Subgroups — #16797702

Order 3328.

990 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1057	C₂	1	2	yes	maximal
2	C₄⋊C₄×#1057	C₂	1	6	yes	maximal
3	C₄×C₄⋊#1057	C₂	1	3	yes	maximal
4	#13130550	C₂	1	3	yes	maximal
5	C₄×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
9	Q₈×#1057	C₄	1	16	yes	complemented
10	C₂×C₄⋊#1057	C₂²	1	6	yes	—
11	C₂×C₄×#1057	C₂²	1	6	yes	—
12	C₂×C₄×#1057	C₂²	1	6	yes	—
13	C₄⋊C₄×Dic₁₃	C₄	1	3	yes	—
14	C₄⋊C₄×Dic₁₃	C₂²	1	3	yes	—
15	C₄×C₄⋊Dic₁₃	C₄	1	3	yes	—
16	C₄²×Dic₁₃	C₂²	1	3	yes	—
17	#11718290	C₂²	1	1	yes	characteristic
18	#11718290	C₂²	1	3	yes	—
19	#11718290	C₂²	1	6	yes	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×#1057	C₂³	1	2	yes	—
22	C₂²×#1057	C₂³	1	6	yes	—
23	C₄×Q₈×C₁₃	C₈	1	8	yes	complemented
24	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
26	Q₈×Dic₁₃	C₈	1	8	yes	complemented
27	Q₈×Dic₁₃	C₂×C₄	1	8	yes	—
28	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
29	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
36	C₄⋊#1057	C₂×C₄	1	24	yes	—
37	C₄×#1057	—	4	6	no	—
38	C₄×#1057	C₂×C₄	1	24	yes	—
39	#88071	C₂×C₄	1	6	yes	—
40	#20569	—	13	1	no	maximal, Sylow
41	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
42	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
43	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
49	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×#1057	—	2	6	no	—
52	C₂×#1057	—	2	2	no	—
53	C₂×#1057	—	2	2	no	—
54	C₂×#1057	—	2	6	no	—
55	C₂×#1057	C₂²×C₄	1	4	yes	—
56	C₂×#1057	C₂²×C₄	1	12	yes	—
57	C₁₃×C₄⋊C₄	C₂×C₈	1	12	yes	—
58	C₄²×C₁₃	C₂×C₈	1	12	yes	—
59	C₄⋊Dic₁₃	C₄²	1	12	yes	—
60	C₄⋊Dic₁₃	C₂×C₈	1	12	yes	—
61	C₄×Dic₁₃	—	2	3	no	—
62	C₄×Dic₁₃	—	2	2	no	—
63	C₄×Dic₁₃	—	2	3	no	—
64	C₄×Dic₁₃	—	4	3	no	—
65	C₄×Dic₁₃	—	4	3	no	—
66	C₄×Dic₁₃	C₂×C₈	1	12	yes	—
67	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
68	C₂×C₈×Q₈	—	13	2	no	—
69	C₄²×Q₈	—	13	1	no	—
70	C₈×C₄⋊C₄	—	13	6	no	—
71	C₄×C₄⋊C₈	—	13	3	no	—
72	#1744	—	13	3	no	—
73	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
74	Q₈×C₁₃	C₄×C₈	1	16	yes	complemented
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	3	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	3	no	—
79	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
84	C₂×Dic₁₃	—	2	1	no	—
85	C₂×Dic₁₃	—	2	3	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	3	no	—
88	C₂×Dic₁₃	—	2	3	no	—
89	C₂×Dic₁₃	—	2	3	no	—
90	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
91	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
92	C₂×Dic₁₃	C₂²×C₈	1	6	yes	—
93	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
94	C₂×Dic₁₃	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂×Dic₁₃	2₋¹⁺⁴	1	1	yes	characteristic
96	C₂×Dic₁₃	2₋¹⁺⁴	1	2	yes	—
97	#1057	—	2	4	no	—
98	#1057	—	2	12	no	—
99	C₂×C₄×Q₈	—	13	1	no	—
100	C₂×C₄×Q₈	—	13	1	no	—
101	C₂×C₄×Q₈	#261	1	1	yes	characteristic
102	C₈×Q₈	—	13	16	no	—
103	C₂×C₄⋊C₈	—	13	6	no	—
104	C₂×C₄×C₈	—	13	6	no	—
105	C₂×C₄×C₈	—	13	6	no	—
106	C₄×C₄⋊C₄	—	13	3	no	—
107	C₄×C₄⋊C₄	—	13	3	no	—
108	C₄×C₄⋊C₄	—	13	3	no	—
109	C₄³	—	13	3	no	—
110	#336	—	13	1	no	—
111	#336	—	13	3	no	—
112	#336	—	13	6	no	—
113	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
114	C₂²×C₁₃	#519	1	1	yes	characteristic
115	C₂²×C₁₃	#520	1	1	yes	characteristic
116	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
117	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
118	C₂²×C₁₃	#568	1	1	yes	characteristic
119	C₂²×C₁₃	#568	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	6	no	—
122	C₄×C₁₃	C₂×C₄×C₈	1	12	yes	—
123	Dic₁₃	—	2	6	no	—
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	—	2	6	no	—
