Subgroups — #15754155

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114705	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
4	#13112677	C₂	1	2	yes	maximal
5	C₄×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#107	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊#1055	C₂²	1	2	yes	—
10	C₂×#88043	—	2	2	no	—
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#340	C₄	1	2	yes	—
14	#11718116	—	2	2	no	—
15	#11718050	C₄	1	4	yes	complemented
16	C₈×#1055	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₁₃	—	2	2	no	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×#1055	—	2	1	no	—
23	C₂²×#1055	—	2	2	no	—
24	C₂²×#1055	D₄	1	1	yes	characteristic
25	C₂²×#1055	D₄	1	1	yes	characteristic
26	C₁₃×#107	—	2	4	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₁₃×#95	—	2	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	C₂×C₄	1	4	yes	—
32	#88043	—	4	4	no	—
33	C₄×#1055	—	2	1	no	—
34	C₄×#1055	C₂×C₄	1	2	yes	—
35	C₄×#1055	C₂×C₄	1	2	yes	—
36	C₄×#1055	Q₈	1	2	yes	—
37	#8282	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	C₄²	1	2	yes	—
44	C₂×C₈×C₁₃	C₄²	1	2	yes	—
45	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
46	C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₄²×C₁₃	—	4	2	no	—
49	C₄²×C₁₃	—	4	2	no	—
50	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₁₃	#41	1	1	yes	characteristic
54	C₂×#1055	—	2	4	no	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	1	no	—
57	C₂×#1055	—	2	1	no	—
58	C₂×#1055	—	2	4	no	—
59	C₂×#1055	—	4	2	no	—
60	C₂×#1055	—	4	2	no	—
61	C₂×#1055	C₄⋊C₄	1	2	yes	—
62	C₂×#1055	C₄⋊C₄	1	2	yes	—
63	C₂×#1055	D₈	1	2	yes	—
64	C₂×#1055	Q₁₆	1	2	yes	—
65	C₄×#107	D₁₃	1	1	yes	characteristic
66	#1770	—	13	2	no	—
67	C₂×#335	—	13	1	no	—
68	C₂×C₈²	—	13	1	no	—
69	#1279	—	13	2	no	—
70	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	4	4	no	—
73	C₂×C₄×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	—	4	4	no	—
75	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	#131	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	#97	1	4	yes	complemented
89	#1055	—	2	4	no	—
90	#1055	—	2	2	no	—
91	#1055	—	4	4	no	—
92	#1055	#107	1	4	yes	complemented
93	C₂×#107	Dic₁₃	1	2	yes	—
94	C₂×C₄⋊C₈	—	13	2	no	—
95	C₂×#97	—	26	2	no	—
96	C₂×C₄×C₈	—	13	1	no	—
97	C₂×C₄×C₈	—	13	1	no	—
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
100	#340	Dic₁₃	1	2	yes	—
101	#336	—	26	2	no	—
102	#335	—	13	4	no	—
103	C₈²	—	13	4	no	—
104	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₃	#436	1	2	yes	—
106	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
107	C₂²×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
108	C₂²×C₁₃	#443	1	2	yes	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	8	4	no	—
112	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
113	C₄×C₁₃	#405	1	1	yes	characteristic
114	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
115	C₄×C₁₃	#427	1	1	yes	characteristic
116	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
117	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
118	C₄×C₁₃	#446	1	2	yes	—
119	C₂²×C₈	—	13	1	no	—
120	C₂²×C₈	—	13	1	no	—
121	C₂²×C₈	—	26	1	no	—
122	C₂²×C₈	—	26	2	no	—
123	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
128	#107	—	2	4	no	—
129	C₄⋊C₈	—	13	4	no	—
130	C₄⋊C₈	—	26	2	no	—
131	#97	—	52	4	no	—
132	C₄×C₈	—	13	2	no	—
133	C₄×C₈	—	13	2	no	—
134	C₄×C₈	—	13	2	no	—
135	C₄×C₈	—	26	1	no	—
136	C₄×C₈	C₄×D₁₃	1	2	yes	—
137	C₄×C₈	C₄×D₁₃	1	2	yes	—
138	#95	—	2	2	no	—
139	C₂×C₁₃	#1589	1	2	yes	—
140	C₂×C₁₃	#1591	1	2	yes	—
141	C₂×C₁₃	#1770	1	1	yes	characteristic
142	C₂×C₁₃	#1772	1	1	yes	characteristic
143	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
146	C₂²×C₄	#6129	1	1	yes	characteristic
147	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	13	2	no	—
150	C₂×C₈	—	13	2	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	—	13	2	no	—
153	C₂×C₈	—	26	4	no	—
154	C₂×C₈	—	26	2	no	—
155	C₂×C₈	—	26	1	no	—
156	C₂×C₈	—	26	1	no	—
157	C₂×C₈	—	26	4	no	—
158	C₂×C₈	—	52	2	no	—
159	C₂×C₈	—	52	2	no	—
160	C₂×C₈	#6099	1	4	yes	—
161	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
162	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄²	—	4	2	no	—
166	C₄²	—	4	2	no	—
167	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
168	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
169	C₄²	Q₈×D₁₃	1	1	yes	characteristic
170	C₄²	#6131	1	1	yes	characteristic
171	C₁₃	#8282	1	1	yes	characteristic, complemented, Sylow
172	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	#88141	1	1	yes	characteristic
178	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
179	C₂×C₄	#88146	1	1	yes	characteristic
180	C₂×C₄	#88147	1	1	yes	characteristic
181	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
182	C₂×C₄	#88166	1	1	yes	characteristic
183	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
184	C₂×C₄	#88173	1	1	yes	characteristic
185	C₂×C₄	C₂×#6099	1	2	yes	—
186	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
187	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
188	C₂×C₄	#88202	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	—	13	4	no	—
191	C₈	—	26	4	no	—
192	C₈	—	26	2	no	—
193	C₈	—	52	4	no	—
194	C₈	#88043	1	4	yes	complemented
195	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
196	C₂²	#11718673	1	2	yes	—
197	C₂²	D₈×Dic₁₃	1	1	yes	characteristic
198	C₂²	Q₁₆×Dic₁₃	1	1	yes	characteristic
199	C₂²	#11718857	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	4	no	—
203	C₄	#11718533	1	1	yes	characteristic
204	C₄	#11718534	1	1	yes	characteristic
205	C₄	D₁₃×#107	1	1	yes	characteristic
206	C₄	#11718541	1	1	yes	characteristic
207	C₄	C₂×#88043	1	1	yes	characteristic
208	C₄	#11718658	1	1	yes	characteristic
209	C₄	#11718685	1	2	yes	—
210	C₂	#13113721	1	2	yes	—
211	C₂	#13113733	1	2	yes	—
212	C₂	#13114705	1	1	yes	characteristic
213	C₂	#13114717	1	1	yes	characteristic
214	C₂	#107×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#107	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#107	D₁₃
hypocenter	C₁₃	#8282
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
15. #11718050	C₄	semidirect
16. C₈×#1055	C₄	semidirect
88. C₈×C₁₃	#97	semidirect
92. #1055	#107	semidirect
171. C₁₃	#8282	semidirect
194. C₈	#88043	semidirect

Maximal subgroups

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

subgroup	index
#13114705	2
C₂×#11718050	2
C₂×C₈×#1055	2
#13112677	2
C₄×C₁₃×#107	2
#8282	13