Subgroups — #16789953

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6092	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#6092	C₂	1	2	yes	maximal
3	#13129874	C₂	1	2	yes	maximal, complemented
4	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
6	C₂²×#6099	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	4	yes	—
9	#11718658	—	2	4	no	—
10	C₂×#88043	C₄	1	1	yes	characteristic
11	C₂×#88043	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂²⋊#6092	—	2	4	no	—
15	C₄⋊#6092	—	2	4	no	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₁₃×#117	—	2	2	no	—
18	C₁₃×#117	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#6099	—	2	2	no	—
24	C₂×#6099	—	2	2	no	—
25	C₂×#6099	—	2	2	no	—
26	C₂²×#1055	—	2	2	no	—
27	C₂²×#1055	—	2	2	no	—
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	D₄	1	2	yes	—
32	C₂²×#1055	Q₈	1	2	yes	—
33	C₂²×#1055	C₂³	1	1	yes	characteristic
34	C₂×#6092	—	2	4	no	—
35	C₂×#6092	—	2	4	no	—
36	C₂×#6092	—	2	4	no	—
37	#88043	—	2	2	no	—
38	#88043	—	2	2	no	—
39	C₄×#1055	—	2	2	no	—
40	C₄×#1055	D₄	1	4	yes	—
41	#12784	—	13	1	no	maximal, Sylow
42	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₂²⋊C₄	—	2	4	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₁₃	M₁₆	1	2	yes	—
59	#6099	—	2	8	no	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	—	2	1	no	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	2	1	no	—
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	—	2	4	no	—
67	C₂×#1055	—	2	4	no	—
68	C₂×#1055	C₂²⋊C₄	1	2	yes	—
69	C₂×#1055	C₄⋊C₄	1	2	yes	—
70	C₂×#1055	M₁₆	1	2	yes	—
71	C₂×#1055	C₂×D₄	1	1	yes	characteristic
72	C₂×#1055	C₂×D₄	1	2	yes	—
73	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
74	C₂×#1055	#41	1	2	yes	—
75	#6092	—	4	4	no	—
76	C₂×#405	—	13	1	no	—
77	C₂×C₄⋊C₁₆	—	13	2	no	—
78	C₂×C₂²⋊C₁₆	—	13	2	no	—
79	#1374	—	13	2	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	—	2	1	no	—
82	C₂³×C₁₃	—	2	4	no	—
83	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
84	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
102	#1055	—	2	2	no	—
103	#1055	—	2	2	no	—
104	#1055	—	2	2	no	—
105	#1055	#121	1	2	yes	—
106	#1055	C₂²⋊Q₈	1	2	yes	—
107	C₂²×M₁₆	—	13	1	no	—
108	C₂³×C₈	—	13	1	no	—
109	C₂×#117	#261	1	1	yes	characteristic
110	C₂²×C₁₆	—	13	4	no	—
111	#405	—	26	4	no	—
112	C₂×#97	—	13	1	no	—
113	C₂×#97	—	13	1	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	C₂×C₄×C₈	—	13	1	no	—
116	C₄⋊C₁₆	—	26	4	no	—
117	C₂²⋊C₁₆	—	26	4	no	—
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	2	4	no	—
120	C₂²×C₁₃	—	4	2	no	—
121	C₂²×C₁₃	—	4	2	no	—
122	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₂²×C₁₃	#431	1	1	yes	characteristic
125	C₂²×C₁₃	#504	1	4	yes	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	—	4	2	no	—
129	C₄×C₁₃	#380	1	1	yes	characteristic
130	C₄×C₁₃	#407	1	1	yes	characteristic
131	C₄×C₁₃	#423	1	1	yes	characteristic
132	C₄×C₁₃	#432	1	1	yes	characteristic
133	C₂³×C₄	C₂×#261	1	1	yes	characteristic
134	C₂×M₁₆	—	26	2	no	—
