Subgroups — #13123821

Order 1664.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719184	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718789	C₂	1	2	yes	maximal, complemented
4	#11718788	C₂	1	2	yes	maximal, complemented
5	#11718587	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718367	C₂	1	2	yes	maximal, complemented
7	#11718362	C₂	1	2	yes	maximal, complemented
8	#11718351	C₂	1	2	yes	maximal, complemented
9	#11718350	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	#88184	C₄	1	1	yes	characteristic
13	#88183	C₄	1	2	yes	—
14	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
15	#88178	C₄	1	1	yes	characteristic
16	C₂×#6094	C₂²	1	2	yes	—
17	C₂×C₈×D₁₃	C₂²	1	2	yes	—
18	#88130	C₄	1	2	yes	—
19	#88129	C₄	1	1	yes	characteristic
20	C₄×D₅₂	C₂²	1	1	yes	characteristic
21	#88126	C₂²	1	2	yes	—
22	#88124	C₄	1	1	yes	characteristic
23	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
24	C₁₃×#97	C₂²	1	2	yes	—
25	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₂²⋊#1055	C₂²	1	1	yes	characteristic
28	C₂²⋊#1055	C₂²	1	1	yes	characteristic
29	#88060	C₂²	1	2	yes	—
30	#88060	C₂²	1	2	yes	—
31	#88055	C₂²	1	2	yes	—
32	#88054	C₂²	1	2	yes	—
33	#88054	C₂²	1	2	yes	—
34	C₈×Dic₁₃	C₂²	1	2	yes	—
35	C₄⋊#1055	C₂²	1	2	yes	—
36	#88043	C₂²	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
41	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₃	C₂³	1	2	yes	—
43	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
44	C₁₃×M₁₆	—	2	4	no	—
45	C₂×C₈×C₁₃	C₂³	1	2	yes	—
46	C₂×C₈×C₁₃	C₂³	1	2	yes	—
47	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
52	#6103	C₂×C₄	1	2	yes	—
53	#6103	C₂×C₄	1	2	yes	—
54	#6103	C₂×C₄	1	2	yes	—
55	#6103	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
58	#6101	C₂×C₄	1	2	yes	—
59	#6101	C₂×C₄	1	2	yes	—
60	#6101	C₂×C₄	1	2	yes	—
61	#6101	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₃	C₂³	1	2	yes	—
63	C₂×#1055	C₂³	1	2	yes	—
64	C₂×#1055	C₂³	1	2	yes	—
65	#6094	—	2	4	no	—
66	C₈×D₁₃	—	2	4	no	—
67	#2967	—	13	1	no	maximal, Sylow
68	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	4	1	no	—
72	C₂×C₄×C₁₃	—	4	1	no	—
73	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²⋊D₁₃	—	4	2	no	—
80	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
83	D₅₂	—	4	1	no	—
84	C₄×D₁₃	—	2	2	no	—
85	C₄×D₁₃	—	2	2	no	—
86	Dic₂₆	—	4	1	no	—
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	#41	1	4	yes	—
89	#1055	—	2	2	no	—
90	#1055	—	2	2	no	—
91	#529	—	13	1	no	—
92	#446	—	13	2	no	—
93	#436	—	13	2	no	—
94	#435	—	13	2	no	—
95	#435	—	13	2	no	—
96	#433	—	13	2	no	—
97	#432	—	13	1	no	—
98	#405	—	13	2	no	—
99	C₄×M₁₆	D₁₃	1	1	yes	characteristic, complemented
100	C₂²×C₁₃	—	2	1	no	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₁₃	—	2	2	no	—
105	C₂×D₁₃	—	2	2	no	—
106	C₂×D₁₃	—	2	2	no	—
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	C₂×#41	1	2	yes	—
113	Dic₁₃	—	2	2	no	—
114	Dic₁₃	—	2	2	no	—
115	Dic₁₃	—	2	2	no	—
116	C₂×M₁₆	—	13	2	no	—
117	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
118	C₂²×C₈	—	13	2	no	—
119	#126	—	13	2	no	—
