Subgroups — C₂²×C₁₃×#555 #15749046

Order 3328.

8710 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#555	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃×#124	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₃×#121	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×D₄×C₁₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×#117	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂³×D₄×C₁₃	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₁₃	C₂²	1	4	yes	—
11	C₁₃×#555	C₂²	1	256	yes	direct product factor
12	C₂×C₁₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
13	C₂×C₁₃×#124	C₂²	1	96	yes	complemented
14	C₂×C₁₃×#121	C₂²	1	96	yes	complemented
15	C₂×C₄×D₄×C₁₃	C₂²	1	48	yes	complemented
16	C₂×C₁₃×#117	C₂²	1	96	yes	complemented
17	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	8	yes	—
20	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
22	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
23	C₂⁵×C₁₃	C₂³	1	4	yes	—
24	C₂²×Q₈×C₁₃	—	4	8	no	—
25	C₂²×Q₈×C₁₃	C₂³	1	24	yes	—
26	C₂²×D₄×C₁₃	—	2	16	no	—
27	C₂²×D₄×C₁₃	—	4	16	no	—
28	C₂²×D₄×C₁₃	C₂³	1	24	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	48	yes	—
30	C₂³×C₄×C₁₃	—	2	32	no	—
31	C₂³×C₄×C₁₃	C₂³	1	48	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
36	C₁₃×C₄⋊Q₈	C₂³	1	64	yes	complemented
37	C₁₃×#124	C₂³	1	256	yes	complemented
38	C₁₃×#121	C₂³	1	256	yes	complemented
39	C₄×D₄×C₁₃	C₂³	1	128	yes	complemented
40	C₁₃×#117	C₂³	1	256	yes	complemented
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
43	C₂×C₁₃×C₂²⋊C₄	C₂³	1	96	yes	—
44	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	48	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
47	C₂²×#555	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₃	—	2	16	no	—
49	C₂⁴×C₁₃	—	2	16	no	—
50	C₂⁴×C₁₃	—	2	16	no	—
51	C₂⁴×C₁₃	C₂⁴	1	24	yes	—
52	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₂×Q₈×C₁₃	—	4	48	no	—
54	C₂×Q₈×C₁₃	C₂⁴	1	32	yes	—
55	C₂×D₄×C₁₃	—	2	96	no	—
56	C₂×D₄×C₁₃	—	4	96	no	—
57	C₂×D₄×C₁₃	C₂⁴	1	32	yes	—
58	C₂×D₄×C₁₃	C₂⁴	1	64	yes	—
59	C₂²×C₄×C₁₃	—	2	16	no	—
60	C₂²×C₄×C₁₃	—	2	16	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	192	no	—
63	C₂²×C₄×C₁₃	#41	1	16	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	64	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
69	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
70	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
71	C₁₃×C₂²⋊C₄	C₂⁴	1	128	yes	—
72	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	—
73	C₄²×C₁₃	C₂⁴	1	64	yes	—
74	C₄²×C₁₃	C₂⁴	1	16	yes	—
75	C₂×#555	C₂×C₁₃	1	48	yes	direct product factor
76	C₂²×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
77	C₂²×#124	C₂×C₁₃	1	4	yes	complemented
78	C₂²×#121	C₂×C₁₃	1	4	yes	complemented
79	C₂²×C₄×D₄	C₂×C₁₃	1	2	yes	complemented
80	C₂²×#117	C₂×C₁₃	1	4	yes	complemented
81	C₂³×C₁₃	—	2	48	no	—
82	C₂³×C₁₃	—	2	48	no	—
83	C₂³×C₁₃	—	2	48	no	—
84	C₂³×C₁₃	—	4	64	no	—
85	C₂³×C₁₃	C₂×#41	1	8	yes	—
86	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
87	C₂³×C₁₃	C₂⁵	1	16	yes	—
88	C₂³×C₁₃	C₂⁵	1	3	yes	—
89	Q₈×C₁₃	—	4	32	no	—