126	Dic₁₃	#519	1	2	yes	—
127	Dic₁₃	#520	1	2	yes	—
128	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
129	C₂²×C₈	—	13	2	no	—
130	C₂²×C₈	—	13	6	no	—
131	C₄×Q₈	—	13	8	no	—
132	C₄×Q₈	—	13	8	no	—
133	C₄×Q₈	#1057	1	8	yes	complemented
134	C₂×C₄⋊C₄	—	13	3	no	—
135	C₂×C₄⋊C₄	—	13	3	no	—
136	C₂×C₄⋊C₄	C₂×#261	1	3	yes	—
137	C₂×C₄²	—	13	3	no	—
138	C₂×C₄²	—	13	3	no	—
139	C₂×C₄²	—	13	1	no	—
140	C₂×C₄²	—	13	3	no	—
141	C₂×C₄²	—	13	3	no	—
142	C₂×C₄²	—	13	3	no	—
143	C₂×C₄²	C₂×#261	1	3	yes	—
144	C₄⋊C₈	—	13	24	no	—
145	C₄×C₈	—	13	24	no	—
146	C₄×C₈	—	52	6	no	—
147	#95	—	13	6	no	—
148	C₂×C₁₃	#2276	1	1	yes	characteristic
149	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
150	C₂×C₁₃	#2875	1	1	yes	characteristic
151	C₂×C₁₃	#2982	1	1	yes	characteristic
152	C₂×C₁₃	#2982	1	1	yes	characteristic
153	C₂×C₁₃	#2988	1	1	yes	characteristic
154	C₂×C₁₃	#2988	1	1	yes	characteristic
155	C₂×Q₈	D₁₃⋊C₈	1	4	yes	—
156	C₂×Q₈	C₄×#261	1	4	yes	—
157	C₂×Q₈	C₂×#1057	1	4	yes	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	13	3	no	—
160	C₂²×C₄	—	13	3	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	C₂²×#261	1	3	yes	—
163	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
164	C₂²×C₄	C₂²×#261	1	3	yes	—
165	C₂×C₈	—	13	4	no	—
166	C₂×C₈	—	13	12	no	—
167	C₂×C₈	—	26	6	no	—
168	C₂×C₈	—	26	2	no	—
169	C₂×C₈	—	26	6	no	—
170	C₂×C₈	—	26	2	no	—
171	C₄⋊C₄	—	13	12	no	—
172	C₄⋊C₄	—	13	12	no	—
173	C₄⋊C₄	C₂×#1057	1	12	yes	—
174	C₄²	—	13	12	no	—
175	C₄²	—	13	12	no	—
176	C₄²	—	26	3	no	—
177	C₄²	—	26	2	no	—
178	C₄²	—	26	3	no	—
179	C₄²	—	52	3	no	—
180	C₄²	—	52	3	no	—
181	C₄²	C₂×#1057	1	12	yes	—
182	C₁₃	#20569	1	1	yes	characteristic, complemented, Sylow
183	C₂³	C₂³×#261	1	1	yes	characteristic
184	Q₈	C₄×#1057	1	16	yes	complemented
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	3	no	—
188	C₂×C₄	—	2	3	no	—
189	C₂×C₄	—	13	2	no	—
190	C₂×C₄	—	13	2	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	2	no	—
194	C₂×C₄	—	13	6	no	—
195	C₂×C₄	—	13	6	no	—
196	C₂×C₄	—	26	3	no	—
197	C₂×C₄	—	26	3	no	—
198	C₂×C₄	—	26	1	no	—
199	C₂×C₄	—	26	1	no	—
200	C₂×C₄	—	26	3	no	—
201	C₂×C₄	—	26	3	no	—
202	C₂×C₄	C₂×D₁₃⋊C₈	1	6	yes	—
203	C₂×C₄	C₂×C₄×#261	1	6	yes	—
204	C₂×C₄	C₂²×#1057	1	2	yes	—
205	C₂×C₄	C₂²×#1057	1	6	yes	—
206	C₂×C₄	C₂²×#1057	1	6	yes	—
207	C₈	—	26	4	no	—
208	C₈	—	26	12	no	—
209	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
210	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
211	C₂²	#11719625	1	1	yes	characteristic
212	C₂²	#11719627	1	1	yes	characteristic
213	C₂²	#11719629	1	1	yes	characteristic
214	C₂²	#11719631	1	1	yes	characteristic
215	C₂²	C₂³×#1057	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	6	no	—
218	C₄	—	13	2	no	—
219	C₄	—	13	2	no	—
220	C₄	—	26	6	no	—
221	C₄	—	26	2	no	—
222	C₄	—	26	6	no	—
223	C₄	C₂×C₄×#1057	1	12	yes	—
224	C₂	#13131276	1	1	yes	characteristic
225	C₂	C₂²×C₄×#1057	1	1	yes	characteristic
226	C₂	#13131408	1	1	yes	characteristic
227	C₂	#13131729	1	1	yes	characteristic
228	C₂	#13131730	1	1	yes	characteristic
229	C₂	#13131751	1	1	yes	characteristic
230	C₂	#13131754	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₂×C₄×Q₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×Dic₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	#20569
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
9. Q₈×#1057	C₄	semidirect
23. C₄×Q₈×C₁₃	C₈	semidirect
26. Q₈×Dic₁₃	C₈	semidirect
74. Q₈×C₁₃	C₄×C₈	semidirect
133. C₄×Q₈	#1057	semidirect
182. C₁₃	#20569	semidirect
184. Q₈	C₄×#1057	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#1057	2
C₄⋊C₄×#1057	2
C₄×C₄⋊#1057	2
#13130550	2
C₄×Q₈×Dic₁₃	2
#20569	13