Subgroups — C₂×C₁₃×#2153 #15719739

Order 3328.

698 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2153	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#364	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
8	C₂×C₁₃×M₃₂	—	2	4	no	—
9	C₂×C₁₃×M₃₂	—	2	2	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
11	C₂×C₁₃×M₃₂	C₂²	1	8	yes	—
12	C₂²×C₁₃×C₁₆	—	2	2	no	—
13	C₁₃×#405	C₄	1	4	yes	—
14	C₁₃×#405	C₂²	1	4	yes	—
15	C₂×C₁₃×#97	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	C₁₃×#364	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	C₈	1	4	yes	—
20	C₂×C₁₃×M₁₆	C₈	1	4	yes	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	D₄	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×#117	—	2	2	no	—
32	C₁₃×#117	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₁₃×M₃₂	—	2	8	no	—
37	C₁₃×M₃₂	—	2	4	no	—
38	C₁₃×M₃₂	C₂³	1	8	yes	—
39	C₂×C₁₃×C₁₆	—	2	8	no	—
40	C₂×C₁₃×C₁₆	—	2	4	no	—
41	C₁₃×#97	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₁₃	C₂³	1	4	yes	—
44	C₂×#2153	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₁₃×M₁₆	C₂×C₈	1	16	yes	—
56	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
58	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
60	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₂²⋊C₄	—	2	4	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
76	C₁₃×C₁₆	—	2	8	no	—
77	C₂²×M₃₂	C₂×C₁₃	1	2	yes	—
78	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
79	#2153	C₂×C₁₃	1	8	yes	direct product factor
80	C₂×#364	C₂×C₁₃	1	4	yes	complemented
81	C₂³×C₁₃	—	4	2	no	—
82	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
83	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
84	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₁₃	—	2	4	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
89	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
90	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
91	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
98	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₈×C₁₃	C₂²×C₈	1	8	yes	—
105	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×M₁₆	C₄×C₁₃	1	1	yes	characteristic
108	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
109	C₂×#117	C₄×C₁₃	1	1	yes	characteristic
110	C₂×M₃₂	—	2	2	no	—
111	C₂×M₃₂	—	2	4	no	—
112	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
113	C₂×M₃₂	C₂²×C₁₃	1	8	yes	—
114	C₂²×C₁₆	—	2	2	no	—
115	#405	C₄×C₁₃	1	4	yes	—
116	#405	C₂²×C₁₃	1	4	yes	—
117	C₂×#97	C₄×C₁₃	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
119	#364	C₂²×C₁₃	1	16	yes	complemented
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	4	2	no	—
123	C₂²×C₁₃	—	4	2	no	—
124	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
126	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
128	C₄×C₁₃	—	4	4	no	—
129	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
131	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
132	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
134	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
135	C₂×M₁₆	C₈×C₁₃	1	4	yes	—
136	C₂×M₁₆	C₈×C₁₃	1	4	yes	—
137	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
138	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
139	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
141	C₂²×C₈	D₄×C₁₃	1	2	yes	—
142	C₂²×C₈	D₄×C₁₃	1	2	yes	—
143	C₂²×C₈	Q₈×C₁₃	1	2	yes	—
144	C₂²×C₈	Q₈×C₁₃	1	2	yes	—
145	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
147	#117	—	2	2	no	—
148	#117	C₂×C₄×C₁₃	1	4	yes	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
152	M₃₂	—	2	8	no	—
153	M₃₂	—	2	4	no	—
154	M₃₂	C₂³×C₁₃	1	8	yes	—
155	C₂×C₁₆	—	2	4	no	—
156	C₂×C₁₆	—	2	8	no	—
157	#97	C₂×C₄×C₁₃	1	8	yes	—
158	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
159	C₄×C₈	C₂³×C₁₃	1	4	yes	—
160	C₂×C₁₃	—	2	1	no	—
161	C₂×C₁₃	—	2	1	no	—
162	C₂×C₁₃	—	2	4	no	—
163	C₂×C₁₃	#2153	1	2	yes	direct product factor
164	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
165	C₂⁴	C₁₃×M₁₆	1	1	yes	characteristic
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
168	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
169	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
170	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
171	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
172	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
175	M₁₆	C₂×C₈×C₁₃	1	16	yes	—
176	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
177	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
178	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
179	C₂×C₈	C₁₃×C₄⋊C₄	1	4	yes	—
180	C₂×C₈	C₂×C₈×C₁₃	1	4	yes	—
181	C₂×C₈	C₂×C₈×C₁₃	1	4	yes	—
182	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
184	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
185	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
187	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
188	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
189	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
190	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
191	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
192	C₄⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
196	C₁₆	—	2	8	no	—
197	C₁₃	C₂×#2153	1	1	yes	characteristic, direct product factor, Sylow
198	C₂³	—	4	2	no	—
199	C₂³	C₁₃×C₄⋊C₈	1	2	yes	—
200	C₂³	C₁₃×C₄⋊C₈	1	2	yes	—
201	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
202	C₂³	C₂×C₁₃×M₁₆	1	2	yes	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
206	C₂×C₄	C₁₃×C₄⋊C₈	1	4	yes	—
207	C₂×C₄	C₁₃×C₄⋊C₈	1	4	yes	—
208	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
209	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
210	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
215	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
216	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
217	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
218	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
219	C₈	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
220	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
221	C₈	C₂²×C₈×C₁₃	1	8	yes	—
222	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
223	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
229	C₂²	C₂×C₁₃×C₄⋊C₈	1	4	yes	—
230	C₂²	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
231	C₂²	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
232	C₄	—	4	4	no	—
233	C₄	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
234	C₄	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
235	C₄	C₂×C₁₃×C₄⋊C₈	1	4	yes	—
236	C₄	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
237	C₄	C₂³×C₈×C₁₃	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	2	4	no	—
241	C₂	C₁₃×#2153	1	2	yes	direct product factor
242	C₂	C₂²×C₁₃×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂×D₄
commutator	C₄	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2153	C₂	direct (N × H)
4. C₂×C₁₃×#364	C₂	semidirect
17. C₁₃×#364	C₂²	semidirect
44. C₂×#2153	C₁₃	direct (N × H)
79. #2153	C₂×C₁₃	direct (N × H)
80. C₂×#364	C₂×C₁₃	semidirect
119. #364	C₂²×C₁₃	semidirect
163. C₂×C₁₃	#2153	direct (N × H)
197. C₁₃	C₂×#2153	direct (N × H)
241. C₂	C₁₃×#2153	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×M₃₂	2
C₂×C₁₃×#405	2
C₁₃×#2153	2
C₂×C₁₃×#364	2
C₂×#2153	13

Subgroups — C2×C13×#2153 #15719739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2153 #15719739