120	#125	—	13	1	no	—
121	#124	—	13	1	no	—
122	#123	—	13	2	no	—
123	C₂²⋊Q₈	—	13	1	no	—
124	#121	—	13	1	no	—
125	C₄×Q₈	—	13	1	no	—
126	C₄×D₄	—	13	2	no	—
127	C₄×D₄	—	13	1	no	—
128	#117	—	13	2	no	—
129	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
130	C₄⋊C₈	—	13	2	no	—
131	C₄⋊C₈	—	13	2	no	—
132	C₄⋊C₈	—	13	2	no	—
133	C₂²⋊C₈	—	13	1	no	—
134	C₂²⋊C₈	—	13	1	no	—
135	C₂²⋊C₈	—	13	2	no	—
136	C₂²⋊C₈	—	13	2	no	—
137	#97	—	13	2	no	—
138	#97	—	13	2	no	—
139	#97	C₂×D₁₃	1	2	yes	—
140	C₄×C₈	—	13	2	no	—
141	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
142	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
143	C₂×C₁₃	—	4	1	no	—
144	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
145	C₂×C₁₃	#568	1	1	yes	characteristic
146	C₂×C₁₃	#568	1	1	yes	characteristic
147	D₁₃	—	4	2	no	—
148	C₂×Q₈	—	13	1	no	—
149	C₂×D₄	—	13	2	no	—
150	C₂×D₄	—	13	1	no	—
151	C₂²×C₄	—	13	2	no	—
152	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
155	M₁₆	—	2	4	no	—
156	M₁₆	—	26	4	no	—
157	C₂×C₈	—	13	2	no	—
158	C₂×C₈	—	13	2	no	—
159	C₂×C₈	—	26	4	no	—
160	C₂×C₈	C₂²×D₁₃	1	2	yes	—
161	C₂×C₈	C₂²×D₁₃	1	2	yes	—
162	C₄⋊C₄	—	13	2	no	—
163	C₄⋊C₄	—	13	2	no	—
164	C₄⋊C₄	—	13	1	no	—
165	C₄⋊C₄	—	13	2	no	—
166	C₄⋊C₄	—	13	1	no	—
167	C₄⋊C₄	—	13	2	no	—
168	C₂²⋊C₄	—	13	2	no	—
169	C₂²⋊C₄	—	13	2	no	—
170	C₂²⋊C₄	—	13	2	no	—
171	C₂²⋊C₄	—	13	2	no	—
172	C₂²⋊C₄	—	13	2	no	—
173	C₄²	—	13	2	no	—
174	C₄²	C₄×D₁₃	1	1	yes	characteristic
175	C₄²	C₄×D₁₃	1	1	yes	characteristic
176	C₄²	C₂²×D₁₃	1	1	yes	characteristic
177	C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₁₃	#2967	1	1	yes	characteristic, complemented, Sylow
179	C₂³	—	13	2	no	—
180	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
181	Q₈	—	52	1	no	—
182	D₄	—	52	1	no	—
183	D₄	—	52	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	13	2	no	—
188	C₂×C₄	—	13	2	no	—
189	C₂×C₄	—	13	2	no	—
190	C₂×C₄	—	26	2	no	—
191	C₂×C₄	—	26	2	no	—
192	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	26	2	no	—
200	C₈	—	26	2	no	—
201	C₈	#6127	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	26	2	no	—
206	C₂²	—	26	2	no	—
207	C₂²	—	26	2	no	—
208	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	26	2	no	—
215	C₄	—	26	2	no	—
216	C₄	—	26	2	no	—
217	C₄	C₂×#6127	1	2	yes	—
218	C₂	—	4	1	no	—
219	C₂	—	52	2	no	—
220	C₂	C₄×#6127	1	1	yes	characteristic
221	C₂	#11719450	1	1	yes	characteristic
222	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×M₁₆	C₂
prime radical	#88184	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×M₁₆	D₁₃
hypocenter	C₁₃	#2967
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₃×M₁₆	C₂	semidirect
3. #11718789	C₂	semidirect
4. #11718788	C₂	semidirect
5. #11718587	C₂	semidirect
6. #11718367	C₂	semidirect
7. #11718362	C₂	semidirect
8. #11718351	C₂	semidirect
9. #11718350	C₂	semidirect
99. C₄×M₁₆	D₁₃	semidirect
178. C₁₃	#2967	semidirect

Maximal subgroups

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

subgroup	index
#11719184	2
C₄×C₁₃×M₁₆	2
#11718789	2
#11718788	2
#11718587	2
#11718367	2
#11718362	2
#11718351	2
#11718350	2
#2967	13