Subgroups — #15781174

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718360	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718088	C₂	1	4	yes	maximal
3	C₄×#88052	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₂	—	2	1	no	—
6	C₂²×Dic₅₂	C₂²	1	2	yes	—
7	C₂×#88123	—	2	1	no	—
8	C₂×C₁₃×#108	—	2	1	no	—
9	C₂×C₁₃×#108	C₂²	1	2	yes	—
10	C₁₃×#423	—	2	1	no	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88052	C₂²	1	2	yes	—
13	#11718360	—	2	1	no	—
14	#11718360	—	2	1	no	—
15	#11718360	C₄	1	2	yes	—
16	#11718360	C₄	1	2	yes	—
17	#11718088	—	2	8	no	—
18	#11718053	—	2	1	no	—
19	C₄×#6090	—	2	1	no	—
20	C₂²×Dic₂₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	—	4	2	no	—
26	C₂×Dic₅₂	—	2	4	no	—
27	C₂×Dic₅₂	—	2	4	no	—
28	C₂×Dic₅₂	—	2	2	no	—
29	C₂×Dic₅₂	—	2	2	no	—
30	C₂×Dic₅₂	—	2	2	no	—
31	C₂×Dic₅₂	—	4	2	no	—
32	C₂×Dic₅₂	—	4	2	no	—
33	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
34	#88123	—	2	2	no	—
35	#88123	—	2	2	no	—
36	#88123	—	4	2	no	—
37	C₁₃×#108	—	2	4	no	—
38	C₁₃×#108	—	4	2	no	—
39	C₁₃×C₄⋊C₈	—	4	2	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	#88052	—	2	2	no	—
45	#88052	D₄	1	4	yes	—
46	C₂×#6090	—	2	2	no	—
47	#13212	—	13	1	no	maximal, Sylow
48	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	4	2	no	—
51	C₂×Dic₂₆	—	2	4	no	—
52	C₂×Dic₂₆	—	4	2	no	—
53	C₂×Dic₂₆	—	4	2	no	—
54	C₂×Dic₂₆	—	4	2	no	—
55	C₂×Dic₂₆	—	4	2	no	—
56	C₂×Dic₂₆	—	4	2	no	—
57	C₁₃×M₁₆	—	4	2	no	—
58	C₁₃×M₁₆	—	4	2	no	—
59	C₂×C₈×C₁₃	—	4	2	no	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	D₈	1	2	yes	—
63	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	#41	1	2	yes	—
68	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₁₃	D₈	1	2	yes	—
71	C₄⋊Dic₁₃	—	4	4	no	—
72	C₄⋊Dic₁₃	—	4	4	no	—
73	Dic₅₂	—	2	8	no	—
74	Dic₅₂	—	4	4	no	—
75	Dic₅₂	—	4	4	no	—
76	Dic₅₂	—	4	4	no	—
77	#6090	—	2	4	no	—
78	C₂×#494	—	13	1	no	—
79	C₂×#362	—	13	4	no	—
80	C₄×M₃₂	—	13	1	no	—
81	#1848	D₁₃	1	1	yes	characteristic
82	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	—	2	4	no	—
84	C₂×C₄×C₁₃	#102	1	2	yes	—
85	C₂×C₄×C₁₃	#102	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×Dic₁₃	—	4	4	no	—
93	C₂×Dic₁₃	—	8	2	no	—
94	C₂×Dic₁₃	—	8	2	no	—
95	Dic₂₆	—	4	4	no	—
96	Dic₂₆	—	4	4	no	—
97	Dic₂₆	—	8	2	no	—
98	Dic₂₆	—	8	2	no	—
99	C₈×C₁₃	—	8	2	no	—
100	C₈×C₁₃	#102	1	2	yes	—
101	C₈×C₁₃	#102	1	2	yes	—
102	C₈×C₁₃	C₄×D₄	1	2	yes	—
103	C₈×C₁₃	#121	1	2	yes	—
104	C₂²×Q₁₆	—	13	2	no	—
105	C₂²×Q₁₆	—	26	1	no	—
106	C₂×C₄⋊Q₈	—	26	1	no	—
107	C₂×M₃₂	—	13	2	no	—
108	#494	—	13	2	no	—
109	#494	—	13	2	no	—
110	#494	—	26	1	no	—
111	#494	—	26	1	no	—
112	C₂×#108	—	2	1	no	—
113	C₂×#108	C₂×D₁₃	1	2	yes	—
114	#423	—	2	1	no	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	#362	—	26	8	no	—
117	#346	—	26	1	no	—
118	C₄×C₁₆	—	26	1	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
121	C₂²×C₁₃	#486	1	1	yes	characteristic
122	C₂²×C₁₃	#492	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	#362	1	4	yes	complemented
125	C₄×C₁₃	#390	1	1	yes	characteristic
126	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
127	C₄×C₁₃	#486	1	1	yes	characteristic
128	C₄×C₁₃	#493	1	1	yes	characteristic
129	Dic₁₃	—	8	4	no	—
130	C₂²×Q₈	—	26	2	no	—
