Subgroups — #16015516

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718363	C₂	1	4	yes	maximal
2	#13123234	C₂	1	1	yes	maximal, characteristic
3	#13123199	C₂	1	1	yes	maximal, characteristic
4	#13115004	C₂	1	4	yes	maximal
5	#13114996	C₂	1	2	yes	maximal
6	#13114993	C₂	1	2	yes	maximal
7	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
8	C₂×#88124	C₂²	1	2	yes	—
9	C₂×#88123	—	2	2	no	—
10	C₂×#88123	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	C₂²	1	2	yes	—
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₄³×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88058	C₂²	1	8	yes	—
16	C₂×#88057	C₂²	1	8	yes	—
17	#11718597	C₂²	1	2	yes	—
18	#11718596	—	2	2	no	—
19	#11718596	C₂²	1	1	yes	characteristic
20	#11718596	C₂²	1	2	yes	—
21	#11718596	C₂²	1	1	yes	characteristic
22	#11718596	C₂²	1	2	yes	—
23	#11718589	—	2	2	no	—
24	#11718589	—	2	2	no	—
25	#11718363	—	2	16	no	—
26	C₂²×Dic₂₆	—	2	2	no	—
27	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
28	C₂×C₄²×C₁₃	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃	D₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	—	2	4	no	—
34	C₂×C₄⋊Dic₁₃	—	2	4	no	—
35	C₂×C₄⋊Dic₁₃	—	2	2	no	—
36	C₂×C₄⋊Dic₁₃	—	2	2	no	—
37	C₂×C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
41	C₂×#6101	—	2	2	no	—
42	C₂×#6101	—	2	2	no	—
43	#88124	—	2	8	no	—
44	#88123	—	2	8	no	—
45	#88123	—	4	8	no	—
46	C₁₃×#97	D₄	1	8	yes	—
47	C₁₃×#97	Q₈	1	8	yes	complemented
48	C₄×C₈×C₁₃	Q₈	1	8	yes	complemented
49	#88071	—	2	4	no	—
50	#88058	—	2	16	no	—
51	#88057	—	2	16	no	—
52	#33316	—	13	1	no	maximal, Sylow
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	2	2	no	—
59	C₂²×Dic₁₃	—	2	2	no	—
60	C₂²×Dic₁₃	—	2	4	no	—
61	C₂×Dic₂₆	—	4	8	no	—
62	C₂×Dic₂₆	—	4	4	no	—
63	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
64	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
65	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
66	C₄²×C₁₃	—	2	4	no	—
67	C₄²×C₁₃	—	2	4	no	—
68	C₄²×C₁₃	C₂×D₄	1	4	yes	—
69	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
70	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
71	C₄⋊Dic₁₃	—	2	8	no	—
72	C₄⋊Dic₁₃	—	2	4	no	—
73	C₄⋊Dic₁₃	—	2	4	no	—
74	C₄⋊Dic₁₃	—	4	8	no	—
75	C₄⋊Dic₁₃	—	4	4	no	—
76	C₄⋊Dic₁₃	—	4	8	no	—
77	C₄⋊Dic₁₃	—	4	4	no	—
78	C₄⋊Dic₁₃	—	4	4	no	—
79	#6101	—	4	4	no	—
80	#6101	—	4	4	no	—
81	C₂×#501	—	13	4	no	—
82	#2868	—	13	1	no	—
83	#2859	—	13	1	no	—
84	#1847	—	13	4	no	—
85	#1845	—	13	2	no	—
86	#1844	—	13	2	no	—
87	C₄×#97	D₁₃	1	1	yes	characteristic
88	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
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	#127	1	2	yes	—
94	C₂×C₄×C₁₃	#127	1	2	yes	—
95	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
96	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×C₄×C₁₃	#137	1	4	yes	—
99	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
102	C₂×Dic₁₃	—	4	2	no	—
103	C₂×Dic₁₃	—	4	2	no	—
104	C₂×Dic₁₃	—	4	4	no	—
105	C₂×Dic₁₃	—	4	8	no	—
106	C₂×Dic₁₃	—	4	4	no	—
107	C₂×Dic₁₃	—	4	4	no	—
108	Dic₂₆	—	8	8	no	—
109	C₈×C₁₃	C₄⋊Q₈	1	16	yes	complemented
110	C₂×C₄⋊Q₈	—	13	2	no	—
111	C₂×C₄⋊Q₈	—	26	2	no	—
112	C₂×#125	—	13	2	no	—
113	#501	—	26	16	no	—
114	C₂×#107	—	13	8	no	—
115	C₂×#106	—	13	8	no	—
116	C₂×#97	C₂×D₁₃	1	2	yes	—
117	C₂×#97	C₂×D₁₃	1	2	yes	—
118	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
119	#389	—	26	2	no	—
120	#389	—	26	2	no	—
121	#384	—	13	1	no	—
122	#384	—	13	1	no	—
123	#384	—	13	2	no	—
124	#384	—	13	2	no	—
125	#384	—	26	2	no	—
126	#382	—	13	2	no	—
127	C₄³	C₂×D₁₃	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
