Subgroups — #16807159

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊M₃₂	C₂	1	2	yes	maximal, complemented
2	#13131222	C₂	1	2	yes	maximal, complemented
3	#13131219	C₂	1	4	yes	maximal, complemented
4	#13131215	C₂	1	2	yes	maximal, complemented
5	#13131168	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131065	C₂	1	2	yes	maximal, complemented
7	C₄×#88106	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11718495	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
10	C₂×C₄×D₅₂	C₄	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	2	yes	—
12	C₂×#88106	C₂²	1	1	yes	characteristic
13	C₂×#88106	C₂²	1	1	yes	characteristic
14	D₁₃⋊M₃₂	—	2	8	no	—
15	C₂×#88098	C₂²	1	4	yes	complemented
16	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	complemented
17	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
18	C₂×#88060	C₄	1	2	yes	—
19	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
20	#11718495	C₄	1	1	yes	characteristic
21	#11718495	C₄	1	2	yes	—
22	#11718495	C₄	1	1	yes	characteristic
23	#11718495	C₂²	1	1	yes	characteristic
24	#11718495	C₂²	1	2	yes	—
25	#11718495	C₂²	1	1	yes	characteristic
26	#11718259	C₂²	1	4	yes	complemented
27	#11718259	C₂²	1	4	yes	complemented
28	#11718231	C₂²	1	2	yes	complemented
29	C₄⋊#6092	C₂²	1	2	yes	complemented
30	C₄⋊#6092	C₂²	1	2	yes	complemented
31	C₄×#6092	C₂²	1	2	yes	complemented
32	C₂²×D₅₂	C₈	1	1	yes	characteristic
33	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#6103	C₈	1	2	yes	—
37	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂²×#1055	C₂³	1	1	yes	characteristic
40	C₂×C₈×D₁₃	—	2	4	no	—
41	C₂×C₈×D₁₃	—	2	4	no	—
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂³	1	2	yes	—
45	C₂×C₈×D₁₃	C₂³	1	2	yes	—
46	C₄×D₅₂	C₈	1	2	yes	—
47	C₄×D₅₂	C₈	1	1	yes	characteristic
48	C₄×D₅₂	C₈	1	1	yes	characteristic
49	C₄×D₅₂	C₂×C₄	1	2	yes	—
50	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
51	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
52	#88106	—	2	4	no	—
53	#88106	—	2	2	no	—
54	#88106	D₄	1	4	yes	complemented
55	C₂×#6092	C₂³	1	2	yes	—
56	C₂×#6092	C₂³	1	4	yes	—
57	C₂×#6092	C₂³	1	2	yes	—
58	#88098	—	2	8	no	—
59	D₁₃⋊C₁₆	—	2	8	no	—
60	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
63	#88060	C₂×C₄	1	2	yes	—
64	#88060	C₂×C₄	1	2	yes	—
65	#88060	C₂×C₄	1	2	yes	—
66	#88060	C₂×C₄	1	2	yes	—
67	C₄×#1055	C₂×C₄	1	1	yes	characteristic
68	C₄×#1055	C₂×C₄	1	1	yes	characteristic
69	C₄×#1055	C₂³	1	1	yes	characteristic
70	C₄×#1055	C₂³	1	1	yes	characteristic
71	D₄×M₃₂	—	13	1	no	maximal, Sylow
72	C₂³×D₁₃	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
77	C₂×D₅₂	—	2	4	no	—
78	C₂×D₅₂	C₂×C₈	1	2	yes	—
79	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
80	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	—	2	4	no	—
82	C₂×C₄×D₁₃	—	2	4	no	—
83	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	—	2	4	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	#6103	C₂×C₈	1	2	yes	—
95	#6103	C₂×C₈	1	2	yes	—
96	#6103	C₂×C₈	1	2	yes	—
97	#6103	C₂×C₈	1	2	yes	—
98	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
100	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
103	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
104	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
105	C₂×#1055	C₂×D₄	1	2	yes	—
106	C₂×#1055	#41	1	2	yes	—
107	C₂×#1055	C₂⁴	1	1	yes	characteristic
108	C₈×D₁₃	—	2	4	no	—
109	C₈×D₁₃	—	2	4	no	—
110	C₈×D₁₃	—	2	2	no	—
111	C₈×D₁₃	—	2	2	no	—
112	C₈×D₁₃	—	2	2	no	—
113	C₈×D₁₃	—	2	2	no	—
114	#6092	—	2	4	no	—
115	#6092	—	2	2	no	—
