Subgroups — C₂²×C₁₃×#361 #15719728

Order 3328.

1390 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#361	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₄×C₁₃	—	2	1	no	—
6	C₂²×C₁₃×D₈	—	2	3	no	—
7	C₂²×C₁₃×D₈	—	2	1	no	—
8	C₂²×C₁₃×D₈	—	2	3	no	—
9	C₂²×C₁₃×D₈	C₄	1	6	yes	—
10	C₂²×C₁₃×D₈	C₄	1	2	yes	—
11	C₂²×C₁₃×D₈	C₂²	1	6	yes	—
12	C₂²×C₁₃×M₁₆	—	2	1	no	—
13	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₃₂	—	2	3	no	—
15	C₂×C₁₃×M₃₂	C₂²	1	6	yes	complemented
16	C₂²×C₁₃×C₁₆	—	2	1	no	—
17	C₂×C₁₃×#108	—	2	3	no	—
18	C₂×C₁₃×#108	C₂²	1	6	yes	complemented
19	C₁₃×#361	C₂²	1	16	yes	direct product factor
20	C₂⁵×C₁₃	—	4	1	no	—
21	C₂²×D₄×C₁₃	—	2	6	no	—
22	C₂²×D₄×C₁₃	—	2	2	no	—
23	C₂²×D₄×C₁₃	—	2	6	no	—
24	C₂²×D₄×C₁₃	—	4	3	no	—
25	C₂²×D₄×C₁₃	—	4	1	no	—
26	C₂²×D₄×C₁₃	—	4	3	no	—
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₁₃×D₈	—	2	6	no	—
29	C₂×C₁₃×D₈	—	2	12	no	—
30	C₂×C₁₃×D₈	—	2	6	no	—
31	C₂×C₁₃×D₈	—	2	6	no	—
32	C₂×C₁₃×D₈	—	2	6	no	—
33	C₂×C₁₃×D₈	—	2	6	no	—
34	C₂×C₁₃×D₈	C₂×C₄	1	12	yes	—
35	C₂×C₁₃×D₈	C₂×C₄	1	12	yes	—
36	C₂×C₁₃×D₈	C₂³	1	4	yes	—
37	C₂×C₁₃×M₁₆	—	2	6	no	—
38	C₂×C₁₃×M₁₆	—	4	3	no	—
39	C₂²×C₈×C₁₃	—	4	1	no	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	D₄	1	3	yes	—
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	D₄	1	3	yes	—
45	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
46	C₁₃×M₃₂	—	2	6	no	—
47	C₁₃×M₃₂	C₂³	1	4	yes	complemented
48	C₂×C₁₃×C₁₆	—	2	6	no	—
49	C₁₃×#108	—	2	6	no	—
50	C₁₃×#108	C₂³	1	4	yes	complemented
51	C₂²×#361	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	4	6	no	—
53	C₂⁴×C₁₃	—	4	2	no	—
54	C₂⁴×C₁₃	—	4	6	no	—
55	C₂⁴×C₁₃	—	8	8	no	—
56	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
57	C₂×D₄×C₁₃	—	2	12	no	—
58	C₂×D₄×C₁₃	—	2	12	no	—
59	C₂×D₄×C₁₃	—	2	4	no	—
60	C₂×D₄×C₁₃	—	4	12	no	—
61	C₂×D₄×C₁₃	—	4	6	no	—
62	C₂×D₄×C₁₃	—	4	6	no	—
63	C₂×D₄×C₁₃	—	4	6	no	—
64	C₂×D₄×C₁₃	—	4	6	no	—
65	C₂×D₄×C₁₃	—	4	6	no	—
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	D₈	1	3	yes	—
69	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	SD₁₆	1	3	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
72	C₁₃×D₈	—	2	12	no	—
73	C₁₃×D₈	—	2	12	no	—
74	C₁₃×D₈	—	2	4	no	—
75	C₁₃×D₈	C₂²×C₄	1	8	yes	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	4	6	no	—
78	C₂×C₈×C₁₃	—	4	6	no	—
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	3	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	6	yes	—
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	3	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₁₆	—	2	4	no	—
89	C₂³×D₈	C₂×C₁₃	1	1	yes	characteristic
90	C₂²×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
91	C₂²×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
92	C₂×#361	C₂×C₁₃	1	12	yes	direct product factor
93	C₂³×C₁₃	—	2	4	no	—
94	C₂³×C₁₃	—	4	4	no	—
95	C₂³×C₁₃	—	4	12	no	—
96	C₂³×C₁₃	—	4	12	no	—
97	C₂³×C₁₃	—	8	24	no	—
98	C₂³×C₁₃	—	8	24	no	—
99	C₂³×C₁₃	—	8	8	no	—
100	C₂³×C₁₃	#102	1	3	yes	—
101	C₂³×C₁₃	#102	1	1	yes	characteristic
102	C₂³×C₁₃	C₂×SD₁₆	1	3	yes	—
103	D₄×C₁₃	—	2	8	no	—
104	D₄×C₁₃	—	4	4	no	—
105	D₄×C₁₃	—	4	12	no	—
106	D₄×C₁₃	—	4	12	no	—
