Subgroups — C₂²×C₁₃×#400 #15720880

Order 3328.

3550 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#400	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₃×#95	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×#95	C₂	1	2	yes	maximal
6	C₂²×C₁₃×#95	C₂	1	2	yes	maximal
7	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₄⋊C₄	—	2	8	no	—
13	C₂²×C₁₃×C₄⋊C₄	—	2	8	no	—
14	C₂²×C₁₃×C₄⋊C₄	C₂²	1	12	yes	—
15	C₂²×C₁₃×C₄⋊C₄	C₂²	1	12	yes	—
16	C₁₃×#400	C₂²	1	64	yes	direct product factor
17	C₂×C₁₃×#95	C₂²	1	24	yes	—
18	C₂×C₁₃×#95	C₂²	1	24	yes	—
19	C₂×C₁₃×#95	C₂²	1	12	yes	—
20	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	—	2	4	no	—
22	C₂³×C₄×C₁₃	—	2	4	no	—
23	C₂³×C₄×C₁₃	—	2	8	no	—
24	C₂³×C₄×C₁₃	—	2	4	no	—
25	C₂³×C₄×C₁₃	—	2	8	no	—
26	C₂³×C₄×C₁₃	—	2	8	no	—
27	C₂³×C₄×C₁₃	—	2	16	no	—
28	C₂³×C₄×C₁₃	—	2	8	no	—
29	C₂³×C₄×C₁₃	—	2	8	no	—
30	C₂³×C₄×C₁₃	—	2	8	no	—
31	C₂³×C₄×C₁₃	D₄	1	8	yes	—
32	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	48	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	48	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
42	C₁₃×#95	C₂³	1	16	yes	—
43	C₁₃×#95	C₂³	1	32	yes	—
44	C₁₃×#95	C₂³	1	32	yes	—
45	C₂²×#400	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂⁴×C₁₃	C₂×D₄	1	4	yes	—
47	C₂⁴×C₁₃	C₂×Q₈	1	4	yes	—
48	C₂⁴×C₁₃	#41	1	4	yes	—
49	C₂⁴×C₁₃	#41	1	8	yes	—
50	C₂⁴×C₁₃	#41	1	8	yes	—
51	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
52	C₂²×C₄×C₁₃	—	2	32	no	—
53	C₂²×C₄×C₁₃	—	2	32	no	—
54	C₂²×C₄×C₁₃	—	2	12	no	—
55	C₂²×C₄×C₁₃	—	2	12	no	—
56	C₂²×C₄×C₁₃	—	2	24	no	—
57	C₂²×C₄×C₁₃	—	2	12	no	—
58	C₂²×C₄×C₁₃	—	2	24	no	—
59	C₂²×C₄×C₁₃	—	2	24	no	—
60	C₂²×C₄×C₁₃	—	2	48	no	—
61	C₂²×C₄×C₁₃	—	2	24	no	—
62	C₂²×C₄×C₁₃	—	2	24	no	—
63	C₂²×C₄×C₁₃	—	2	24	no	—
64	C₂²×C₄×C₁₃	—	4	16	no	—
65	C₂²×C₄×C₁₃	—	4	32	no	—
66	C₂²×C₄×C₁₃	—	4	32	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
74	C₁₃×C₄⋊C₄	—	2	32	no	—
75	C₁₃×C₄⋊C₄	—	2	32	no	—
76	C₂³×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
77	C₂³×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
78	C₂×#400	C₂×C₁₃	1	24	yes	direct product factor
79	C₂²×#95	C₂×C₁₃	1	2	yes	—
80	C₂²×#95	C₂×C₁₃	1	2	yes	—
81	C₂²×#95	C₂×C₁₃	1	1	yes	characteristic
82	C₂³×C₁₃	C₂²⋊Q₈	1	16	yes	—
83	C₂³×C₁₃	#123	1	16	yes	—
84	C₂³×C₁₃	#124	1	16	yes	—
85	C₂³×C₁₃	#125	1	32	yes	—
86	C₂³×C₁₃	#126	1	32	yes	—
87	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
88	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
89	C₂³×C₁₃	C₂×#41	1	6	yes	—
90	C₂³×C₁₃	C₂×#41	1	12	yes	—
91	C₂³×C₁₃	C₂×#41	1	12	yes	—
92	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	48	no	—
94	C₂×C₄×C₁₃	—	2	48	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	8	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	8	no	—
100	C₂×C₄×C₁₃	—	2	8	no	—
101	C₂×C₄×C₁₃	—	2	16	no	—