131	C₂×Q₁₆	—	13	4	no	—
132	C₂×Q₁₆	—	26	2	no	—
133	C₂×Q₁₆	—	26	2	no	—
134	C₂×Q₁₆	—	26	2	no	—
135	C₂×Q₁₆	—	26	4	no	—
136	C₂×Q₁₆	—	26	4	no	—
137	C₂×Q₁₆	—	52	2	no	—
138	C₂×Q₁₆	—	52	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
142	C₄⋊Q₈	—	26	2	no	—
143	C₄⋊Q₈	—	26	2	no	—
144	C₄⋊Q₈	—	52	2	no	—
145	C₂×C₄⋊C₄	—	52	2	no	—
146	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
147	M₃₂	—	13	4	no	—
148	M₃₂	—	26	2	no	—
149	C₂×C₁₆	—	26	2	no	—
150	#108	—	2	4	no	—
151	#108	—	4	2	no	—
152	C₄⋊C₈	—	4	2	no	—
153	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
154	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
155	C₄×C₈	D₅₂	1	1	yes	characteristic
156	C₄×C₈	D₅₂	1	1	yes	characteristic
157	C₂×C₁₃	—	2	1	no	—
158	C₂×C₁₃	—	2	1	no	—
159	C₂×C₁₃	#1957	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#362	1	2	yes	—
161	C₂×Q₈	—	26	4	no	—
162	C₂×Q₈	—	52	2	no	—
163	C₂×Q₈	—	52	2	no	—
164	C₂×Q₈	—	52	2	no	—
165	C₂×Q₈	—	52	2	no	—
166	C₂×Q₈	—	52	2	no	—
167	C₂²×C₄	—	52	2	no	—
168	C₂²×C₄	#6106	1	2	yes	—
169	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
170	Q₁₆	—	26	8	no	—
171	Q₁₆	—	52	4	no	—
172	Q₁₆	—	52	4	no	—
173	Q₁₆	—	52	4	no	—
174	M₁₆	—	4	2	no	—
175	M₁₆	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	#6103	1	1	yes	characteristic
178	C₂×C₈	#6103	1	1	yes	characteristic
179	C₂×C₈	#6104	1	2	yes	—
180	C₂×C₈	#6106	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
183	C₂×C₈	D₄×D₁₃	1	2	yes	—
184	C₂×C₈	#6131	1	2	yes	—
185	C₄⋊C₄	—	52	4	no	—
186	C₄⋊C₄	—	52	4	no	—
187	C₄²	#6103	1	1	yes	characteristic
188	C₄²	#6103	1	1	yes	characteristic
189	C₄²	#6104	1	2	yes	—
190	C₁₆	—	26	4	no	—
191	C₁₃	#13212	1	1	yes	characteristic, complemented, Sylow
192	C₂³	C₂×#6106	1	1	yes	characteristic
193	Q₈	—	52	4	no	—
194	Q₈	—	52	4	no	—
195	Q₈	—	104	2	no	—
196	Q₈	—	104	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	52	4	no	—
199	C₂×C₄	—	104	2	no	—
200	C₂×C₄	—	104	2	no	—
201	C₂×C₄	#88049	1	2	yes	—
202	C₂×C₄	#88049	1	2	yes	—
203	C₂×C₄	C₂×#6103	1	1	yes	characteristic
204	C₂×C₄	C₂×#6104	1	1	yes	characteristic
205	C₂×C₄	C₂×#6104	1	1	yes	characteristic
206	C₂×C₄	#88194	1	1	yes	characteristic
207	C₂×C₄	C₂×#6106	1	1	yes	characteristic
208	C₂×C₄	#88201	1	1	yes	characteristic
209	C₈	—	8	2	no	—
210	C₈	#88049	1	2	yes	—
211	C₈	#88049	1	2	yes	—
212	C₈	#88147	1	2	yes	—
213	C₈	#88149	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#88049	1	1	yes	characteristic
216	C₂²	#11718746	1	1	yes	characteristic
217	C₂²	#11718749	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	104	4	no	—
220	C₄	#11718088	1	4	yes	complemented
221	C₄	C₂×#88049	1	1	yes	characteristic
222	C₄	#11718649	1	1	yes	characteristic
223	C₄	#11718734	1	1	yes	characteristic
224	C₄	#11718749	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#13115641	1	1	yes	characteristic
228	C₂	C₂×#11718088	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88049
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#13212
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
124. C₄×C₁₃	#362	semidirect
191. C₁₃	#13212	semidirect
220. C₄	#11718088	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718360	2
C₂×#11718088	2
C₄×#88052	2
C₁₃×#1848	2
#13212	13