116	#6092	C₂×D₄	1	4	yes	complemented
117	C₂²×M₃₂	—	13	2	no	—
118	C₂×C₈×D₄	—	13	1	no	—
119	#2169	—	13	2	no	—
120	#2168	—	13	4	no	—
121	D₄×C₁₆	—	13	2	no	—
122	#2150	—	13	1	no	—
123	#2112	—	13	2	no	—
124	C₄×M₃₂	—	13	1	no	—
125	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂²×D₁₃	—	2	4	no	—
127	C₂²×D₁₃	—	2	4	no	—
128	C₂²×D₁₃	—	4	2	no	—
129	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
130	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#131	1	2	yes	—
140	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×Dic₁₃	—	2	4	no	—
142	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
143	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
144	D₅₂	—	2	4	no	—
145	D₅₂	—	2	2	no	—
146	D₅₂	M₃₂	1	4	yes	complemented
147	C₄×D₁₃	—	2	2	no	—
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	2	2	no	—
150	C₄×D₁₃	—	2	2	no	—
151	C₄×D₁₃	—	2	4	no	—
152	C₄×D₁₃	—	2	4	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	2	2	no	—
155	#1055	—	2	1	no	—
156	#1055	—	2	1	no	—
157	#1055	C₄×D₄	1	2	yes	—
158	#1055	C₂²×D₄	1	1	yes	characteristic
159	#1055	C₂×#41	1	1	yes	characteristic
160	C₂³×C₈	—	13	2	no	—
161	C₂×C₄×D₄	—	13	1	no	—
162	C₂×M₃₂	—	13	1	no	—
163	C₂×M₃₂	—	13	1	no	—
164	C₂×M₃₂	—	13	2	no	—
165	C₂×M₃₂	—	13	4	no	—
166	C₂×M₃₂	—	26	8	no	—
167	C₂²×C₁₆	—	13	4	no	—
168	C₈×D₄	—	13	2	no	—
169	C₈×D₄	—	13	1	no	—
170	C₈×D₄	—	13	1	no	—
171	C₈×D₄	—	13	1	no	—
172	C₈×D₄	—	13	2	no	—
173	C₈×D₄	—	13	1	no	—
174	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
175	C₂×C₂²⋊C₈	—	13	2	no	—
176	C₂×C₄×C₈	—	13	1	no	—
177	C₄⋊C₁₆	—	13	2	no	—
178	C₄⋊C₁₆	—	13	2	no	—
179	C₂²⋊C₁₆	—	13	4	no	—
180	C₂²⋊C₁₆	—	13	4	no	—
181	#346	—	13	2	no	—
182	C₄×C₁₆	—	13	2	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	C₈×D₄	1	2	yes	—
186	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
187	C₂×D₁₃	—	2	2	no	—
188	C₂×D₁₃	—	2	2	no	—
189	C₂×D₁₃	—	2	2	no	—
190	C₂×D₁₃	—	2	2	no	—
191	C₂×D₁₃	—	2	2	no	—
192	C₂×D₁₃	—	4	8	no	—
193	C₂×D₁₃	C₂×M₃₂	1	4	yes	complemented
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	C₈×D₄	1	2	yes	—
198	C₄×C₁₃	C₂×M₃₂	1	2	yes	complemented
199	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
201	Dic₁₃	—	2	2	no	—
202	Dic₁₃	—	2	2	no	—
203	C₂²×D₄	—	13	1	no	—
204	C₂³×C₄	—	13	2	no	—
205	C₂²×C₈	—	13	1	no	—
206	C₂²×C₈	—	13	1	no	—
207	C₂²×C₈	—	13	2	no	—
208	C₂²×C₈	—	13	2	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	—	13	2	no	—
211	C₂²×C₈	—	26	4	no	—
212	C₂²×C₈	—	26	4	no	—
213	C₂²×C₈	C₂×#261	1	2	yes	—
214	C₄×D₄	—	13	1	no	—
215	C₄×D₄	—	13	2	no	—
216	C₄×D₄	—	13	1	no	—
217	C₄×D₄	—	13	1	no	—
218	C₄×D₄	—	13	2	no	—
219	C₄×D₄	—	13	1	no	—
220	C₂×C₄⋊C₄	—	13	1	no	—
221	C₂×C₂²⋊C₄	—	13	2	no	—
222	C₂×C₄²	C₂×#261	1	1	yes	characteristic
223	M₃₂	—	13	4	no	—
224	M₃₂	—	26	8	no	—
225	M₃₂	—	26	2	no	—
226	M₃₂	—	26	4	no	—
227	C₂×C₁₆	—	13	2	no	—
228	C₂×C₁₆	—	13	2	no	—
229	C₂×C₁₆	—	13	4	no	—
230	C₂×C₁₆	—	26	8	no	—
231	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
232	C₄⋊C₈	C₂×#261	1	2	yes	—
233	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
234	C₂²⋊C₈	—	13	2	no	—
235	C₂²⋊C₈	—	13	2	no	—
236	C₂²⋊C₈	—	13	2	no	—
237	C₂²⋊C₈	—	13	2	no	—
238	C₄×C₈	—	13	1	no	—
239	C₄×C₈	—	13	1	no	—
240	C₄×C₈	—	13	1	no	—
241	C₄×C₈	—	13	1	no	—
242	C₂×C₁₃	—	2	2	no	—
243	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
244	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
245	C₂×C₁₃	#3407	1	1	yes	characteristic
246	D₁₃	—	2	4	no	—
247	D₁₃	—	4	2	no	—
248	C₂⁴	—	13	2	no	—
249	C₂×D₄	—	13	1	no	—
250	C₂×D₄	—	13	2	no	—
251	C₂×D₄	—	13	1	no	—
252	C₂×D₄	—	26	4	no	—
