Subgroups — #16807260

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 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	#13131171	C₂	1	2	yes	maximal
5	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic
6	#13131045	C₂	1	2	yes	maximal
7	C₄×#88106	C₂	1	1	yes	maximal, characteristic
8	#13123771	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₄	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	1	yes	characteristic
12	C₂²×#6092	C₂²	1	2	yes	—
13	#11718685	C₄	1	2	yes	—
14	#11718685	C₄	1	2	yes	—
15	Q₈×#1055	C₂²	1	2	yes	—
16	Q₈×#1055	C₂²	1	2	yes	—
17	#11718660	C₄	1	2	yes	—
18	#11718658	C₂²	1	2	yes	—
19	#11718585	C₄	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	#11718231	C₂²	1	2	yes	—
22	#11718231	C₂²	1	4	yes	—
23	C₄⋊#6092	C₂²	1	1	yes	characteristic
24	C₄⋊#6092	C₂²	1	2	yes	—
25	C₄⋊#6092	C₂²	1	1	yes	characteristic
26	C₄⋊#6092	C₂²	1	4	yes	—
27	C₄⋊#6092	C₂²	1	4	yes	—
28	C₄×#6092	C₂²	1	2	yes	—
29	C₄×#6092	C₂²	1	4	yes	—
30	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
31	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
32	C₁₃×#125	C₈	1	2	yes	—
33	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
34	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
35	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₁₃×#117	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×#6099	C₂×C₄	1	2	yes	—
40	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
41	C₂²×#1055	C₂³	1	1	yes	characteristic
42	#88106	—	2	2	no	—
43	#88106	Q₈	1	4	yes	complemented
44	C₂×#6092	—	2	4	no	—
45	C₂×#6092	C₂³	1	2	yes	—
46	C₂×#6092	C₂³	1	2	yes	—
47	C₂×#6092	C₂³	1	2	yes	—
48	C₂×#6092	C₂³	1	2	yes	—
49	C₄⋊#1055	C₂×C₄	1	4	yes	—
50	C₄⋊#1055	C₂×C₄	1	4	yes	—
51	C₄⋊#1055	C₂×C₄	1	2	yes	—
52	C₄⋊#1055	C₂×C₄	1	2	yes	—
53	#88043	C₂×C₄	1	4	yes	—
54	C₄×#1055	C₂×C₄	1	1	yes	characteristic
55	C₄×#1055	C₂×C₄	1	1	yes	characteristic
56	C₄×#1055	C₂³	1	4	yes	—
57	C₄×#1055	C₂³	1	1	yes	characteristic
58	C₄×#1055	C₂³	1	1	yes	characteristic
59	#33476	—	13	1	no	maximal, Sylow
60	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
61	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
66	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
67	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
68	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
69	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
70	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
71	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
72	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
77	#6099	—	2	4	no	—
78	C₂×#1055	—	2	2	no	—
79	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
80	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
81	C₂×#1055	C₂²×C₄	1	4	yes	—
82	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
83	C₂×#1055	C₂×Q₈	1	2	yes	—
84	C₂×#1055	#41	1	2	yes	—
85	C₂×#1055	C₂⁴	1	1	yes	characteristic
86	#6092	—	2	2	no	—
87	#6092	—	2	2	no	—
88	#6092	—	2	2	no	—
89	#6092	C₂×Q₈	1	4	yes	—
90	#2960	—	13	1	no	—
91	#2183	—	13	2	no	—
92	#2183	—	13	4	no	—
93	Q₈×C₁₆	—	13	2	no	—
94	#2151	—	13	2	no	—
95	C₂×C₄⋊C₁₆	—	13	1	no	—
96	#2108	—	13	2	no	—
97	C₄×M₃₂	—	13	1	no	—
98	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	Q₈×C₁₃	—	2	4	no	—
100	Q₈×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#131	1	2	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	#1055	—	2	1	no	—
114	#1055	—	2	1	no	—
115	#1055	—	2	4	no	—
116	#1055	C₄×Q₈	1	2	yes	—
117	#1055	C₂²×Q₈	1	1	yes	characteristic
118	#1055	C₂×#41	1	1	yes	characteristic
119	#533	#261	1	1	yes	characteristic
120	C₂×M₃₂	—	13	1	no	—
121	C₂×M₃₂	—	13	1	no	—
122	C₂²×C₁₆	—	13	2	no	—
123	#446	—	13	2	no	—
124	#446	—	13	2	no	—
