Subgroups — #16807372

Order 3328.

452 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131251	C₂	1	2	yes	maximal, complemented
2	#13131251	C₂	1	4	yes	maximal, complemented
3	Q₈×#6092	C₂	1	2	yes	maximal, complemented
4	#13131201	C₂	1	4	yes	maximal
5	#13131197	C₂	1	1	yes	maximal, characteristic
6	#13131193	C₂	1	1	yes	maximal, characteristic
7	#13123771	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	C₄	1	1	yes	characteristic
9	#11718685	C₄	1	2	yes	—
10	#11718685	C₄	1	2	yes	—
11	Q₈×#1055	C₂²	1	2	yes	—
12	Q₈×#1055	C₂²	1	2	yes	—
13	#11718660	C₄	1	2	yes	—
14	#11718658	C₂²	1	2	yes	—
15	#11718585	C₄	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂²⋊#6092	C₂²	1	2	yes	—
18	C₂²⋊#6092	C₂²	1	2	yes	—
19	#11718231	C₂²	1	2	yes	—
20	#11718231	C₂²	1	4	yes	—
21	C₄⋊#6092	C₂²	1	2	yes	—
22	C₄⋊#6092	C₂²	1	2	yes	—
23	C₄⋊#6092	C₂²	1	4	yes	—
24	C₄⋊#6092	C₂²	1	4	yes	—
25	C₄×#6092	C₂²	1	1	yes	characteristic
26	C₄×#6092	C₂²	1	1	yes	characteristic
27	C₄×#6092	C₂²	1	4	yes	—
28	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
29	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
30	C₁₃×#125	C₈	1	2	yes	—
31	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
32	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
33	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₁₃×#117	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#6099	C₂×C₄	1	2	yes	—
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂²×#1055	C₂³	1	1	yes	characteristic
40	C₂×#6092	C₂³	1	2	yes	—
41	C₂×#6092	C₂³	1	2	yes	—
42	C₂×#6092	C₂³	1	4	yes	—
43	C₄⋊#1055	C₂×C₄	1	4	yes	—
44	C₄⋊#1055	C₂×C₄	1	4	yes	—
45	C₄⋊#1055	C₂×C₄	1	2	yes	—
46	C₄⋊#1055	C₂×C₄	1	2	yes	—
47	#88043	C₂×C₄	1	4	yes	—
48	C₄×#1055	C₂×C₄	1	1	yes	characteristic
49	C₄×#1055	C₂×C₄	1	1	yes	characteristic
50	C₄×#1055	C₂³	1	4	yes	—
51	C₄×#1055	C₂³	1	1	yes	characteristic
52	C₄×#1055	C₂³	1	1	yes	characteristic
53	#33499	—	13	1	no	maximal, Sylow
54	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
55	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
60	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
61	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
62	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
63	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
64	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
65	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
66	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
71	#6099	—	4	2	no	—
72	C₂×#1055	—	2	2	no	—
73	C₂×#1055	—	4	1	no	—
74	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
75	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
76	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
77	C₂×#1055	C₂²×C₄	1	4	yes	—
78	C₂×#1055	C₂⁴	1	1	yes	characteristic
79	#6092	—	2	2	no	—
80	#6092	—	2	2	no	—
81	#6092	—	2	4	no	—
82	#2960	—	13	1	no	—
83	#2183	—	13	2	no	—
84	#2183	—	13	4	no	—
85	Q₈×C₁₆	—	13	2	no	—
86	#2164	—	13	4	no	—
87	#2163	—	13	1	no	—
88	#2162	—	13	1	no	—
89	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
90	Q₈×C₁₃	—	2	4	no	—
91	Q₈×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	4	1	no	—
96	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
104	#1055	—	2	1	no	—
105	#1055	—	2	1	no	—
106	#1055	—	2	1	no	—
107	#1055	—	2	4	no	—
108	#1055	2₋¹⁺⁴	1	2	yes	—
109	#533	#261	1	1	yes	characteristic
110	#446	—	13	2	no	—
111	#446	—	13	2	no	—
112	C₈×Q₈	—	13	2	no	—
113	C₈×Q₈	—	13	2	no	—
114	#424	—	13	2	no	—
115	#423	—	13	1	no	—
116	#405	—	13	2	no	—