102	C₂×C₄×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	—	2	8	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	4	24	no	—
106	C₂×C₄×C₁₃	—	4	48	no	—
107	C₂×C₄×C₁₃	—	4	48	no	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
109	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
110	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
111	C₂⁴×C₄	C₂²×C₁₃	1	2	yes	—
112	C₂⁴×C₄	C₂²×C₁₃	1	2	yes	—
113	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
114	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
115	C₂²×C₄⋊C₄	—	2	8	no	—
116	C₂²×C₄⋊C₄	—	2	8	no	—
117	C₂²×C₄⋊C₄	C₂²×C₁₃	1	12	yes	—
118	C₂²×C₄⋊C₄	C₂²×C₁₃	1	12	yes	—
119	#400	C₂²×C₁₃	1	64	yes	direct product factor
120	C₂×#95	C₂²×C₁₃	1	24	yes	—
121	C₂×#95	C₂²×C₁₃	1	24	yes	—
122	C₂×#95	C₂²×C₁₃	1	12	yes	—
123	C₂²×C₁₃	#400	1	64	yes	direct product factor
124	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	12	yes	—
125	C₂²×C₁₃	C₂×#123	1	12	yes	—
126	C₂²×C₁₃	C₂×#124	1	12	yes	—
127	C₂²×C₁₃	C₂×#125	1	24	yes	—
128	C₂²×C₁₃	C₂×#126	1	24	yes	—
129	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
130	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
131	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
132	C₂²×C₁₃	C₂²×#41	1	2	yes	—
133	C₂²×C₁₃	C₂²×#41	1	2	yes	—
134	C₄×C₁₃	—	2	8	no	—
135	C₄×C₁₃	—	2	8	no	—
136	C₄×C₁₃	—	4	4	no	—
137	C₄×C₁₃	—	4	8	no	—
138	C₄×C₁₃	—	4	8	no	—
139	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
140	C₂³×C₄	—	2	4	no	—
141	C₂³×C₄	—	2	4	no	—
142	C₂³×C₄	—	2	8	no	—
143	C₂³×C₄	—	2	4	no	—
144	C₂³×C₄	—	2	8	no	—
145	C₂³×C₄	—	2	8	no	—
146	C₂³×C₄	—	2	16	no	—
147	C₂³×C₄	—	2	8	no	—
148	C₂³×C₄	—	2	8	no	—
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	D₄×C₁₃	1	8	yes	—
151	C₂³×C₄	Q₈×C₁₃	1	8	yes	—
152	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
153	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
154	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
155	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
156	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
157	C₂×C₄⋊C₄	—	2	48	no	—
158	C₂×C₄⋊C₄	—	2	48	no	—
159	C₂×C₄⋊C₄	C₂³×C₁₃	1	16	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₃	1	16	yes	—
161	#95	C₂³×C₁₃	1	16	yes	—
162	#95	C₂³×C₁₃	1	32	yes	—
163	#95	C₂³×C₁₃	1	32	yes	—
164	C₂×C₁₃	C₂×#400	1	24	yes	direct product factor
165	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
166	C₂×C₁₃	C₂²×#123	1	1	yes	characteristic
167	C₂×C₁₃	C₂²×#124	1	1	yes	characteristic
168	C₂×C₁₃	C₂²×#125	1	2	yes	—
169	C₂×C₁₃	C₂²×#126	1	2	yes	—
170	C₂⁴	C₂×D₄×C₁₃	1	4	yes	—
171	C₂⁴	C₂×Q₈×C₁₃	1	4	yes	—
172	C₂⁴	C₁₃×#41	1	4	yes	—
173	C₂⁴	C₁₃×#41	1	8	yes	—
174	C₂⁴	C₁₃×#41	1	8	yes	—
175	C₂⁴	C₂⁴×C₁₃	1	3	yes	—
176	C₂²×C₄	—	2	32	no	—
177	C₂²×C₄	—	2	32	no	—
178	C₂²×C₄	—	2	12	no	—
179	C₂²×C₄	—	2	12	no	—