125	C₈×Q₈	—	13	2	no	—
126	C₈×Q₈	—	13	2	no	—
127	#424	—	13	2	no	—
128	#423	—	13	1	no	—
129	#405	—	13	2	no	—
130	C₂×C₄×C₈	—	13	1	no	—
131	C₄⋊C₁₆	—	13	1	no	—
132	C₄⋊C₁₆	—	13	1	no	—
133	C₄⋊C₁₆	—	13	2	no	—
134	C₄⋊C₁₆	—	13	4	no	—
135	C₄⋊C₁₆	—	13	4	no	—
136	#346	—	13	2	no	—
137	#346	—	13	4	no	—
138	C₄×C₁₆	—	13	2	no	—
139	C₄×C₁₆	—	13	4	no	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
143	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	C₈×Q₈	1	2	yes	—
150	C₄×C₁₃	#504	1	2	yes	—
151	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
153	C₂×M₁₆	—	13	2	no	—
154	C₂²×C₈	—	13	1	no	—
155	C₂²×C₈	—	13	1	no	—
156	C₄⋊Q₈	#1057	1	1	yes	characteristic
157	C₄⋊Q₈	#1057	1	1	yes	characteristic
158	#125	#1057	1	2	yes	—
159	C₂²⋊Q₈	#1057	1	2	yes	—
160	C₂²⋊Q₈	#1057	1	2	yes	—
161	C₄×Q₈	C₂×#261	1	2	yes	—
162	C₄×Q₈	C₂×#261	1	2	yes	—
163	#117	C₂×#261	1	2	yes	—
164	C₂×C₄²	C₂×#261	1	1	yes	characteristic
165	M₃₂	—	13	4	no	—
166	M₃₂	—	26	2	no	—
167	C₂×C₁₆	—	13	2	no	—
168	C₂×C₁₆	—	13	2	no	—
169	C₂×C₁₆	—	13	2	no	—
170	C₂×C₁₆	—	13	2	no	—
171	C₂×C₁₆	—	26	4	no	—
172	C₄⋊C₈	—	13	2	no	—
173	C₄⋊C₈	—	13	2	no	—
174	C₄⋊C₈	—	13	4	no	—
175	C₄⋊C₈	—	13	4	no	—
176	#97	—	13	4	no	—
177	C₄×C₈	—	13	1	no	—
178	C₄×C₈	—	13	4	no	—
179	C₄×C₈	—	13	1	no	—
180	C₄×C₈	—	13	1	no	—
181	C₄×C₈	—	13	1	no	—
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
185	C₂×C₁₃	#3407	1	1	yes	characteristic
186	C₂×Q₈	C₂×#1057	1	2	yes	—
187	C₂×Q₈	C₂×#1057	1	2	yes	—
188	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
189	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
190	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
191	M₁₆	—	26	4	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	13	2	no	—
194	C₂×C₈	—	13	1	no	—
195	C₂×C₈	—	13	1	no	—
196	C₂×C₈	—	13	1	no	—
197	C₂×C₈	—	13	1	no	—
198	C₂×C₈	—	13	4	no	—
199	C₂×C₈	—	26	2	no	—
200	C₄⋊C₄	C₂×#1057	1	2	yes	—
201	C₄⋊C₄	C₂×#1057	1	2	yes	—
202	C₄⋊C₄	C₂×#1057	1	4	yes	—
203	C₄⋊C₄	C₂×#1057	1	4	yes	—
204	C₄⋊C₄	C₂×#1057	1	2	yes	—
205	C₄⋊C₄	C₂×#1057	1	2	yes	—
206	C₂²⋊C₄	C₂×#1057	1	4	yes	—
207	C₄²	C₂×#1057	1	1	yes	characteristic
208	C₄²	C₂×#1057	1	1	yes	characteristic
209	C₄²	C₂²×#261	1	1	yes	characteristic
210	C₄²	C₂²×#261	1	1	yes	characteristic
211	C₄²	C₂²×#261	1	4	yes	—
212	C₁₆	—	13	4	no	—
213	C₁₆	—	26	2	no	—
214	C₁₆	—	26	2	no	—
215	C₁₆	—	26	2	no	—
216	C₁₃	#33476	1	1	yes	characteristic, complemented, Sylow
217	C₂³	C₂²×#1057	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	#88240	1	2	yes	—
223	C₂×C₄	Q₈×#261	1	2	yes	—
224	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
225	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
226	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
227	C₂×C₄	C₂²×#1057	1	4	yes	—
228	C₂×C₄	C₂²×#1057	1	4	yes	—
229	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
230	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
231	C₂×C₄	C₂³×#261	1	1	yes	characteristic
232	C₈	—	13	1	no	—
233	C₈	—	13	1	no	—
234	C₈	—	13	2	no	—
235	C₈	—	26	1	no	—
236	C₈	—	26	1	no	—
237	C₈	—	26	4	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	Q₈×#1057	1	2	yes	—
241	C₂²	C₂³×#1057	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	Q₈×#1057	1	2	yes	—
248	C₄	#11719162	1	2	yes	—
249	C₄	C₂×#88240	1	1	yes	characteristic
250	C₄	C₂×Q₈×#261	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×Q₈×#1057	1	1	yes	characteristic
253	C₂	C₂×#11719162	1	1	yes	characteristic
254	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	#33476
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
43. #88106	Q₈	semidirect
216. C₁₃	#33476	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
#13131171	2
C₂×C₄⋊#6092	2
#13131045	2
C₄×#88106	2
#13123771	2
#33476	13