
141	C₂³×C₁₃	—	4	2	no	—
142	C₂³×C₁₃	—	8	4	no	—
143	C₂³×C₁₃	—	8	4	no	—
144	C₂³×C₁₃	—	16	2	no	—
145	C₂³×C₁₃	C₂×D₈	1	2	yes	—
146	D₄×C₁₃	—	2	2	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	4	1	no	—
149	D₄×C₁₃	—	4	2	no	—
150	D₄×C₁₃	—	8	4	no	—
151	D₄×C₁₃	—	8	4	no	—
152	D₄×C₁₃	—	8	2	no	—
153	D₄×C₁₃	—	16	1	no	—
154	D₄×C₁₃	SD₃₂	1	4	yes	complemented
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	4	1	no	—
158	C₂×C₄×C₁₃	—	8	2	no	—
159	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×Dic₁₃	—	4	1	no	—
164	C₂×Dic₁₃	—	4	1	no	—
165	C₂×Dic₁₃	—	4	2	no	—
166	C₂×Dic₁₃	—	16	2	no	—
167	Dic₂₆	—	4	1	no	—
168	Dic₂₆	—	4	1	no	—
169	Dic₂₆	—	16	1	no	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	—	2	1	no	—
172	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
173	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂²×D₈	C₂×D₁₃	1	2	yes	—
175	#554	—	26	1	no	—
176	D₄²	—	2	1	no	—
177	C₂×SD₃₂	—	13	1	no	—
178	C₂×SD₃₂	—	13	2	no	—
179	C₂²×C₁₆	—	13	2	no	—
180	#500	—	13	1	no	—
181	#493	C₂×D₁₃	1	1	yes	characteristic
182	#484	—	26	2	no	—
183	#477	—	26	1	no	—
184	#467	—	13	2	no	—
185	#466	C₂×D₁₃	1	2	yes	—
186	#459	—	2	1	no	—
187	#447	—	2	2	no	—
188	C₄×Q₁₆	—	13	1	no	—
189	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
190	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
191	C₂×#107	—	13	2	no	—
192	#367	—	13	1	no	—
193	#367	—	13	2	no	—
194	C₄⋊C₁₆	—	13	1	no	—
195	#358	—	13	1	no	—
196	#357	—	13	1	no	—
197	#357	—	13	2	no	—
198	#357	—	13	2	no	—
199	#357	—	13	2	no	—
200	#357	—	13	2	no	—
201	C₂²⋊C₁₆	—	13	2	no	—
202	C₄×C₁₆	—	13	1	no	—
203	C₂²×C₁₃	—	2	2	no	—
204	C₂²×C₁₃	—	2	2	no	—
205	C₂²×C₁₃	—	4	2	no	—
206	C₂²×C₁₃	—	8	1	no	—
207	C₂²×C₁₃	—	8	1	no	—
208	C₂²×C₁₃	—	8	2	no	—
209	C₂²×C₁₃	—	8	2	no	—
210	C₂²×C₁₃	—	8	2	no	—
211	C₂²×C₁₃	—	16	8	no	—
212	C₂²×C₁₃	C₂×SD₃₂	1	4	yes	—
213	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	—	2	2	no	—
216	C₄×C₁₃	—	8	1	no	—
217	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
218	C₄×C₁₃	#546	1	1	yes	characteristic
219	C₄×C₁₃	#577	1	1	yes	characteristic
220	Dic₁₃	—	8	2	no	—
221	Dic₁₃	—	8	1	no	—
222	Dic₁₃	—	8	1	no	—
223	C₂²×D₄	—	2	2	no	—
224	C₂²×D₄	—	4	1	no	—
225	C₂×Q₁₆	—	13	1	no	—
226	C₂×D₈	—	2	4	no	—
227	C₂×D₈	—	2	2	no	—
228	C₂×D₈	—	2	1	no	—
229	C₂×D₈	—	4	2	no	—
230	C₂×D₈	C₂²×D₁₃	1	2	yes	—
231	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
233	C₄⋊Q₈	—	26	1	no	—
234	#127	—	2	1	no	—
235	#125	—	52	1	no	—
236	C₂²⋊Q₈	—	26	2	no	—
237	C₂²⋊Q₈	—	52	2	no	—
238	#121	—	2	2	no	—
239	#121	—	4	1	no	—
240	#120	—	4	2	no	—
241	C₄×Q₈	—	26	1	no	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	—	52	1	no	—
244	C₄×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
245	C₂×C₄⋊C₄	—	26	2	no	—
246	SD₃₂	—	26	2	no	—
247	SD₃₂	—	52	2	no	—
248	C₂×C₁₆	—	13	1	no	—
249	C₂×C₁₆	—	13	1	no	—
250	C₂×C₁₆	—	13	2	no	—
251	C₂×C₁₆	—	52	2	no	—
252	#107	—	13	1	no	—
253	#107	—	13	2	no	—
254	#107	—	26	2	no	—
255	#107	—	26	1	no	—
256	#107	C₂²×D₁₃	1	1	yes	characteristic
257	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
258	#103	—	26	1	no	—
259	#103	—	26	2	no	—
260	#102	—	2	1	no	—
261	#102	—	2	2	no	—
262	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₂×C₁₃	—	2	4	no	—
265	C₂×C₁₃	—	8	2	no	—
266	C₂×C₁₃	—	16	2	no	—
267	C₂×C₁₃	#3294	1	1	yes	characteristic
268	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
269	C₂×C₁₃	#3415	1	1	yes	characteristic
270	C₂⁴	—	4	2	no	—
271	C₂×Q₈	—	26	1	no	—
272	C₂×Q₈	—	52	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	4	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	8	2	no	—
281	C₂×D₄	—	8	4	no	—
282	C₂×D₄	#6104	1	1	yes	characteristic
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	52	2	no	—
285	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
286	Q₁₆	—	26	1	no	—
287	Q₁₆	—	26	1	no	—
288	D₈	—	2	2	no	—
289	D₈	—	2	1	no	—
290	D₈	—	2	2	no	—
291	D₈	—	4	4	no	—
292	D₈	—	8	1	no	—
293	D₈	#6129	1	2	yes	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
298	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	26	1	no	—
301	C₄⋊C₄	—	26	2	no	—
302	C₄⋊C₄	—	52	2	no	—
303	C₄⋊C₄	—	52	1	no	—
304	C₄⋊C₄	—	52	1	no	—
305	C₄⋊C₄	—	52	2	no	—
306	C₄⋊C₄	#6104	1	1	yes	characteristic
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	52	2	no	—
310	C₂²⋊C₄	#6104	1	2	yes	—
311	C₄²	—	52	1	no	—
312	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
313	C₁₆	—	26	1	no	—
314	C₁₆	—	26	1	no	—
315	C₁₆	—	26	2	no	—
316	C₁₃	#33862	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	4	1	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	8	4	no	—
320	C₂³	—	8	4	no	—
321	C₂³	—	16	2	no	—
322	C₂³	C₂×#6104	1	2	yes	—
323	Q₈	—	52	1	no	—
324	Q₈	—	52	1	no	—
325	Q₈	—	208	1	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	1	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	4	no	—
331	D₄	—	8	4	no	—
332	D₄	—	8	2	no	—
333	D₄	—	16	1	no	—
334	D₄	#88067	1	4	yes	complemented
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	52	1	no	—
340	C₂×C₄	—	52	1	no	—
341	C₂×C₄	—	52	2	no	—
342	C₂×C₄	—	208	2	no	—
343	C₂×C₄	C₂×#6104	1	1	yes	characteristic
344	C₂×C₄	C₂×#6104	1	2	yes	—
345	C₂×C₄	C₂×#6104	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	2	1	no	—
349	C₈	C₂×#6129	1	1	yes	characteristic
350	C₈	#88251	1	1	yes	characteristic
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	16	8	no	—
360	C₂²	C₂×#88067	1	4	yes	—
361	C₂²	C₂²×#6104	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	104	1	no	—
366	C₄	—	104	2	no	—
367	C₄	—	104	1	no	—
368	C₄	C₂×#88067	1	2	yes	—
369	C₄	#11719511	1	1	yes	characteristic
370	C₄	#11719541	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	—	8	2	no	—
373	C₂	—	16	2	no	—
374	C₂	#13126649	1	1	yes	characteristic
375	C₂	C₂²×#88067	1	1	yes	characteristic
376	C₂	#13127975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	D₄×C₁₃×D₈	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	D₄×D₈	D₁₃
hypocenter	C₁₃	#33862
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×D₈