180	C₂²×C₄	—	2	24	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	2	24	no	—
183	C₂²×C₄	—	2	24	no	—
184	C₂²×C₄	—	2	48	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	2	24	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	—	4	16	no	—
189	C₂²×C₄	—	4	32	no	—
190	C₂²×C₄	—	4	32	no	—
191	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
192	C₂²×C₄	C₂×Q₈×C₁₃	1	24	yes	—
193	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
194	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
195	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
196	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
197	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
198	C₄⋊C₄	—	2	32	no	—
199	C₄⋊C₄	—	2	32	no	—
200	C₁₃	C₂²×#400	1	1	yes	characteristic, direct product factor, Sylow
201	C₂³	C₁₃×C₂²⋊Q₈	1	16	yes	—
202	C₂³	C₁₃×#123	1	16	yes	—
203	C₂³	C₁₃×#124	1	16	yes	—
204	C₂³	C₁₃×#125	1	32	yes	—
205	C₂³	C₁₃×#126	1	32	yes	—
206	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
207	C₂³	C₂²×Q₈×C₁₃	1	6	yes	—
208	C₂³	C₂×C₁₃×#41	1	6	yes	—
209	C₂³	C₂×C₁₃×#41	1	12	yes	—
210	C₂³	C₂×C₁₃×#41	1	12	yes	—
211	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
212	C₂×C₄	—	2	48	no	—
213	C₂×C₄	—	2	48	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	24	no	—
225	C₂×C₄	—	4	48	no	—
226	C₂×C₄	—	4	48	no	—
227	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
228	C₂×C₄	C₂²×Q₈×C₁₃	1	8	yes	—
229	C₂²	C₁₃×#400	1	64	yes	direct product factor
230	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	12	yes	—
231	C₂²	C₂×C₁₃×#123	1	12	yes	—
232	C₂²	C₂×C₁₃×#124	1	12	yes	—
233	C₂²	C₂×C₁₃×#125	1	24	yes	—
234	C₂²	C₂×C₁₃×#126	1	24	yes	—
235	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
236	C₂²	C₂³×Q₈×C₁₃	1	1	yes	characteristic
237	C₂²	C₂²×C₁₃×#41	1	1	yes	characteristic
238	C₂²	C₂²×C₁₃×#41	1	2	yes	—
239	C₂²	C₂²×C₁₃×#41	1	2	yes	—
240	C₄	—	2	8	no	—
241	C₄	—	2	8	no	—
242	C₄	—	4	4	no	—
243	C₄	—	4	8	no	—
244	C₄	—	4	8	no	—
245	C₂	C₂×C₁₃×#400	1	24	yes	direct product factor
246	C₂	C₂²×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂	C₂²×C₁₃×#123	1	1	yes	characteristic
248	C₂	C₂²×C₁₃×#124	1	1	yes	characteristic
249	C₂	C₂²×C₁₃×#125	1	2	yes	—
250	C₂	C₂²×C₁₃×#126	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₁₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₃×#400	C₂	direct (N × H)
16. C₁₃×#400	C₂²	direct (N × H)
45. C₂²×#400	C₁₃	direct (N × H)
78. C₂×#400	C₂×C₁₃	direct (N × H)
119. #400	C₂²×C₁₃	direct (N × H)
123. C₂²×C₁₃	#400	direct (N × H)
164. C₂×C₁₃	C₂×#400	direct (N × H)
200. C₁₃	C₂²×#400	direct (N × H)
229. C₂²	C₁₃×#400	direct (N × H)
245. C₂	C₂×C₁₃×#400	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×C₄⋊C₄	2
C₂×C₁₃×#400	2
C₂²×C₁₃×#95	2
C₂²×#400	13

Subgroups — C22×C13×#400 #15720880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#400 #15720880