Subgroups — #16807001

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊M₃₂	C₂	1	2	yes	maximal
2	#13131172	C₂	1	8	yes	maximal, complemented
3	#13131170	C₂	1	2	yes	maximal, complemented
4	#13131168	C₂	1	2	yes	maximal, complemented
5	C₂×#11718369	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88126	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	4	yes	—
10	D₁₃⋊M₃₂	—	2	8	no	—
11	C₂×#88098	C₂²	1	4	yes	—
12	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	—
13	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
14	C₂×#88055	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718369	C₄	1	4	yes	—
18	#11718369	C₂²	1	4	yes	—
19	#11718260	C₂²	1	8	yes	complemented
20	C₄⋊#6092	C₂²	1	8	yes	complemented
21	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×#6103	C₈	1	2	yes	—
26	C₂×#6101	C₈	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂³	1	1	yes	characteristic
30	C₂×#6094	—	2	4	no	—
31	C₂×#6094	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₁₃	D₄	1	4	yes	—
34	C₂×C₈×D₁₃	Q₈	1	4	yes	—
35	C₂×C₈×D₁₃	C₂³	1	2	yes	—
36	#88126	C₈	1	4	yes	—
37	#88126	C₂×C₄	1	4	yes	—
38	#88106	—	2	8	no	—
39	C₂×#6092	C₂³	1	8	yes	—
40	#88098	—	2	8	no	—
41	D₁₃⋊C₁₆	—	2	8	no	—
42	C₁₃×#97	C₂×C₄	1	4	yes	—
43	#88055	C₂×C₄	1	4	yes	—
44	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₈×Dic₁₃	C₂³	1	2	yes	—
46	C₄×#1055	C₂×C₄	1	2	yes	—
47	C₄×#1055	C₂³	1	2	yes	—
48	#33397	—	13	1	no	maximal, Sylow
49	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
53	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
57	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₁₃	C₂×C₈	1	2	yes	—
63	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
64	#6103	C₂×C₈	1	8	yes	—
65	#6101	C₂×C₈	1	8	yes	—
66	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
67	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
68	C₂×#1055	—	2	2	no	—
69	C₂×#1055	C₂²×C₄	1	2	yes	—
70	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
71	C₂×#1055	C₂×D₄	1	2	yes	—
72	C₂×#1055	C₂×Q₈	1	2	yes	—
73	C₂×#1055	C₂⁴	1	1	yes	characteristic
74	#6094	—	2	8	no	—
75	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
76	C₈×D₁₃	C₂×D₄	1	4	yes	—
77	C₈×D₁₃	C₂×Q₈	1	4	yes	—
78	#6092	—	2	8	no	—
79	C₂²×M₃₂	—	13	2	no	—
80	C₂×#405	—	13	1	no	—
81	#2151	—	13	8	no	—
82	#2150	—	13	2	no	—
83	#2150	—	13	2	no	—
84	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂²×D₁₃	—	4	2	no	—
86	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
87	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×Dic₁₃	—	4	2	no	—
95	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
96	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
97	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
98	C₄×D₁₃	C₄⋊C₈	1	8	yes	—
99	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
103	#1055	—	2	2	no	—
104	#1055	C₂×C₄⋊C₄	1	2	yes	—
105	#1055	C₂²×D₄	1	1	yes	characteristic
106	#1055	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×M₁₆	—	13	1	no	—
108	C₂³×C₈	—	13	1	no	—
109	C₂×#117	—	13	1	no	—
110	C₂×M₃₂	—	13	4	no	—
111	C₂×M₃₂	—	13	4	no	—
112	C₂×M₃₂	—	26	8	no	—
113	C₂²×C₁₆	—	13	4	no	—
114	#405	—	13	4	no	—
115	#405	—	13	4	no	—
116	C₂×#97	—	13	1	no	—
117	C₂×#97	#261	1	1	yes	characteristic
118	C₂×C₄×C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₄⋊C₁₆	—	13	8	no	—
121	C₄⋊C₁₆	—	13	8	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
124	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
125	C₂×D₁₃	—	2	4	no	—
126	C₂×D₁₃	—	4	4	no	—
127	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