107	C₂×C₄×C₁₃	#102	1	6	yes	—
108	C₂×C₄×C₁₃	#102	1	3	yes	—
109	C₂×C₄×C₁₃	#102	1	3	yes	—
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
114	C₂×C₄×C₁₃	C₂×SD₁₆	1	3	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₁₃	—	4	4	no	—
117	C₈×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
118	C₈×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
119	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂³×D₄	—	2	1	no	—
122	C₂²×D₈	—	2	3	no	—
123	C₂²×D₈	—	2	1	no	—
124	C₂²×D₈	—	2	3	no	—
125	C₂²×D₈	C₄×C₁₃	1	6	yes	—
126	C₂²×D₈	C₄×C₁₃	1	2	yes	—
127	C₂²×D₈	C₂²×C₁₃	1	6	yes	—
128	C₂²×M₁₆	—	2	1	no	—
129	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×M₃₂	—	2	3	no	—
131	C₂×M₃₂	C₂²×C₁₃	1	6	yes	complemented
132	C₂²×C₁₆	—	2	1	no	—
133	C₂×#108	—	2	3	no	—
134	C₂×#108	C₂²×C₁₃	1	6	yes	complemented
135	#361	C₂²×C₁₃	1	16	yes	direct product factor
136	C₂²×C₁₃	—	2	6	no	—
137	C₂²×C₁₃	—	2	6	no	—
138	C₂²×C₁₃	—	4	8	no	—
139	C₂²×C₁₃	—	8	8	no	—
140	C₂²×C₁₃	—	8	24	no	—
141	C₂²×C₁₃	—	8	24	no	—
142	C₂²×C₁₃	#361	1	4	yes	direct product factor
143	C₂²×C₁₃	C₂×#102	1	3	yes	—
144	C₂²×C₁₃	C₂×#102	1	3	yes	—
145	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#102	1	3	yes	—
147	C₄×C₁₃	C₂×#102	1	3	yes	—
148	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₂⁵	—	4	1	no	—
151	C₂²×D₄	—	2	6	no	—
152	C₂²×D₄	—	2	2	no	—
153	C₂²×D₄	—	2	6	no	—
154	C₂²×D₄	—	4	1	no	—
155	C₂²×D₄	—	4	3	no	—
156	C₂²×D₄	—	4	3	no	—
157	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
158	C₂×D₈	—	2	6	no	—
159	C₂×D₈	—	2	12	no	—
160	C₂×D₈	—	2	6	no	—
161	C₂×D₈	—	2	6	no	—
162	C₂×D₈	—	2	6	no	—
163	C₂×D₈	—	2	6	no	—
164	C₂×D₈	C₂×C₄×C₁₃	1	12	yes	—
165	C₂×D₈	C₂×C₄×C₁₃	1	12	yes	—
166	C₂×D₈	C₂³×C₁₃	1	4	yes	—
167	C₂×M₁₆	—	2	6	no	—
168	C₂×M₁₆	—	4	3	no	—
169	C₂²×C₈	—	4	1	no	—
170	C₂²×C₈	C₂×C₄×C₁₃	1	3	yes	—
171	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₁₃	1	3	yes	—
173	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
174	C₂²×C₈	D₄×C₁₃	1	3	yes	—
175	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
176	M₃₂	—	2	6	no	—
177	M₃₂	C₂³×C₁₃	1	4	yes	complemented
178	C₂×C₁₆	—	2	6	no	—
179	#108	—	2	6	no	—
180	#108	C₂³×C₁₃	1	4	yes	complemented
181	C₂×C₁₃	—	2	1	no	—
182	C₂×C₁₃	—	2	3	no	—
183	C₂×C₁₃	—	8	8	no	—
184	C₂×C₁₃	C₂×#361	1	6	yes	direct product factor
185	C₂×C₁₃	C₂²×#102	1	1	yes	characteristic
186	C₂⁴	—	4	6	no	—
187	C₂⁴	—	4	2	no	—
188	C₂⁴	—	4	6	no	—
189	C₂⁴	—	8	8	no	—
190	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
191	C₂×D₄	—	2	12	no	—
192	C₂×D₄	—	2	12	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	4	12	no	—
195	C₂×D₄	—	4	6	no	—
196	C₂×D₄	—	4	6	no	—
197	C₂×D₄	—	4	6	no	—
198	C₂×D₄	—	4	6	no	—
199	C₂×D₄	—	4	6	no	—
200	C₂²×C₄	C₁₃×C₂²⋊C₄	1	3	yes	—
201	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
203	C₂²×C₄	C₁₃×D₈	1	3	yes	—
204	C₂²×C₄	C₁₃×SD₁₆	1	3	yes	—
205	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
206	D₈	—	2	12	no	—
207	D₈	—	2	12	no	—
208	D₈	—	2	4	no	—
209	D₈	C₂²×C₄×C₁₃	1	8	yes	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	4	6	no	—
212	C₂×C₈	—	4	6	no	—
213	C₂×C₈	C₁₃×C₂²⋊C₄	1	3	yes	—
