Subgroups — #97×D₅₂ #15861380

Order 3328.

1274 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅₂	C₂	1	2	yes	maximal
2	C₂×D₁₃×#97	C₂	1	2	yes	maximal, complemented
3	C₄²×D₅₂	C₂	1	1	yes	maximal, characteristic
4	#13114703	C₂	1	2	yes	maximal, complemented
5	#13114689	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114613	C₂	1	4	yes	maximal, complemented
7	#13114599	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₁₃	C₂²	1	4	yes	—
10	C₂×C₄×D₅₂	C₄	1	2	yes	—
11	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₁₃	C₂²	1	2	yes	—
13	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88060	C₂²	1	4	yes	—
19	C₂×#88055	C₂²	1	4	yes	—
20	C₄×#6103	C₄	1	4	yes	—
21	C₄×#6103	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
23	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#1055	C₂²	1	2	yes	—
25	C₂×#88043	C₂²	1	2	yes	—
26	D₁₃×#97	—	2	8	no	—
27	C₈×D₅₂	C₄	1	16	yes	complemented
28	#11718142	C₂²	1	4	yes	—
29	#11718116	C₂²	1	2	yes	—
30	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₁₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×D₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×#6103	C₂×C₄	1	4	yes	—
41	C₂×#6103	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×#1055	C₂³	1	4	yes	—
47	C₂×C₈×D₁₃	—	2	8	no	—
48	C₂×C₈×D₁₃	—	2	8	no	—
49	C₂×C₈×D₁₃	C₂×C₄	1	16	yes	complemented
50	C₄×D₅₂	—	2	8	no	—
51	C₄×D₅₂	C₂×C₄	1	8	yes	—
52	C₄²×D₁₃	—	2	4	no	—
53	C₄²×D₁₃	—	2	4	no	—
54	C₁₃×#97	—	2	4	no	—
55	C₁₃×#97	—	2	2	no	—
56	C₁₃×#97	D₄	1	4	yes	complemented
57	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
58	#88071	C₂×C₄	1	4	yes	—
59	#88071	C₂×C₄	1	2	yes	—
60	#88060	C₂×C₄	1	16	yes	complemented
61	#88055	—	2	8	no	—
62	C₄⋊#1055	C₂×C₄	1	8	yes	complemented
63	#88043	—	2	4	no	—
64	D₄×#97	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
72	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
73	C₂×D₅₂	C₄²	1	4	yes	—
74	C₂×D₅₂	M₁₆	1	8	yes	—
75	C₂×C₄×D₁₃	—	2	2	no	—
76	C₂×C₄×D₁₃	—	2	2	no	—
77	C₂×C₄×D₁₃	—	2	4	no	—
78	C₂×C₄×D₁₃	—	2	8	no	—
79	C₂×C₄×D₁₃	—	2	8	no	—
80	C₂×C₄×D₁₃	—	4	4	no	—
81	C₂×C₄×D₁₃	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₁₃	#41	1	4	yes	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	4	2	no	—
91	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃	#41	1	2	yes	—
94	#6103	—	2	8	no	—
95	#6103	C₄²	1	8	yes	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	C₄²	1	4	yes	—
98	C₄×Dic₁₃	—	2	8	no	—
99	C₄×Dic₁₃	—	2	2	no	—
100	C₄×Dic₁₃	—	2	2	no	—
101	C₂×#1055	—	2	4	no	—
102	C₂×#1055	—	2	4	no	—
103	C₂×#1055	C₂²×C₄	1	8	yes	—
104	C₈×D₁₃	—	2	32	no	—
105	C₂×C₈×D₄	—	13	2	no	—
106	C₂²×#97	—	13	2	no	—
107	C₄²×D₄	—	13	1	no	—
108	#1770	—	13	2	no	—
109	#1767	—	13	1	no	—
110	#1752	—	13	4	no	—
111	#1749	—	13	2	no	—
112	C₄×#97	D₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂²×D₁₃	—	2	2	no	—
115	C₂²×D₁₃	—	2	2	no	—
116	C₂²×D₁₃	—	2	4	no	—
117	C₂²×D₁₃	C₂×C₄²	1	4	yes	—
118	C₂²×D₁₃	C₂×M₁₆	1	8	yes	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
128	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	2	no	—