128	C₄×C₁₃	—	2	4	no	—
129	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
130	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
132	Dic₁₃	—	2	4	no	—
133	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
134	C₂³×C₄	—	13	1	no	—
135	C₂×M₁₆	—	13	4	no	—
136	C₂×M₁₆	—	26	4	no	—
137	C₂²×C₈	—	13	1	no	—
138	C₂²×C₈	—	13	4	no	—
139	C₂²×C₈	—	13	4	no	—
140	C₂²×C₈	—	13	2	no	—
141	C₂²×C₈	—	13	2	no	—
142	C₂²×C₈	—	13	1	no	—
143	C₂²×C₈	C₂×#261	1	1	yes	characteristic
144	C₂²×C₈	C₂×#261	1	1	yes	characteristic
145	#117	—	13	4	no	—
146	#117	—	13	4	no	—
147	C₂×C₄⋊C₄	—	13	2	no	—
148	C₂×C₂²⋊C₄	—	13	2	no	—
149	C₂×C₄²	—	13	1	no	—
150	C₂×C₄²	C₂×#261	1	1	yes	characteristic
151	M₃₂	—	26	8	no	—
152	M₃₂	—	26	8	no	—
153	C₂×C₁₆	—	13	8	no	—
154	C₂×C₁₆	—	26	8	no	—
155	#97	—	13	4	no	—
156	#97	C₂×#261	1	4	yes	—
157	C₄×C₈	—	13	2	no	—
158	C₄×C₈	—	13	2	no	—
159	C₄×C₈	—	13	2	no	—
160	C₄×C₈	—	13	2	no	—
161	C₂×C₁₃	—	2	2	no	—
162	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₁₃	#3407	1	2	yes	—
164	D₁₃	—	2	4	no	—
165	C₂⁴	—	13	1	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	2	no	—
168	C₂²×C₄	—	13	4	no	—
169	C₂²×C₄	—	13	4	no	—
170	C₂²×C₄	—	13	2	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
173	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
174	M₁₆	—	26	8	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	13	2	no	—
177	C₂×C₈	—	13	2	no	—
178	C₂×C₈	—	13	1	no	—
179	C₂×C₈	—	13	1	no	—
180	C₂×C₈	—	13	4	no	—
181	C₂×C₈	—	13	8	no	—
182	C₂×C₈	—	13	4	no	—
183	C₂×C₈	—	13	2	no	—
184	C₂×C₈	—	26	2	no	—
185	C₂×C₈	C₄⋊#261	1	4	yes	—
186	C₂×C₈	C₂²×#261	1	2	yes	—
187	C₂×C₈	C₂²×#261	1	2	yes	—
188	C₄⋊C₄	—	13	8	no	—
189	C₂²⋊C₄	—	13	8	no	—
190	C₄²	—	13	2	no	—
191	C₄²	—	13	2	no	—
192	C₄²	D₁₃⋊C₈	1	2	yes	—
193	C₄²	C₂²×#261	1	2	yes	—
194	C₁₆	—	26	8	no	—
195	C₁₃	#33397	1	1	yes	characteristic, complemented, Sylow
196	C₂³	—	13	2	no	—
197	C₂³	—	13	4	no	—
198	C₂³	—	52	2	no	—
199	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	13	4	no	—
202	C₂×C₄	—	13	2	no	—
203	C₂×C₄	—	13	8	no	—
204	C₂×C₄	—	13	4	no	—
205	C₂×C₄	—	13	4	no	—
206	C₂×C₄	—	13	4	no	—
207	C₂×C₄	—	52	2	no	—
208	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
209	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
210	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
211	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
212	C₂×C₄	C₂³×#261	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	13	1	no	—
215	C₈	—	13	2	no	—
216	C₈	—	13	1	no	—
217	C₈	—	26	2	no	—
218	C₈	C₂×C₄⋊#261	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	13	4	no	—
221	C₂²	—	26	4	no	—
222	C₂²	—	52	4	no	—
223	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
224	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	13	4	no	—
227	C₄	—	26	4	no	—
228	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
229	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
230	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	26	4	no	—
233	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
234	C₂	#13131805	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×#97	C₄
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#33397
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #13131172	C₂	semidirect
3. #13131170	C₂	semidirect
4. #13131168	C₂	semidirect
19. #11718260	C₂²	semidirect
20. C₄⋊#6092	C₂²	semidirect
195. C₁₃	#33397	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃⋊M₃₂	2
#13131172	2
#13131170	2
#13131168	2
C₂×#11718369	2
#33397	13