132	C₂²×C₁₃	C₂×#137	1	2	yes	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	4	no	—
136	C₄×C₁₃	#501	1	8	yes	complemented
137	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
138	Dic₁₃	—	8	4	no	—
139	Dic₁₃	—	8	2	no	—
140	Dic₁₃	—	8	2	no	—
141	C₂²×Q₈	—	26	2	no	—
142	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
143	C₄⋊Q₈	—	26	8	no	—
144	C₄⋊Q₈	—	52	8	no	—
145	#125	—	26	8	no	—
146	C₂×C₄⋊C₄	—	13	2	no	—
147	C₂×C₄⋊C₄	—	13	2	no	—
148	C₂×C₄⋊C₄	—	13	4	no	—
149	C₂×C₄⋊C₄	—	26	2	no	—
150	C₂×C₄⋊C₄	—	26	2	no	—
151	C₂×C₄⋊C₄	—	26	2	no	—
152	C₂×C₄⋊C₄	—	26	4	no	—
153	C₂×C₄⋊C₄	—	26	4	no	—
154	C₂×C₄⋊C₄	—	26	2	no	—
155	C₂×C₄⋊C₄	—	26	2	no	—
156	C₂×C₄²	D₅₂	1	2	yes	—
157	C₂×C₄²	D₅₂	1	2	yes	—
158	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
159	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
161	#107	—	26	16	no	—
162	#106	—	26	16	no	—
163	#97	Dic₂₆	1	8	yes	complemented
164	#97	D₅₂	1	8	yes	—
165	C₄×C₈	Dic₂₆	1	8	yes	complemented
166	#95	—	26	4	no	—
167	C₂×C₁₃	#2868	1	1	yes	characteristic
168	C₂×C₁₃	#3151	1	1	yes	characteristic
169	C₂×C₁₃	#3152	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#501	1	2	yes	—
171	C₂×C₁₃	#3165	1	2	yes	—
172	C₂×Q₈	—	52	8	no	—
173	C₂×Q₈	—	52	4	no	—
174	C₂²×C₄	—	26	2	no	—
175	C₂²×C₄	—	26	2	no	—
176	C₂²×C₄	—	26	4	no	—
177	C₂²×C₄	C₂×D₅₂	1	2	yes	—
178	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₅₂	1	2	yes	—
181	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
183	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
184	C₂×C₈	C₂×D₅₂	1	8	yes	—
185	C₄⋊C₄	—	26	4	no	—
186	C₄⋊C₄	—	26	8	no	—
187	C₄⋊C₄	—	26	4	no	—
188	C₄⋊C₄	—	52	8	no	—
189	C₄⋊C₄	—	52	4	no	—
190	C₄⋊C₄	—	52	4	no	—
191	C₄⋊C₄	—	52	4	no	—
192	C₄⋊C₄	—	52	8	no	—
193	C₄⋊C₄	—	52	4	no	—
194	C₄⋊C₄	—	52	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	C₂×Dic₂₆	1	4	yes	—
198	C₄²	C₂×Dic₂₆	1	4	yes	—
199	C₄²	C₂×D₅₂	1	4	yes	—
200	C₁₃	#33316	1	1	yes	characteristic, complemented, Sylow
201	C₂³	C₂²×D₅₂	1	1	yes	characteristic
202	Q₈	—	104	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	52	2	no	—
208	C₂×C₄	—	52	2	no	—
209	C₂×C₄	—	52	4	no	—
210	C₂×C₄	—	52	8	no	—
211	C₂×C₄	—	52	4	no	—
212	C₂×C₄	—	52	4	no	—
213	C₂×C₄	#88123	1	4	yes	—
214	C₂×C₄	#88123	1	4	yes	—
215	C₂×C₄	#88128	1	2	yes	—
216	C₂×C₄	#88128	1	2	yes	—
217	C₂×C₄	#88162	1	4	yes	—
218	C₂×C₄	#88163	1	4	yes	—
219	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
220	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
221	C₂×C₄	C₂²×D₅₂	1	2	yes	—
222	C₈	#88123	1	16	yes	complemented
223	C₂²	C₂×#88123	1	1	yes	characteristic
224	C₂²	C₂×#88123	1	1	yes	characteristic
225	C₂²	C₂×#88128	1	1	yes	characteristic
226	C₂²	C₂×#88162	1	2	yes	—
227	C₂²	C₂×#88163	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	104	2	no	—
232	C₄	—	104	4	no	—
233	C₄	—	104	2	no	—
234	C₄	#11718363	1	8	yes	complemented
235	C₄	C₂×#88123	1	4	yes	—
236	C₂	#13123234	1	1	yes	characteristic
237	C₂	#13125288	1	1	yes	characteristic
238	C₂	#13125295	1	1	yes	characteristic
239	C₂	C₂×#11718363	1	2	yes	—
240	C₂	#13125382	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#97	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	#33316
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
47. C₁₃×#97	Q₈	semidirect
48. C₄×C₈×C₁₃	Q₈	semidirect
109. C₈×C₁₃	C₄⋊Q₈	semidirect
136. C₄×C₁₃	#501	semidirect
163. #97	Dic₂₆	semidirect
165. C₄×C₈	Dic₂₆	semidirect
200. C₁₃	#33316	semidirect
222. C₈	#88123	semidirect
234. C₄	#11718363	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718363	2
#13123234	2
#13123199	2
#13115004	2
#13114996	2
#13114993	2
C₄×C₁₃×#97	2
#33316	13