Subgroups — #15881802

Order 3328.

730 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
2	#13117794	C₂	1	2	yes	maximal
3	#13115607	C₂	1	2	yes	maximal
4	#13115604	C₂	1	2	yes	maximal
5	#13115593	C₂	1	2	yes	maximal
6	#13115003	C₂	1	2	yes	maximal
7	#13115000	C₂	1	1	yes	maximal, characteristic
8	#13114735	C₂	1	2	yes	maximal
9	C₄×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×#88058	C₂²	1	1	yes	characteristic
16	C₂×#88058	C₂²	1	1	yes	characteristic
17	C₂×#88057	C₂²	1	2	yes	—
18	C₂×#88055	C₂²	1	2	yes	—
19	C₂×#88054	C₂²	1	2	yes	—
20	#11718638	C₂²	1	2	yes	—
21	#11718637	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
24	C₂×#88048	C₂²	1	2	yes	—
25	C₂×#88047	C₂²	1	2	yes	—
26	#11718597	C₂²	1	2	yes	—
27	#11718594	—	2	2	no	—
28	C₄×#6101	—	2	2	no	—
29	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
30	#11718340	C₄	1	8	yes	complemented
31	#11718315	—	2	2	no	—
32	#11718307	—	2	2	no	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	#11718121	C₂²	1	2	yes	—
35	#11718121	C₂²	1	2	yes	—
36	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	—	2	2	no	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
46	C₂×#6101	—	2	2	no	—
47	C₂×#6101	—	2	2	no	—
48	C₂×#6101	D₄	1	2	yes	—
49	C₂×C₄×Dic₁₃	—	2	2	no	—
50	C₂×C₄×Dic₁₃	—	2	2	no	—
51	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
52	C₂²×#1055	C₂³	1	2	yes	—
53	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₆	Q₈	1	4	yes	—
55	C₁₃×#107	—	2	2	no	—
56	C₁₃×#107	Q₈	1	4	yes	complemented
57	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
58	C₁₃×#95	—	2	2	no	—
59	#88071	—	2	2	no	—
60	#88071	—	2	2	no	—
61	#88071	—	2	2	no	—
62	#88071	—	2	2	no	—
63	#88071	—	2	2	no	—
64	#88071	—	2	2	no	—
65	#88058	—	4	1	no	—
66	#88058	—	4	1	no	—
67	#88057	—	2	4	no	—
68	#88055	C₂×C₄	1	8	yes	—
69	#88054	C₂×C₄	1	8	yes	—
70	#88048	—	4	2	no	—
71	#88047	—	4	2	no	—
72	C₄⋊#1055	C₂×C₄	1	4	yes	—
73	#22731	—	13	1	no	maximal, Sylow
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	2	2	no	—
80	C₂²×Dic₁₃	—	2	2	no	—
81	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
82	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
83	C₂×Dic₂₆	—	4	1	no	—
84	C₂×Dic₂₆	—	4	1	no	—
85	C₂×Dic₂₆	C₄⋊C₄	1	4	yes	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₁₃	#41	1	2	yes	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
96	C₄²×C₁₃	—	8	2	no	—
97	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	2	2	no	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
104	#6101	—	4	4	no	—
105	#6101	—	8	2	no	—
106	#6101	C₄⋊C₄	1	8	yes	—
107	C₄×Dic₁₃	—	4	4	no	—
108	C₄×Dic₁₃	—	8	2	no	—
109	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
110	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	C₂²×C₄	1	4	yes	—
114	C₂×#446	—	13	1	no	—
115	#2355	—	13	2	no	—
116	#1949	—	13	2	no	—
117	#1949	—	13	2	no	—
118	#1947	—	13	2	no	—
119	#1847	—	13	2	no	—
120	#1846	—	13	1	no	—
121	#1776	—	13	2	no	—
122	C₄×#107	D₁₃	1	1	yes	characteristic
123	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×Dic₁₃	—	2	2	no	—
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	4	2	no	—
147	C₂×Dic₁₃	—	4	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	4	2	no	—
151	C₂×Dic₁₃	—	4	2	no	—
152	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
153	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
154	Dic₂₆	—	4	4	no	—
155	C₈×C₁₃	—	2	2	no	—
156	C₈×C₁₃	C₄×Q₈	1	4	yes	—
157	#1055	—	2	4	no	—
158	C₂×C₄×Q₈	—	13	1	no	—
159	#446	—	13	8	no	—
160	C₂×#107	—	13	1	no	—
161	C₂×#107	—	13	1	no	—
162	C₂×#107	—	13	2	no	—
163	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
164	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
165	C₂×#106	—	13	2	no	—
166	C₂×#106	—	13	2	no	—
167	C₂×C₄⋊C₈	—	13	2	no	—
168	C₂×C₄⋊C₈	—	13	1	no	—
169	C₂×#97	—	13	2	no	—
170	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
171	#391	—	26	2	no	—
172	#387	—	26	2	no	—
173	#387	—	26	2	no	—
174	#384	—	13	2	no	—
175	#382	—	13	2	no	—
176	#382	—	13	2	no	—
177	C₄×C₄⋊C₄	—	13	2	no	—
178	C₄×C₄⋊C₄	—	13	2	no	—
179	C₄×C₄⋊C₄	—	26	2	no	—
180	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
181	#340	—	13	2	no	—
182	#340	—	13	2	no	—
183	#340	C₂×D₁₃	1	2	yes	—
184	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
186	C₂²×C₁₃	#550	1	1	yes	characteristic
187	C₂²×C₁₃	#575	1	1	yes	characteristic
188	C₂²×C₁₃	#575	1	1	yes	characteristic
189	C₂²×C₁₃	#576	1	1	yes	characteristic
190	C₂²×C₁₃	#578	1	1	yes	characteristic