90	D₄×C₁₃	—	2	64	no	—
91	D₄×C₁₃	—	4	64	no	—
92	C₂×C₄×C₁₃	—	2	48	no	—
93	C₂×C₄×C₁₃	—	2	48	no	—
94	C₂×C₄×C₁₃	—	2	12	no	—
95	C₂×C₄×C₁₃	—	2	128	no	—
96	C₂×C₄×C₁₃	C₂×#41	1	48	yes	—
97	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
98	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
99	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
100	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
101	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
102	C₂³×D₄	C₂²×C₁₃	1	2	yes	—
103	C₂³×D₄	C₂²×C₁₃	1	4	yes	—
104	C₂⁴×C₄	C₂²×C₁₃	1	4	yes	—
105	#555	C₂²×C₁₃	1	256	yes	direct product factor
106	C₂×C₄⋊Q₈	C₂²×C₁₃	1	24	yes	complemented
107	C₂×#124	C₂²×C₁₃	1	96	yes	complemented
108	C₂×#121	C₂²×C₁₃	1	96	yes	complemented
109	C₂×C₄×D₄	C₂²×C₁₃	1	48	yes	complemented
110	C₂×#117	C₂²×C₁₃	1	96	yes	complemented
111	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
112	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
113	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	8	yes	—
114	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
115	C₂²×C₄²	C₂²×C₁₃	1	4	yes	—
116	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
117	C₂²×C₁₃	—	2	16	no	—
118	C₂²×C₁₃	—	2	16	no	—
119	C₂²×C₁₃	—	2	16	no	—
120	C₂²×C₁₃	—	4	96	no	—
121	C₂²×C₁₃	#555	1	16	yes	direct product factor
122	C₂²×C₁₃	C₂²×#41	1	12	yes	—
123	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	6	yes	—
124	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
125	C₄×C₁₃	—	2	16	no	—
126	C₄×C₁₃	—	2	16	no	—
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	C₂²×#41	1	16	yes	—
129	C₂⁵	C₂³×C₁₃	1	4	yes	—
130	C₂²×Q₈	—	4	8	no	—
131	C₂²×Q₈	C₂³×C₁₃	1	24	yes	—
132	C₂²×D₄	—	2	16	no	—
133	C₂²×D₄	—	4	16	no	—
134	C₂²×D₄	C₂³×C₁₃	1	24	yes	—
135	C₂²×D₄	C₂³×C₁₃	1	48	yes	—
136	C₂³×C₄	—	2	32	no	—
137	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
138	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
139	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
140	C₂³×C₄	C₂³×C₁₃	1	48	yes	—
141	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
142	C₄⋊Q₈	C₂³×C₁₃	1	64	yes	complemented
143	#124	C₂³×C₁₃	1	256	yes	complemented
144	#121	C₂³×C₁₃	1	256	yes	complemented
145	C₄×D₄	C₂³×C₁₃	1	128	yes	complemented
146	#117	C₂³×C₁₃	1	256	yes	complemented
147	C₂×C₄⋊C₄	C₂³×C₁₃	1	24	yes	—
148	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
149	C₂×C₂²⋊C₄	C₂³×C₁₃	1	96	yes	—
150	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
151	C₂×C₄²	C₂³×C₁₃	1	48	yes	—
152	C₂×C₄²	C₂³×C₁₃	1	12	yes	—
153	C₂×C₁₃	—	4	16	no	—
154	C₂×C₁₃	C₂×#555	1	12	yes	direct product factor
155	C₂×C₁₃	C₂³×#41	1	2	yes	—
156	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
157	C₂⁴	—	2	16	no	—
158	C₂⁴	—	2	16	no	—
159	C₂⁴	—	2	16	no	—
160	C₂⁴	C₂⁴×C₁₃	1	24	yes	—
161	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
162	C₂×Q₈	—	4	48	no	—
163	C₂×Q₈	C₂⁴×C₁₃	1	32	yes	—
164	C₂×D₄	—	2	96	no	—
165	C₂×D₄	—	4	96	no	—
166	C₂×D₄	C₂⁴×C₁₃	1	32	yes	—
167	C₂×D₄	C₂⁴×C₁₃	1	64	yes	—
168	C₂²×C₄	—	2	16	no	—
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	192	no	—
