Subgroups — #15867353

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123638	C₂	1	1	yes	maximal, characteristic
2	#13123485	C₂	1	2	yes	maximal, complemented
3	#13115729	C₂	1	1	yes	maximal, characteristic
4	#13115724	C₂	1	1	yes	maximal, characteristic
5	#13115589	C₂	1	1	yes	maximal, characteristic, complemented
6	#13115584	C₂	1	1	yes	maximal, characteristic, complemented
7	#13115547	C₂	1	2	yes	maximal, complemented
8	#13115540	C₂	1	2	yes	maximal, complemented
9	#13114742	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#88045	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88158	C₂²	1	1	yes	characteristic
13	C₄×#6127	C₂²	1	2	yes	—
14	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₄³×C₁₃	C₂²	1	1	yes	characteristic
21	#11718774	C₂²	1	2	yes	—
22	#11718771	C₂²	1	2	yes	—
23	#11718771	C₂²	1	2	yes	—
24	C₂×#88059	C₂²	1	1	yes	characteristic
25	C₂×#88059	C₂²	1	1	yes	characteristic
26	#11718767	C₂²	1	1	yes	characteristic
27	#11718763	C₂²	1	2	yes	—
28	#11718763	C₂²	1	2	yes	—
29	C₂×#88045	C₂²	1	2	yes	—
30	C₂×#88045	C₂²	1	2	yes	—
31	C₄×#6099	C₂²	1	2	yes	—
32	#11718377	C₄	1	2	yes	complemented
33	#11718372	C₄	1	2	yes	complemented
34	#11718371	C₄	1	4	yes	complemented
35	#11718370	—	2	4	no	—
36	#11718366	—	2	4	no	—
37	#11718144	C₂²	1	2	yes	—
38	#11718115	C₂²	1	2	yes	—
39	#11718115	C₂²	1	2	yes	—
40	C₂×#6127	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₄×C₂²⋊D₁₃	—	2	4	no	—
50	#88180	C₂³	1	2	yes	—
51	C₂×#6099	C₂³	1	2	yes	—
52	C₂×#6099	C₂³	1	2	yes	—
53	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
54	#88158	—	2	2	no	—
55	#88158	—	2	2	no	—
56	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
57	C₂×#6095	C₂×C₄	1	2	yes	—
58	C₄×D₅₂	C₂×C₄	1	4	yes	—
59	#88126	—	2	2	no	—
60	C₄²×D₁₃	—	2	2	no	—
61	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
62	C₁₃×#97	—	2	2	no	—
63	C₁₃×#97	—	2	2	no	—
64	C₁₃×#97	C₂×C₄	1	2	yes	—
65	C₁₃×#97	C₂×C₄	1	2	yes	—
66	C₁₃×#97	D₄	1	4	yes	—
67	C₄×C₈×C₁₃	—	2	2	no	—
68	C₄×C₈×C₁₃	—	2	2	no	—
69	#88061	C₂×C₄	1	4	yes	—
70	#88060	—	2	4	no	—
71	#88060	—	2	4	no	—
72	#88059	—	2	2	no	—
73	#88059	—	2	2	no	—
74	#88058	C₂×C₄	1	2	yes	—
75	#88057	C₂×C₄	1	2	yes	—
76	#88056	C₂×C₄	1	4	yes	—
77	#88054	—	2	4	no	—
78	#88054	—	2	4	no	—
79	#88045	—	2	4	no	—
80	#88045	—	2	4	no	—
81	#88043	—	2	2	no	—
82	C₄×#1055	—	2	2	no	—
83	#21297	—	13	1	no	maximal, Sylow
84	C₂²×C₄×C₁₃	—	2	1	no	—
85	C₂²×C₄×C₁₃	—	2	1	no	—
86	C₂²×C₄×C₁₃	#41	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₃	—	2	2	no	—
89	#6127	—	2	2	no	—
90	#6127	—	2	2	no	—
91	#6127	—	2	2	no	—
92	#6127	—	2	2	no	—
93	C₂×D₅₂	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₁₃	—	2	2	no	—
95	C₂×C₄×D₁₃	—	2	4	no	—
96	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	—	2	1	no	—
98	C₂×C₈×C₁₃	—	2	1	no	—
99	C₂×C₈×C₁₃	—	2	1	no	—
100	C₂×C₈×C₁₃	—	2	1	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₁₃	#41	1	2	yes	—
109	C₂×C₈×C₁₃	#41	1	2	yes	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	—	2	1	no	—
112	C₄²×C₁₃	—	4	4	no	—
113	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
114	C₄²×C₁₃	C₂×D₄	1	2	yes	—
115	C₄²×C₁₃	#41	1	2	yes	—
116	C₄²×C₁₃	#41	1	2	yes	—
117	C₂²⋊Dic₁₃	—	2	2	no	—
118	#6103	—	2	4	no	—
119	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
120	#6101	—	2	4	no	—
121	C₄×Dic₁₃	—	2	4	no	—
122	C₄×Dic₁₃	—	2	2	no	—
123	#6099	—	2	2	no	—