214	C₂×C₈	C₁₃×C₂²⋊C₄	1	6	yes	—
215	C₂×C₈	C₁₃×C₂²⋊C₄	1	3	yes	—
216	C₂×C₈	C₂²×C₄×C₁₃	1	3	yes	—
217	C₂×C₈	C₂²×C₄×C₁₃	1	3	yes	—
218	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
219	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
220	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
221	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
222	C₁₆	—	2	4	no	—
223	C₁₃	C₂²×#361	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	—	2	4	no	—
225	C₂³	—	4	12	no	—
226	C₂³	—	4	12	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	8	24	no	—
229	C₂³	—	8	24	no	—
230	C₂³	—	8	8	no	—
231	C₂³	C₁₃×#102	1	3	yes	—
232	C₂³	C₁₃×#102	1	1	yes	characteristic
233	C₂³	C₂×C₁₃×SD₁₆	1	3	yes	—
234	D₄	—	2	8	no	—
235	D₄	—	4	12	no	—
236	D₄	—	4	12	no	—
237	D₄	—	4	4	no	—
238	C₂×C₄	C₁₃×#102	1	6	yes	—
239	C₂×C₄	C₁₃×#102	1	3	yes	—
240	C₂×C₄	C₁₃×#102	1	3	yes	—
241	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
242	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
243	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
244	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
245	C₂×C₄	C₂×C₁₃×SD₁₆	1	3	yes	—
246	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
247	C₈	—	4	4	no	—
248	C₈	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
249	C₈	C₂×C₁₃×C₂²⋊C₄	1	3	yes	—
250	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
251	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
252	C₂²	—	2	6	no	—
253	C₂²	—	2	6	no	—
254	C₂²	—	4	8	no	—
255	C₂²	—	8	24	no	—
256	C₂²	—	8	8	no	—
257	C₂²	—	8	24	no	—
258	C₂²	C₁₃×#361	1	4	yes	direct product factor
259	C₂²	C₂×C₁₃×#102	1	3	yes	—
260	C₂²	C₂×C₁₃×#102	1	3	yes	—
261	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
262	C₄	C₂×C₁₃×#102	1	3	yes	—
263	C₄	C₂×C₁₃×#102	1	3	yes	—
264	C₄	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
265	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
266	C₂	—	2	3	no	—
267	C₂	—	2	1	no	—
268	C₂	—	8	8	no	—
269	C₂	C₂×C₁₃×#361	1	6	yes	direct product factor
270	C₂	C₂²×C₁₃×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	#102
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂³×C₁₃×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₃×M₃₂	C₂	semidirect
3. C₂²×C₁₃×#108	C₂	semidirect
4. C₂×C₁₃×#361	C₂	direct (N × H)
15. C₂×C₁₃×M₃₂	C₂²	semidirect
18. C₂×C₁₃×#108	C₂²	semidirect
19. C₁₃×#361	C₂²	direct (N × H)
47. C₁₃×M₃₂	C₂³	semidirect
50. C₁₃×#108	C₂³	semidirect
51. C₂²×#361	C₁₃	direct (N × H)
90. C₂²×M₃₂	C₂×C₁₃	semidirect
91. C₂²×#108	C₂×C₁₃	semidirect
92. C₂×#361	C₂×C₁₃	direct (N × H)
131. C₂×M₃₂	C₂²×C₁₃	semidirect
134. C₂×#108	C₂²×C₁₃	semidirect
135. #361	C₂²×C₁₃	direct (N × H)
142. C₂²×C₁₃	#361	direct (N × H)
177. M₃₂	C₂³×C₁₃	semidirect
180. #108	C₂³×C₁₃	semidirect
184. C₂×C₁₃	C₂×#361	direct (N × H)
223. C₁₃	C₂²×#361	direct (N × H)
258. C₂²	C₁₃×#361	direct (N × H)
269. C₂	C₂×C₁₃×#361	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×D₈	2
C₂²×C₁₃×M₃₂	2
C₂²×C₁₃×#108	2
C₂×C₁₃×#361	2
C₂²×#361	13

Subgroups — C22×C13×#361 #15719728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#361 #15719728