134	C₂×Dic₁₃	—	2	2	no	—
135	C₂×Dic₁₃	—	2	4	no	—
136	C₂×Dic₁₃	—	2	4	no	—
137	C₂×Dic₁₃	—	2	4	no	—
138	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
139	D₅₂	#97	1	16	yes	direct product factor
140	C₄×D₁₃	—	2	8	no	—
141	C₄×D₁₃	—	2	8	no	—
142	C₄×D₁₃	—	4	16	no	—
143	C₈×C₁₃	—	2	4	no	—
144	C₈×C₁₃	C₄×D₄	1	8	yes	complemented
145	#1055	—	2	8	no	—
146	C₂³×C₈	—	13	4	no	—
147	C₂×C₄×D₄	—	13	2	no	—
148	C₂×C₄×D₄	—	13	1	no	—
149	C₂²×C₄²	—	13	2	no	—
150	C₈×D₄	—	13	16	no	—
151	C₂×C₄⋊C₈	—	13	2	no	—
152	C₂×C₂²⋊C₈	—	13	4	no	—
153	C₂×#97	—	13	4	no	—
154	C₂×#97	—	13	2	no	—
155	C₂×#97	—	26	8	no	—
156	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#97	C₂×D₁₃	1	2	yes	—
158	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
160	C₄×C₄⋊C₄	—	13	2	no	—
161	C₄×C₄⋊C₄	—	13	1	no	—
162	C₄×C₂²⋊C₄	—	13	4	no	—
163	C₄×C₂²⋊C₄	—	13	2	no	—
164	C₄³	C₂×D₁₃	1	1	yes	characteristic
165	#336	—	13	4	no	—
166	#336	—	13	2	no	—
167	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
170	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
171	C₂²×C₁₃	#568	1	2	yes	—
172	C₂×D₁₃	—	2	4	no	—
173	C₂×D₁₃	—	2	8	no	—
174	C₂×D₁₃	—	2	8	no	—
175	C₂×D₁₃	—	2	4	no	—
176	C₂×D₁₃	C₂×#97	1	8	yes	complemented
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	2	4	no	—
179	C₄×C₁₃	—	4	2	no	—
180	C₄×C₁₃	C₂×#97	1	4	yes	complemented
181	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
182	C₄×C₁₃	C₄×#41	1	2	yes	—
183	Dic₁₃	—	2	4	no	—
184	Dic₁₃	—	4	4	no	—
185	C₂²×D₄	—	13	1	no	—
186	C₂³×C₄	—	13	2	no	—
187	C₂³×C₄	—	13	4	no	—
188	C₂²×C₈	—	13	4	no	—
189	C₂²×C₈	—	13	16	no	—
190	C₂²×C₈	—	26	8	no	—
191	C₂²×C₈	—	26	8	no	—
192	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
193	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
194	C₄×D₄	—	13	8	no	—
195	C₄×D₄	—	26	8	no	—
196	C₂×C₄⋊C₄	—	13	1	no	—
197	C₂×C₄⋊C₄	—	13	2	no	—
198	C₂×C₂²⋊C₄	—	13	2	no	—
199	C₂×C₂²⋊C₄	—	13	4	no	—
200	C₂×C₄²	—	13	4	no	—
201	C₂×C₄²	—	13	2	no	—
202	C₂×C₄²	—	26	4	no	—
203	C₂×C₄²	—	26	4	no	—
204	C₂×C₄²	C₄×D₁₃	1	2	yes	—
205	C₂×C₄²	C₄×D₁₃	1	2	yes	—
206	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊C₈	—	13	8	no	—
210	C₂²⋊C₈	—	13	16	no	—
211	#97	—	2	4	no	—
212	#97	—	2	2	no	—
213	#97	—	26	8	no	—
214	#97	—	26	4	no	—
215	#97	D₅₂	1	4	yes	direct product factor
216	C₄×C₈	C₄×D₁₃	1	8	yes	complemented
217	#95	—	13	4	no	—
218	#95	—	13	2	no	—
219	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
220	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
221	C₂×C₁₃	#2875	1	1	yes	characteristic
222	C₂×C₁₃	D₄×M₁₆	1	2	yes	—
223	C₂×C₁₃	#2970	1	2	yes	—
224	D₁₃	—	2	8	no	—
225	C₂⁴	—	13	2	no	—
226	C₂×D₄	—	13	4	no	—
227	C₂×D₄	—	13	8	no	—
228	C₂²×C₄	—	13	2	no	—
229	C₂²×C₄	—	13	8	no	—
230	C₂²×C₄	—	13	4	no	—
231	C₂²×C₄	—	26	8	no	—
232	C₂²×C₄	—	26	2	no	—
233	C₂²×C₄	—	26	2	no	—
234	C₂²×C₄	—	26	4	no	—
235	C₂²×C₄	—	26	8	no	—
236	C₂²×C₄	—	52	4	no	—
237	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
238	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	13	8	no	—
245	C₂×C₈	—	26	4	no	—
246	C₂×C₈	—	26	4	no	—