124	#6099	—	2	2	no	—
125	#6099	—	2	2	no	—
126	#6099	—	2	2	no	—
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	—	2	2	no	—
129	Dic₅₂	—	2	2	no	—
130	D₁₀₄	—	2	2	no	—
131	#6095	—	2	4	no	—
132	#2943	—	13	1	no	—
133	#2923	—	13	2	no	—
134	#1975	—	13	1	no	—
135	#1974	—	13	1	no	—
136	#1946	—	13	1	no	—
137	#1945	—	13	1	no	—
138	#1938	—	13	2	no	—
139	#1937	—	13	2	no	—
140	#1778	—	13	1	no	—
141	C₄×#104	—	13	2	no	—
142	C₄×#97	D₁₃	1	1	yes	characteristic, complemented
143	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
144	C₂²×D₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	4	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂²⋊D₁₃	—	4	2	no	—
158	C₂²⋊D₁₃	—	4	2	no	—
159	C₂×Dic₁₃	—	2	2	no	—
160	C₂×Dic₁₃	—	2	4	no	—
161	D₅₂	—	2	2	no	—
162	D₅₂	—	2	2	no	—
163	C₄×D₁₃	—	4	2	no	—
164	C₄×D₁₃	—	4	2	no	—
165	C₄×D₁₃	—	8	4	no	—
166	Dic₂₆	—	2	2	no	—
167	Dic₂₆	—	2	2	no	—
168	C₈×C₁₃	—	2	2	no	—
169	C₈×C₁₃	—	2	2	no	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	#117	1	2	yes	—
172	C₈×C₁₃	#117	1	2	yes	—
173	#1055	—	4	2	no	—
174	#1055	—	4	2	no	—
175	C₂×#135	—	13	1	no	—
176	C₄×#41	—	13	2	no	—
177	#442	—	13	2	no	—
178	#441	—	13	2	no	—
179	#440	—	13	4	no	—
180	#433	—	26	4	no	—
181	#432	—	26	4	no	—
182	C₂×#108	—	13	1	no	—
183	C₂×#108	—	13	1	no	—
184	#427	—	13	1	no	—
185	#424	—	13	2	no	—
186	#424	—	13	2	no	—
187	C₂×#104	—	13	2	no	—
188	C₂×#104	—	13	2	no	—
189	#416	—	13	2	no	—
190	#408	—	13	2	no	—
191	#408	—	13	2	no	—
192	C₄×M₁₆	—	13	2	no	—
193	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
194	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
195	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
196	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
199	C₄³	C₂×D₁₃	1	1	yes	characteristic
200	#341	—	13	2	no	—
201	#339	—	13	2	no	—
202	#339	—	13	2	no	—
203	C₂²×C₁₃	—	2	2	no	—
204	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂²×C₁₃	#529	1	1	yes	characteristic
206	C₂²×C₁₃	#529	1	1	yes	characteristic
207	C₂×D₁₃	—	4	4	no	—
208	C₂×D₁₃	—	4	2	no	—
209	C₄×C₁₃	—	2	4	no	—
210	C₄×C₁₃	—	2	4	no	—
211	C₄×C₁₃	—	4	2	no	—
212	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
213	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
214	C₄×C₁₃	#532	1	1	yes	characteristic
215	C₄×C₁₃	#532	1	1	yes	characteristic
216	Dic₁₃	—	4	4	no	—
217	Dic₁₃	—	4	2	no	—
218	C₂×#41	—	13	2	no	—
219	#135	—	26	2	no	—
220	#135	—	26	2	no	—
221	C₂×Q₁₆	—	13	1	no	—
222	C₂×SD₁₆	—	13	2	no	—
223	C₂×D₈	—	13	1	no	—
224	C₂×M₁₆	—	13	2	no	—
225	C₂×M₁₆	—	13	2	no	—
226	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
230	C₄×Q₈	—	13	4	no	—
231	C₄×D₄	—	13	4	no	—
232	C₄×D₄	—	26	4	no	—
233	#117	—	13	2	no	—
234	#117	—	26	2	no	—
235	C₂×C₄²	—	2	2	no	—
236	C₂×C₄²	—	26	2	no	—
237	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
240	#108	—	26	2	no	—
241	#108	—	26	2	no	—
242	#107	—	13	2	no	—
243	#106	—	13	2	no	—
244	C₄⋊C₈	—	26	4	no	—
245	C₄⋊C₈	—	26	4	no	—
246	#104	—	26	4	no	—
247	#104	—	26	4	no	—
248	#103	—	13	4	no	—
249	#102	—	13	4	no	—
250	C₂²⋊C₈	—	26	4	no	—
251	C₂²⋊C₈	—	26	4	no	—
252	#97	—	2	2	no	—
253	#97	—	2	2	no	—
254	#97	—	26	2	no	—
255	#97	C₄×D₁₃	1	2	yes	—
256	#97	C₄×D₁₃	1	2	yes	—
257	#97	D₅₂	1	4	yes	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	2	2	no	—
260	C₄×C₈	—	26	2	no	—
261	C₂×C₁₃	—	2	1	no	—
262	C₂×C₁₃	—	2	1	no	—