Complements

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

N	H ≅ G/N	type
1. #13127229	C₂	semidirect
2. #13117059	C₂	semidirect
3. #13117019	C₂	semidirect
4. #13116983	C₂	semidirect
5. #13116930	C₂	semidirect
6. #13116873	C₂	semidirect
7. #13116765	C₂	semidirect
8. C₄×#88067	C₂	semidirect
9. D₄×#6090	C₂	semidirect
10. C₂×#11718154	C₂	semidirect
22. C₂×#88067	C₂²	semidirect
23. #11718777	C₂²	semidirect
24. C₂×#88058	C₂²	semidirect
25. C₂²×#6090	C₂²	semidirect
28. C₄×Dic₅₂	C₂²	semidirect
29. #11718343	C₂²	semidirect
32. #11718154	C₂²	semidirect
33. #11718154	C₂²	semidirect
34. #11718154	C₂²	semidirect
35. C₂²⋊#6090	C₂²	semidirect
36. #11718084	C₂²	semidirect
37. #11718082	C₂²	semidirect
38. #11718082	C₂²	semidirect
39. C₄⋊#6090	C₂²	semidirect
40. C₄×#6090	C₂²	semidirect
154. D₄×C₁₃	SD₃₂	semidirect
316. C₁₃	#33862	semidirect
334. D₄	#88067	semidirect

Maximal subgroups

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

subgroup	index
#13127229	2
#13117059	2
#13117019	2
#13116983	2
#13116930	2
#13116873	2
#13116765	2
C₄×#88067	2
D₄×#6090	2
C₂×#11718154	2
D₄×C₁₃×D₈	2
#33862	13