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	4	4	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₄×C₁₃	#520	1	1	yes	characteristic
197	C₄×C₁₃	#557	1	1	yes	characteristic
198	C₄×C₁₃	Q₈²	1	1	yes	characteristic
199	Dic₁₃	—	2	4	no	—
200	Dic₁₃	—	2	4	no	—
201	Dic₁₃	—	8	2	no	—
202	Dic₁₃	—	8	2	no	—
203	C₂²×Q₈	—	13	1	no	—
204	C₂²×C₈	—	13	2	no	—
205	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₄×Q₈	—	13	4	no	—
208	C₄×Q₈	—	13	4	no	—
209	C₂×C₄⋊C₄	—	13	2	no	—
210	C₂×C₄⋊C₄	—	13	1	no	—
211	C₂×C₄⋊C₄	—	13	2	no	—
212	C₂×C₄⋊C₄	—	13	2	no	—
213	C₂×C₄⋊C₄	—	26	2	no	—
214	C₂×C₄⋊C₄	—	26	2	no	—
215	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	13	2	no	—
219	C₂×C₄²	—	26	2	no	—
220	C₂×C₄²	—	26	2	no	—
221	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	#107	—	2	2	no	—
223	#107	—	52	2	no	—
224	#107	—	52	1	no	—
225	#107	—	52	1	no	—
226	#107	Dic₂₆	1	4	yes	complemented
227	#106	—	26	4	no	—
228	#106	—	52	2	no	—
229	C₄⋊C₈	—	13	8	no	—
230	C₄⋊C₈	—	13	4	no	—
231	#97	—	13	8	no	—
232	C₄×C₈	C₄×D₁₃	1	4	yes	—
233	#95	—	2	2	no	—
234	#95	—	26	2	no	—
235	#95	—	26	2	no	—
236	#95	—	26	2	no	—
237	#95	—	26	2	no	—
238	#95	—	26	2	no	—
239	#95	—	26	2	no	—
240	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₁₃	#2911	1	1	yes	characteristic
242	C₂×C₁₃	#2913	1	1	yes	characteristic
243	C₂×C₁₃	#3389	1	1	yes	characteristic
244	C₂×C₁₃	#3390	1	1	yes	characteristic
245	C₂×C₁₃	#3402	1	1	yes	characteristic
246	C₂×C₁₃	#3403	1	1	yes	characteristic
247	C₂×Q₈	—	13	4	no	—
248	C₂×Q₈	—	52	1	no	—
249	C₂×Q₈	—	52	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	13	2	no	—
254	C₂²×C₄	—	26	2	no	—
255	C₂²×C₄	—	26	2	no	—
256	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
258	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	13	4	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
265	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
267	C₂×C₈	#6127	1	2	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	13	8	no	—
272	C₄⋊C₄	—	13	4	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄⋊C₄	—	52	4	no	—
278	C₄⋊C₄	—	104	2	no	—
279	C₄⋊C₄	C₂×Dic₂₆	1	4	yes	—
280	C₄²	—	8	2	no	—
281	C₄²	—	13	4	no	—
282	C₄²	—	13	4	no	—
283	C₄²	—	52	4	no	—
284	C₄²	—	104	2	no	—
285	C₄²	C₂×C₄×D₁₃	1	2	yes	—
286	C₄²	Q₈×D₁₃	1	2	yes	—
287	C₁₃	#22731	1	1	yes	characteristic, complemented, Sylow
288	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
289	Q₈	—	52	4	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	13	4	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	26	2	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	52	2	no	—
308	C₂×C₄	—	52	2	no	—
309	C₂×C₄	—	52	2	no	—
310	C₂×C₄	—	52	2	no	—
311	C₂×C₄	—	52	2	no	—
312	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
313	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
314	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂×#6127	1	1	yes	characteristic
317	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
318	C₂×C₄	#88256	1	1	yes	characteristic
319	C₂×C₄	#88257	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	26	4	no	—
322	C₈	C₄×Dic₂₆	1	4	yes	—
323	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
324	C₂²	D₄×Dic₂₆	1	1	yes	characteristic
325	C₂²	#11719258	1	1	yes	characteristic
326	C₂²	#11719462	1	1	yes	characteristic
327	C₂²	#11719471	1	1	yes	characteristic
328	C₂²	#11719473	1	1	yes	characteristic
329	C₂²	#11719473	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	2	no	—
334	C₄	—	26	4	no	—
335	C₄	—	26	4	no	—
336	C₄	—	104	2	no	—
337	C₄	—	104	2	no	—
338	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
339	C₄	#11719228	1	1	yes	characteristic
340	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
341	C₄	#11719272	1	1	yes	characteristic
342	C₂	C₄⋊C₄×Dic₂₆	1	1	yes	characteristic
343	C₂	#13123424	1	1	yes	characteristic
344	C₂	#13123432	1	1	yes	characteristic
345	C₂	#13127792	1	1	yes	characteristic
346	C₂	#13127801	1	1	yes	characteristic
347	C₂	#13127913	1	1	yes	characteristic
348	C₂	#13127922	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#107	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#107	D₁₃
hypocenter	C₁₃	#22731
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
30. #11718340	C₄	semidirect
56. C₁₃×#107	Q₈	semidirect
226. #107	Dic₂₆	semidirect
287. C₁₃	#22731	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718340	2
#13117794	2
#13115607	2
#13115604	2
#13115593	2
#13115003	2
#13115000	2
#13114735	2
C₄×C₁₃×#107	2
#22731	13