247	C₂×C₈	—	26	32	no	—
248	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
249	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
250	C₂×C₈	C₂×D₅₂	1	4	yes	—
251	C₂×C₈	#6127	1	4	yes	—
252	C₄⋊C₄	—	13	4	no	—
253	C₄⋊C₄	—	26	4	no	—
254	C₂²⋊C₄	—	13	8	no	—
255	C₂²⋊C₄	—	26	8	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	26	8	no	—
260	C₄²	—	26	2	no	—
261	C₄²	—	26	2	no	—
262	C₄²	C₂×C₄×D₁₃	1	4	yes	—
263	C₄²	C₂×D₅₂	1	2	yes	—
264	C₄²	#6127	1	2	yes	—
265	C₁₃	D₄×#97	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	13	4	no	—
267	C₂³	—	13	8	no	—
268	C₂³	—	26	2	no	—
269	C₂³	—	26	2	no	—
270	C₂³	—	26	4	no	—
271	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
272	D₄	—	13	16	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	13	4	no	—
280	C₂×C₄	—	26	2	no	—
281	C₂×C₄	—	26	2	no	—
282	C₂×C₄	—	26	8	no	—
283	C₂×C₄	—	26	8	no	—
284	C₂×C₄	—	26	4	no	—
285	C₂×C₄	—	26	4	no	—
286	C₂×C₄	—	26	4	no	—
287	C₂×C₄	—	52	16	no	—
288	C₂×C₄	C₄²×D₁₃	1	2	yes	—
289	C₂×C₄	C₄²×D₁₃	1	2	yes	—
290	C₂×C₄	C₄×D₅₂	1	4	yes	—
291	C₂×C₄	M₁₆×D₁₃	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
293	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
294	C₂×C₄	C₂×#6127	1	2	yes	—
295	C₂×C₄	C₂×#6127	1	1	yes	characteristic
296	C₈	—	2	4	no	—
297	C₈	—	26	8	no	—
298	C₈	C₄×D₅₂	1	8	yes	complemented
299	C₂²	—	13	8	no	—
300	C₂²	—	26	4	no	—
301	C₂²	—	26	8	no	—
302	C₂²	—	26	4	no	—
303	C₂²	—	26	8	no	—
304	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
305	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
306	C₂²	C₄×#6127	1	1	yes	characteristic
307	C₂²	C₂×M₁₆×D₁₃	1	2	yes	—
308	C₂²	#11719450	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	26	4	no	—
313	C₄	—	52	4	no	—
314	C₄	D₁₃×#97	1	4	yes	complemented
315	C₄	C₂×C₄×D₅₂	1	2	yes	—
316	C₄	C₄×#6127	1	2	yes	—
317	C₂	—	26	8	no	—
318	C₂	C₄²×D₅₂	1	1	yes	characteristic
319	C₂	C₂×D₁₃×#97	1	1	yes	characteristic
320	C₂	#13123261	1	1	yes	characteristic
321	C₂	M₁₆×D₅₂	1	2	yes	—
322	C₂	#13123851	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₄×C₁₃×#97	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	D₄×#97
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. C₂×D₁₃×#97	C₂	semidirect
4. #13114703	C₂	semidirect
5. #13114689	C₂	semidirect
6. #13114613	C₂	semidirect
7. #13114599	C₂	semidirect
8. C₄×C₁₃×#97	C₂	semidirect
27. C₈×D₅₂	C₄	semidirect
49. C₂×C₈×D₁₃	C₂×C₄	semidirect
56. C₁₃×#97	D₄	semidirect
57. C₄×C₈×C₁₃	C₂×C₄	semidirect
60. #88060	C₂×C₄	semidirect
62. C₄⋊#1055	C₂×C₄	semidirect
112. C₄×#97	D₁₃	semidirect
139. D₅₂	#97	direct (N × H)
144. C₈×C₁₃	C₄×D₄	semidirect
176. C₂×D₁₃	C₂×#97	semidirect
180. C₄×C₁₃	C₂×#97	semidirect
215. #97	D₅₂	direct (N × H)
216. C₄×C₈	C₄×D₁₃	semidirect
265. C₁₃	D₄×#97	semidirect
298. C₈	C₄×D₅₂	semidirect
314. C₄	D₁₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅₂	2
C₂×D₁₃×#97	2
C₄²×D₅₂	2
#13114703	2
#13114689	2
#13114613	2
#13114599	2
C₄×C₁₃×#97	2
D₄×#97	13

Subgroups — #97×D52 #15861380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #97×D₅₂ #15861380