Subgroups — C₂×C₁₃×#2225 #15719962

Order 3328.

574 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	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2225	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#367	C₂	1	2	yes	maximal
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#358	C₂	1	2	yes	maximal
7	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#500	—	2	2	no	—
12	C₁₃×#500	C₂²	1	4	yes	—
13	C₄×C₁₃×Q₁₆	—	2	2	no	—
14	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
15	C₂×C₁₃×#107	—	2	1	no	—
16	C₂×C₁₃×#107	C₂²	1	2	yes	—
17	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#103	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#367	C₂²	1	8	yes	—
21	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
22	C₁₃×#358	C₂²	1	8	yes	—
23	C₂²×Q₈×C₁₃	—	2	1	no	—
24	C₂²×Q₈×C₁₃	—	4	1	no	—
25	C₂×C₁₃×Q₁₆	—	2	1	no	—
26	C₂×C₁₃×Q₁₆	—	2	2	no	—
27	C₂×C₁₃×Q₁₆	Q₈	1	4	yes	complemented
28	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×C₄⋊Q₈	—	2	4	no	—
32	C₁₃×C₄⋊Q₈	—	4	2	no	—
33	C₄×Q₈×C₁₃	—	2	4	no	—
34	C₄×Q₈×C₁₃	—	4	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
39	C₂×C₄²×C₁₃	—	4	1	no	—
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	—	2	4	no	—
42	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
43	C₁₃×#107	—	2	4	no	—
44	C₁₃×#107	—	2	1	no	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	C₂³	1	4	yes	—
47	C₁₃×#107	C₂³	1	2	yes	—
48	C₁₃×#103	—	2	4	no	—
49	C₄×C₈×C₁₃	D₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂×#2225	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×Q₈×C₁₃	—	2	4	no	—
53	C₂×Q₈×C₁₃	—	2	2	no	—
54	C₂×Q₈×C₁₃	—	4	1	no	—
55	C₂×Q₈×C₁₃	—	4	4	no	—
56	C₂×Q₈×C₁₃	—	4	2	no	—
57	C₂×Q₈×C₁₃	—	8	4	no	—
58	C₂²×C₄×C₁₃	—	4	1	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×Q₁₆	—	2	4	no	—
65	C₁₃×Q₁₆	—	2	2	no	—
66	C₁₃×Q₁₆	C₂×Q₈	1	4	yes	complemented
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₁₃	#41	1	2	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	4	1	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	4	4	no	—
78	C₁₃×C₄⋊C₄	—	4	4	no	—
79	C₄²×C₁₃	—	4	4	no	—
80	C₄²×C₁₃	D₈	1	2	yes	—
81	C₄²×C₁₃	C₂×D₄	1	2	yes	—
82	C₁₃×C₁₆	—	2	4	no	—
83	C₂×#500	C₂×C₁₃	1	1	yes	characteristic
84	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
85	#2225	C₂×C₁₃	1	8	yes	direct product factor
86	C₂×#367	C₂×C₁₃	1	2	yes	—
87	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
88	C₂×#358	C₂×C₁₃	1	2	yes	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₁₃	—	2	4	no	—
91	Q₈×C₁₃	—	4	2	no	—
92	Q₈×C₁₃	—	4	4	no	—
93	Q₈×C₁₃	—	8	8	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	8	4	no	—
100	C₂×C₄×C₁₃	—	8	2	no	—
101	C₂×C₄×C₁₃	SD₃₂	1	4	yes	—
102	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₃	#137	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
110	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
111	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂×C₄×Q₈	—	2	1	no	—
115	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
116	#500	—	2	2	no	—
117	#500	C₂²×C₁₃	1	4	yes	—
118	C₄×Q₁₆	—	2	2	no	—
119	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	—
120	C₂×#107	—	2	1	no	—
121	C₂×#107	C₂²×C₁₃	1	2	yes	—
122	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
123	C₂×#103	—	2	1	no	—
124	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
125	#367	C₂²×C₁₃	1	8	yes	—
126	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
127	#358	C₂²×C₁₃	1	8	yes	—
128	C₂²×C₁₃	#474	1	2	yes	—
129	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
130	C₂²×C₁₃	#510	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	4	4	no	—
134	C₄×C₁₃	—	8	4	no	—
135	C₄×C₁₃	—	8	2	no	—
136	C₄×C₁₃	#474	1	2	yes	—
137	C₄×C₁₃	C₂×SD₃₂	1	4	yes	—
138	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
140	C₂²×Q₈	—	2	1	no	—
141	C₂²×Q₈	—	4	1	no	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	Q₈×C₁₃	1	4	yes	complemented
145	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
146	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
148	C₄⋊Q₈	—	2	4	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	C₄×Q₈	—	2	4	no	—
151	C₄×Q₈	—	4	2	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	1	no	—
155	C₂×C₄⋊C₄	—	4	1	no	—
156	C₂×C₄²	—	4	1	no	—
157	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
160	#107	—	2	4	no	—
161	#107	—	2	1	no	—
162	#107	—	2	2	no	—
163	#107	C₂³×C₁₃	1	4	yes	—
164	#107	C₂³×C₁₃	1	2	yes	—
165	#103	—	2	4	no	—
166	C₄×C₈	D₄×C₁₃	1	2	yes	—
167	C₄×C₈	C₂³×C₁₃	1	2	yes	—
168	C₂×C₁₃	#2225	1	4	yes	direct product factor
169	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
170	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	8	4	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
182	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
184	Q₁₆	—	2	4	no	—
185	Q₁₆	—	2	2	no	—
186	Q₁₆	C₂×Q₈×C₁₃	1	4	yes	complemented
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
189	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
190	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
191	C₂×C₈	C₁₃×#41	1	2	yes	—
192	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄²	—	4	4	no	—
200	C₄²	C₁₃×D₈	1	2	yes	—
201	C₄²	C₂×D₄×C₁₃	1	2	yes	—
202	C₁₆	—	2	4	no	—
203	C₁₃	C₂×#2225	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	4	4	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	8	8	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	C₁₃×SD₃₂	1	4	yes	—
217	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
218	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
219	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
220	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
221	C₂×C₄	C₁₃×#137	1	2	yes	—
222	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
225	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
226	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
227	C₂²	C₁₃×#474	1	2	yes	—
228	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
229	C₂²	C₁₃×#510	1	2	yes	—
230	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	4	4	no	—
233	C₄	—	8	4	no	—
234	C₄	—	8	2	no	—
235	C₄	C₁₃×#474	1	2	yes	—
236	C₄	C₂×C₁₃×SD₃₂	1	4	yes	—
237	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
238	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
239	C₂	C₁₃×#2225	1	4	yes	direct product factor
240	C₂	C₂×C₁₃×#474	1	1	yes	characteristic
241	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
242	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2225	C₂	direct (N × H)
27. C₂×C₁₃×Q₁₆	Q₈	semidirect
51. C₂×#2225	C₁₃	direct (N × H)
66. C₁₃×Q₁₆	C₂×Q₈	semidirect
85. #2225	C₂×C₁₃	direct (N × H)
144. C₂×Q₁₆	Q₈×C₁₃	semidirect
168. C₂×C₁₃	#2225	direct (N × H)
186. Q₁₆	C₂×Q₈×C₁₃	semidirect
203. C₁₃	C₂×#2225	direct (N × H)
239. C₂	C₁₃×#2225	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#500	2
C₂×C₄×C₁₃×Q₁₆	2
C₁₃×#2225	2
C₂×C₁₃×#367	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#358	2
C₂×#2225	13

Subgroups — C2×C13×#2225 #15719962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2225 #15719962