117	C₂×C₄×C₈	—	13	1	no	—
118	C₄⋊C₁₆	—	13	2	no	—
119	C₄⋊C₁₆	—	13	2	no	—
120	C₄⋊C₁₆	—	13	4	no	—
121	C₄⋊C₁₆	—	13	4	no	—
122	C₂²⋊C₁₆	—	13	2	no	—
123	C₂²⋊C₁₆	—	13	2	no	—
124	#346	—	13	2	no	—
125	#346	—	13	4	no	—
126	C₄×C₁₆	—	13	1	no	—
127	C₄×C₁₆	—	13	1	no	—
128	C₄×C₁₆	—	13	4	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	#504	1	2	yes	—
140	C₄×C₁₃	#519	1	1	yes	characteristic
141	C₄×C₁₃	#568	1	1	yes	characteristic
142	C₂×M₁₆	—	13	2	no	—
143	C₂²×C₈	—	13	1	no	—
144	C₂²×C₈	—	13	1	no	—
145	C₄⋊Q₈	#1057	1	1	yes	characteristic
146	C₄⋊Q₈	#1057	1	1	yes	characteristic
147	#125	#1057	1	2	yes	—
148	C₂²⋊Q₈	#1057	1	2	yes	—
149	C₂²⋊Q₈	#1057	1	2	yes	—
150	C₄×Q₈	C₂×#261	1	2	yes	—
151	C₄×Q₈	C₂×#261	1	2	yes	—
152	#117	C₂×#261	1	2	yes	—
153	C₂×C₄²	C₂×#261	1	1	yes	characteristic
154	C₂×C₁₆	—	13	2	no	—
155	C₂×C₁₆	—	13	2	no	—
156	C₂×C₁₆	—	13	4	no	—
157	C₄⋊C₈	—	13	2	no	—
158	C₄⋊C₈	—	13	2	no	—
159	C₄⋊C₈	—	13	4	no	—
160	C₄⋊C₈	—	13	4	no	—
161	#97	—	13	4	no	—
162	C₄×C₈	—	13	1	no	—
163	C₄×C₈	—	13	4	no	—
164	C₄×C₈	—	13	1	no	—
165	C₄×C₈	—	13	1	no	—
166	C₄×C₈	—	13	1	no	—
167	C₂×C₁₃	—	4	1	no	—
168	C₂×C₁₃	#2982	1	1	yes	characteristic
169	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
170	C₂×C₁₃	#3407	1	1	yes	characteristic
171	C₂×Q₈	C₂×#1057	1	2	yes	—
172	C₂×Q₈	C₂×#1057	1	2	yes	—
173	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
174	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
175	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
176	M₁₆	—	52	2	no	—
177	C₂×C₈	—	13	1	no	—
178	C₂×C₈	—	13	1	no	—
179	C₂×C₈	—	13	1	no	—
180	C₂×C₈	—	13	1	no	—
181	C₂×C₈	—	13	4	no	—
182	C₂×C₈	—	26	2	no	—
183	C₂×C₈	—	52	1	no	—
184	C₄⋊C₄	C₂×#1057	1	2	yes	—
185	C₄⋊C₄	C₂×#1057	1	2	yes	—
186	C₄⋊C₄	C₂×#1057	1	4	yes	—
187	C₄⋊C₄	C₂×#1057	1	4	yes	—
188	C₄⋊C₄	C₂×#1057	1	2	yes	—
189	C₄⋊C₄	C₂×#1057	1	2	yes	—
190	C₂²⋊C₄	C₂×#1057	1	4	yes	—
191	C₄²	C₂×#1057	1	1	yes	characteristic
192	C₄²	C₂×#1057	1	1	yes	characteristic
193	C₄²	C₂²×#261	1	1	yes	characteristic
194	C₄²	C₂²×#261	1	1	yes	characteristic
195	C₄²	C₂²×#261	1	4	yes	—
196	C₁₆	—	26	2	no	—
197	C₁₆	—	26	2	no	—
198	C₁₆	—	26	4	no	—
199	C₁₃	#33499	1	1	yes	characteristic, complemented, Sylow
200	C₂³	C₂²×#1057	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
208	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
209	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
210	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
211	C₂×C₄	C₂²×#1057	1	4	yes	—
212	C₂×C₄	C₂²×#1057	1	4	yes	—
213	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
214	C₂×C₄	C₂³×#261	1	1	yes	characteristic
215	C₈	—	13	2	no	—
216	C₈	—	26	1	no	—
217	C₈	—	26	1	no	—
218	C₈	—	26	4	no	—
219	C₈	—	26	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	C₂³×#1057	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	4	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	4	no	—
229	C₄	—	2	1	no	—
230	C₄	#11719162	1	2	yes	—
231	C₄	#11719625	1	1	yes	characteristic
232	C₄	#11719627	1	1	yes	characteristic
233	C₂	—	4	1	no	—
234	C₂	#13131729	1	1	yes	characteristic
235	C₂	C₂×#11719162	1	1	yes	characteristic
236	C₂	#13131803	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#533	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#533	#261
hypocenter	C₁₃	#33499
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
2. #13131251	C₂	semidirect
3. Q₈×#6092	C₂	semidirect
199. C₁₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
Q₈×#6092	2
#13131201	2
#13131197	2
#13131193	2
#13123771	2
#33499	13