135	C₂×M₁₆	—	26	2	no	—
136	C₂×M₁₆	—	26	2	no	—
137	C₂²×C₈	—	13	1	no	—
138	C₂²×C₈	—	13	2	no	—
139	C₂²×C₈	—	13	2	no	—
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	1	no	—
142	C₂²×C₈	—	13	1	no	—
143	C₂²×C₈	—	26	2	no	—
144	C₂²×C₈	—	26	2	no	—
145	#117	—	2	2	no	—
146	#117	—	2	2	no	—
147	C₂×C₄⋊C₄	#1057	1	2	yes	—
148	C₂×C₂²⋊C₄	#1057	1	2	yes	—
149	C₂×C₄²	C₂×#261	1	1	yes	characteristic
150	C₂×C₄²	C₂×#261	1	1	yes	characteristic
151	C₂×C₁₆	—	26	4	no	—
152	C₂×C₁₆	—	26	4	no	—
153	C₂×C₁₆	—	26	4	no	—
154	#97	—	26	2	no	—
155	#97	—	26	2	no	—
156	C₄×C₈	—	13	4	no	—
157	C₄×C₈	—	26	2	no	—
158	C₂×C₁₃	—	4	2	no	—
159	C₂×C₁₃	#1797	1	1	yes	characteristic
160	C₂×C₁₃	#2113	1	2	yes	—
161	C₂×C₁₃	#2151	1	2	yes	—
162	C₂×C₁₃	#2164	1	2	yes	—
163	C₂⁴	C₂×#1057	1	1	yes	characteristic
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	#6118	1	2	yes	—
169	C₂²×C₄	C₄⋊#261	1	2	yes	—
170	C₂²×C₄	C₂×#1057	1	2	yes	—
171	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
172	C₂²×C₄	C₂×#1057	1	2	yes	—
173	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
174	M₁₆	—	26	8	no	—
175	C₂×C₈	—	13	1	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	2	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	13	2	no	—
181	C₂×C₈	—	13	2	no	—
182	C₂×C₈	—	26	4	no	—
183	C₂×C₈	—	26	4	no	—
184	C₂×C₈	—	26	1	no	—
185	C₂×C₈	—	26	1	no	—
186	C₂×C₈	—	26	1	no	—
187	C₂×C₈	—	26	1	no	—
188	C₂×C₈	—	26	1	no	—
189	C₂×C₈	—	26	1	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	#6122	1	2	yes	—
195	C₄²	C₂²⋊#261	1	2	yes	—
196	C₁₆	—	52	4	no	—
197	C₁₃	#12784	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	4	no	—
201	C₂³	C₄⋊#1057	1	2	yes	—
202	C₂³	C₂²×#1057	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	C₄⋊#1057	1	2	yes	—
213	C₂×C₄	C₂²⋊#1057	1	4	yes	—
214	C₂×C₄	C₂×#6118	1	1	yes	characteristic
215	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
217	C₂×C₄	#88237	1	1	yes	characteristic
218	C₂×C₄	C₂×#6122	1	1	yes	characteristic
219	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
220	C₈	—	13	2	no	—
221	C₈	—	13	2	no	—
222	C₈	—	26	2	no	—
223	C₈	—	26	2	no	—
224	C₈	—	26	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
230	C₂²	#11719119	1	1	yes	characteristic
231	C₂²	#11719162	1	4	yes	—
232	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	#11719127	1	1	yes	characteristic
237	C₄	#11719129	1	1	yes	characteristic
238	C₄	#11719130	1	1	yes	characteristic
239	C₄	#11719136	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	#13130690	1	1	yes	characteristic
242	C₂	#13131066	1	2	yes	—
243	C₂	#13131171	1	2	yes	—
244	C₂	#13131201	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator, socle	C₂³×C₁₃	C₂²×C₈
Fitting	C₂×C₁₃×#117	C₄
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#12784
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#6092	C₂	semidirect
3. #13129874	C₂	semidirect
197. C₁₃	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
C₂×C₂²⋊#6092	2
#13129874	2
C₂×#11718658	2
#12784	13