172	C₂²×C₄	C₁₃×#41	1	16	yes	—
173	C₂²×C₄	C₂⁴×C₁₃	1	24	yes	—
174	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
175	C₂²×C₄	C₂⁴×C₁₃	1	24	yes	—
176	C₂²×C₄	C₂⁴×C₁₃	1	64	yes	—
177	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
178	C₄⋊C₄	C₂⁴×C₁₃	1	32	yes	—
179	C₄⋊C₄	C₂⁴×C₁₃	1	64	yes	—
180	C₂²⋊C₄	C₂⁴×C₁₃	1	128	yes	—
181	C₂²⋊C₄	C₂⁴×C₁₃	1	64	yes	—
182	C₄²	C₂⁴×C₁₃	1	64	yes	—
183	C₄²	C₂⁴×C₁₃	1	16	yes	—
184	C₁₃	C₂²×#555	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³	—	2	48	no	—
186	C₂³	—	2	48	no	—
187	C₂³	—	2	48	no	—
188	C₂³	—	4	64	no	—
189	C₂³	C₂×C₁₃×#41	1	8	yes	—
190	C₂³	C₁₃×H(5,𝔽₂)	1	4	yes	—
191	C₂³	C₂⁵×C₁₃	1	16	yes	—
192	C₂³	C₂⁵×C₁₃	1	3	yes	—
193	Q₈	—	4	32	no	—
194	D₄	—	2	64	no	—
195	D₄	—	4	64	no	—
196	C₂×C₄	—	2	48	no	—
197	C₂×C₄	—	2	48	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	2	128	no	—
200	C₂×C₄	C₂×C₁₃×#41	1	48	yes	—
201	C₂×C₄	C₂⁵×C₁₃	1	16	yes	—
202	C₂×C₄	C₂⁵×C₁₃	1	4	yes	—
203	C₂×C₄	C₂⁵×C₁₃	1	16	yes	—
204	C₂×C₄	C₂⁵×C₁₃	1	8	yes	—
205	C₂²	—	2	16	no	—
206	C₂²	—	2	16	no	—
207	C₂²	—	2	16	no	—
208	C₂²	—	4	96	no	—
209	C₂²	C₁₃×#555	1	16	yes	direct product factor
210	C₂²	C₂²×C₁₃×#41	1	12	yes	—
211	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	6	yes	—
212	C₂²	C₂⁶×C₁₃	1	1	yes	characteristic
213	C₄	—	2	16	no	—
214	C₄	—	2	16	no	—
215	C₄	—	2	4	no	—
216	C₄	C₂²×C₁₃×#41	1	16	yes	—
217	C₂	—	4	16	no	—
218	C₂	C₂×C₁₃×#555	1	12	yes	direct product factor
219	C₂	C₂³×C₁₃×#41	1	2	yes	—
220	C₂	C₂²×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#555	C₂	direct (N × H)
2. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₁₃×#124	C₂	semidirect
4. C₂²×C₁₃×#121	C₂	semidirect
5. C₂²×C₄×D₄×C₁₃	C₂	semidirect
6. C₂²×C₁₃×#117	C₂	semidirect
11. C₁₃×#555	C₂²	direct (N × H)
12. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
13. C₂×C₁₃×#124	C₂²	semidirect
14. C₂×C₁₃×#121	C₂²	semidirect
15. C₂×C₄×D₄×C₁₃	C₂²	semidirect
16. C₂×C₁₃×#117	C₂²	semidirect
36. C₁₃×C₄⋊Q₈	C₂³	semidirect
37. C₁₃×#124	C₂³	semidirect
38. C₁₃×#121	C₂³	semidirect
39. C₄×D₄×C₁₃	C₂³	semidirect
40. C₁₃×#117	C₂³	semidirect
47. C₂²×#555	C₁₃	direct (N × H)
75. C₂×#555	C₂×C₁₃	direct (N × H)
76. C₂²×C₄⋊Q₈	C₂×C₁₃	semidirect
77. C₂²×#124	C₂×C₁₃	semidirect
78. C₂²×#121	C₂×C₁₃	semidirect
79. C₂²×C₄×D₄	C₂×C₁₃	semidirect
80. C₂²×#117	C₂×C₁₃	semidirect
105. #555	C₂²×C₁₃	direct (N × H)
106. C₂×C₄⋊Q₈	C₂²×C₁₃	semidirect
107. C₂×#124	C₂²×C₁₃	semidirect
108. C₂×#121	C₂²×C₁₃	semidirect
109. C₂×C₄×D₄	C₂²×C₁₃	semidirect
110. C₂×#117	C₂²×C₁₃	semidirect
121. C₂²×C₁₃	#555	direct (N × H)
142. C₄⋊Q₈	C₂³×C₁₃	semidirect
143. #124	C₂³×C₁₃	semidirect
144. #121	C₂³×C₁₃	semidirect
145. C₄×D₄	C₂³×C₁₃	semidirect
146. #117	C₂³×C₁₃	semidirect
154. C₂×C₁₃	C₂×#555	direct (N × H)
184. C₁₃	C₂²×#555	direct (N × H)
209. C₂²	C₁₃×#555	direct (N × H)
218. C₂	C₂×C₁₃×#555	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×#555	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×C₁₃×#124	2
C₂²×C₁₃×#121	2
C₂²×C₄×D₄×C₁₃	2
C₂²×C₁₃×#117	2
C₂²×#555	13

Subgroups — C22×C13×#555 #15749046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#555 #15749046