263	C₂×C₁₃	#2293	1	1	yes	characteristic
264	C₂×C₁₃	#2956	1	2	yes	—
265	D₁₃	—	8	2	no	—
266	#41	—	26	2	no	—
267	#41	—	26	2	no	—
268	#41	—	26	2	no	—
269	#41	—	26	2	no	—
270	C₂×Q₈	—	13	2	no	—
271	C₂×D₄	—	13	2	no	—
272	C₂×D₄	—	26	2	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	26	2	no	—
276	C₂²×C₄	—	26	4	no	—
277	C₂²×C₄	#6127	1	2	yes	—
278	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
279	Q₁₆	—	26	2	no	—
280	SD₁₆	—	26	4	no	—
281	D₈	—	26	2	no	—
282	M₁₆	—	26	2	no	—
283	M₁₆	—	26	2	no	—
284	M₁₆	—	26	2	no	—
285	M₁₆	—	26	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	26	2	no	—
293	C₂×C₈	—	26	2	no	—
294	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
295	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
297	C₂×C₈	C₂×D₅₂	1	2	yes	—
298	C₂×C₈	C₂×D₅₂	1	2	yes	—
299	C₂×C₈	#6127	1	2	yes	—
300	C₂×C₈	#6127	1	2	yes	—
301	C₄⋊C₄	—	13	4	no	—
302	C₄⋊C₄	—	26	4	no	—
303	C₂²⋊C₄	—	26	2	no	—
304	C₂²⋊C₄	—	26	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	4	4	no	—
308	C₄²	—	26	4	no	—
309	C₄²	—	26	2	no	—
310	C₄²	C₂×C₄×D₁₃	1	2	yes	—
311	C₄²	C₂×D₅₂	1	2	yes	—
312	C₄²	#6127	1	2	yes	—
313	C₄²	#6127	1	2	yes	—
314	C₁₃	#21297	1	1	yes	characteristic, complemented, Sylow
315	C₂³	—	26	2	no	—
316	C₂³	C₂×#6127	1	1	yes	characteristic
317	Q₈	—	26	2	no	—
318	Q₈	—	26	2	no	—
319	D₄	—	26	2	no	—
320	D₄	—	26	2	no	—
321	D₄	—	52	2	no	—
322	D₄	—	52	2	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	26	4	no	—
329	C₂×C₄	—	26	2	no	—
330	C₂×C₄	—	52	2	no	—
331	C₂×C₄	—	52	2	no	—
332	C₂×C₄	—	104	4	no	—
333	C₂×C₄	C₄×D₅₂	1	4	yes	—
334	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
336	C₂×C₄	C₂×#6127	1	1	yes	characteristic
337	C₂×C₄	C₂×#6127	1	1	yes	characteristic
338	C₂×C₄	C₂×#6127	1	1	yes	characteristic
339	C₂×C₄	C₂×#6127	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	2	2	no	—
342	C₈	—	2	2	no	—
343	C₈	—	52	2	no	—
344	C₈	—	52	2	no	—
345	C₈	#88126	1	2	yes	—
346	C₈	#88126	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	52	2	no	—
349	C₂²	—	52	4	no	—
350	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
351	C₂²	#11719184	1	1	yes	characteristic
352	C₂²	#11719184	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	2	4	no	—
355	C₄	—	4	2	no	—
356	C₄	—	52	2	no	—
357	C₄	—	52	4	no	—
358	C₄	C₂×#88126	1	1	yes	characteristic
359	C₄	C₄×#6127	1	1	yes	characteristic
360	C₄	#11719182	1	1	yes	characteristic
361	C₄	#11719182	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	104	2	no	—
365	C₂	#13117300	1	1	yes	characteristic
366	C₂	#13123747	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#97	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88158	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	#21297
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. #13123485	C₂	semidirect
5. #13115589	C₂	semidirect
6. #13115584	C₂	semidirect
7. #13115547	C₂	semidirect
8. #13115540	C₂	semidirect
9. #13114742	C₂	semidirect
10. C₄×#88045	C₂	semidirect
11. C₄×C₁₃×#97	C₂	semidirect
32. #11718377	C₄	semidirect
33. #11718372	C₄	semidirect
34. #11718371	C₄	semidirect
142. C₄×#97	D₁₃	semidirect
314. C₁₃	#21297	semidirect

Maximal subgroups

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

subgroup	index
#13123638	2
#13123485	2
#13115729	2
#13115724	2
#13115589	2
#13115584	2
#13115547	2
#13115540	2
#13114742	2
C₄×#88045	2
C₄×C₁₃×#97	2
#21297	13