253	C₂²×C₄	—	13	2	no	—
254	C₂²×C₄	—	13	2	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	26	4	no	—
259	C₂²×C₄	—	26	4	no	—
260	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
261	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
262	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	13	1	no	—
269	C₂×C₈	—	13	1	no	—
270	C₂×C₈	—	26	2	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	—	26	2	no	—
274	C₂×C₈	—	26	4	no	—
275	C₂×C₈	—	26	4	no	—
276	C₂×C₈	—	26	2	no	—
277	C₂×C₈	C₂²×#261	1	2	yes	—
278	C₂×C₈	C₂²×#261	1	2	yes	—
279	C₄⋊C₄	—	13	2	no	—
280	C₄⋊C₄	—	13	1	no	—
281	C₄⋊C₄	—	13	1	no	—
282	C₂²⋊C₄	—	13	2	no	—
283	C₂²⋊C₄	—	13	2	no	—
284	C₂²⋊C₄	—	13	2	no	—
285	C₂²⋊C₄	—	13	2	no	—
286	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
287	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
288	C₄²	C₂²×#261	1	1	yes	characteristic
289	C₄²	C₂²×#261	1	1	yes	characteristic
290	C₁₆	—	13	4	no	—
291	C₁₆	—	26	4	no	—
292	C₁₆	—	26	2	no	—
293	C₁₃	D₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	13	2	no	—
295	C₂³	—	13	2	no	—
296	C₂³	—	26	4	no	—
297	C₂³	—	26	4	no	—
298	C₂³	—	52	2	no	—
299	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
300	D₄	—	13	4	no	—
301	D₄	—	26	4	no	—
302	D₄	—	26	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	26	4	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	26	4	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	2	no	—
312	C₂×C₄	—	26	2	no	—
313	C₂×C₄	—	26	2	no	—
314	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
315	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
316	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
317	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
318	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
319	C₂×C₄	D₄×#261	1	2	yes	—
320	C₂×C₄	#88240	1	2	yes	—
321	C₂×C₄	C₂³×#261	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	13	1	no	—
325	C₈	—	13	1	no	—
326	C₈	—	13	2	no	—
327	C₈	—	26	1	no	—
328	C₈	—	26	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	13	4	no	—
332	C₂²	—	26	2	no	—
333	C₂²	—	26	2	no	—
334	C₂²	—	26	2	no	—
335	C₂²	—	26	2	no	—
336	C₂²	—	26	2	no	—
337	C₂²	—	52	8	no	—
338	C₂²	#11719073	1	2	yes	—
339	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	26	2	no	—
344	C₄	—	26	2	no	—
345	C₄	#11719073	1	2	yes	—
346	C₄	D₁₃⋊M₃₂	1	2	yes	complemented
347	C₄	C₂×D₄×#261	1	1	yes	characteristic
348	C₄	C₂×#88240	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	26	4	no	—
351	C₂	—	52	2	no	—
352	C₂	C₂×#11719073	1	1	yes	characteristic
353	C₂	C₂×D₁₃⋊M₃₂	1	1	yes	characteristic
354	C₂	#13131805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
prime radical	C₂²×D₅₂	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	D₄×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₃⋊M₃₂	C₂	semidirect
2. #13131222	C₂	semidirect
3. #13131219	C₂	semidirect
4. #13131215	C₂	semidirect
5. #13131168	C₂	semidirect
6. #13131065	C₂	semidirect
7. C₄×#88106	C₂	semidirect
15. C₂×#88098	C₂²	semidirect
16. C₂×D₁₃⋊C₁₆	C₂²	semidirect
26. #11718259	C₂²	semidirect
27. #11718259	C₂²	semidirect
28. #11718231	C₂²	semidirect
29. C₄⋊#6092	C₂²	semidirect
30. C₄⋊#6092	C₂²	semidirect
31. C₄×#6092	C₂²	semidirect
54. #88106	D₄	semidirect
116. #6092	C₂×D₄	semidirect
146. D₅₂	M₃₂	semidirect
193. C₂×D₁₃	C₂×M₃₂	semidirect
198. C₄×C₁₃	C₂×M₃₂	semidirect
293. C₁₃	D₄×M₃₂	semidirect
346. C₄	D₁₃⋊M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃⋊M₃₂	2
#13131222	2
#13131219	2
#13131215	2
#13131168	2
#13131065	2
C₄×#88106	2
C₂×#11718495	2
D₄×M₃₂	13