Subgroups — #16726556

Order 3328.

2666 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719524	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88202	C₂	1	1	yes	maximal, characteristic
3	#13123404	C₂	1	16	yes	maximal, complemented
4	#13123400	C₂	1	6	yes	maximal, complemented
5	C₂×#11718660	C₂	1	6	yes	maximal, complemented
6	C₁₃×#3428	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
8	C₁₃×#520	C₄	1	8	yes	—
9	C₁₃×#519	C₄	1	2	yes	—
10	C₁₃×#518	C₄	1	6	yes	—
11	C₄×C₁₃×#41	C₂²	1	8	yes	—
12	C₂×C₁₃×#117	C₂²	1	6	yes	—
13	#11719524	—	2	8	no	—
14	C₂×#88202	—	2	8	no	—
15	C₂×#88202	C₂²	1	8	yes	—
16	C₂×#88202	C₂²	1	8	yes	—
17	C₂²×#6099	C₂²	1	6	yes	—
18	C₂²×#6099	C₂²	1	3	yes	—
19	C₂³×#1055	C₂²	1	3	yes	—
20	#11718660	C₂²	1	4	yes	—
21	#11718660	C₂²	1	12	yes	—
22	#11718660	C₂²	1	24	yes	complemented
23	#11718660	C₂²	1	24	yes	complemented
24	#11718585	C₂²	1	24	yes	complemented
25	C₂×C₄⋊#1055	C₂²	1	24	yes	complemented
26	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×#41	D₄	1	8	yes	—
28	C₂×C₁₃×#41	Q₈	1	8	yes	—
29	C₂×C₁₃×#41	C₂³	1	4	yes	—
30	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₄×C₁₃	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
33	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	—
34	C₄×D₄×C₁₃	C₂×C₄	1	48	yes	—
35	C₁₃×#117	C₂×C₄	1	12	yes	—
36	C₁₃×#117	C₂×C₄	1	12	yes	—
37	C₁₃×#117	C₂³	1	4	yes	—
38	C₁₃×#117	C₂³	1	12	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	12	yes	—
40	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	12	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
42	#88202	—	2	32	no	—
43	#88202	—	2	32	no	—
44	C₂×#6099	—	2	12	no	—
45	C₂×#6099	—	2	24	no	—
46	C₂×#6099	C₂³	1	4	yes	—
47	C₂×#6099	C₂³	1	12	yes	—
48	C₂×#6099	C₂³	1	12	yes	—
49	C₂²×#1055	—	2	12	no	—
50	C₂²×#1055	C₂³	1	1	yes	characteristic
51	C₂²×#1055	C₂³	1	3	yes	—
52	C₂²×#1055	C₂³	1	12	yes	—
53	C₂²×#1055	C₂³	1	12	yes	—
54	C₄⋊#1055	C₂³	1	16	yes	—
55	C₄⋊#1055	C₂³	1	48	yes	—
56	#60100	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂²×C₄	1	3	yes	—
58	C₁₃×#41	C₄⋊C₄	1	32	yes	—
59	C₁₃×#41	C₂×D₄	1	16	yes	—
60	C₁₃×#41	C₂×Q₈	1	16	yes	—
61	C₂×Q₈×C₁₃	C₄⋊C₄	1	8	yes	—
62	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
63	C₂×D₄×C₁₃	C₄⋊C₄	1	24	yes	—
64	C₂×D₄×C₁₃	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
69	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
73	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
74	C₁₃×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
75	C₁₃×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	24	yes	—
77	C₄²×C₁₃	C₂⁴	1	8	yes	—
78	#6099	—	2	8	no	—
79	#6099	—	2	24	no	—
80	#6099	—	2	24	no	—
81	C₂×#1055	—	2	2	no	—
82	C₂×#1055	—	2	6	no	—
83	C₂×#1055	—	2	24	no	—
84	C₂×#1055	—	2	24	no	—
85	C₂×#1055	C₂⁴	1	2	yes	—
86	C₂×#1055	C₂⁴	1	6	yes	—
87	C₂×#1055	C₂⁴	1	8	yes	—
88	C₂×#568	—	13	1	no	—
89	C₂²×#131	—	13	1	no	—
90	#3428	D₁₃	1	1	yes	characteristic
91	#2906	—	13	16	no	—
92	#2905	—	13	6	no	—
93	C₂×#424	—	13	6	no	—
94	C₂³×C₁₃	—	4	6	no	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
96	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	C₂³×C₁₃	C₂³×C₄	1	6	yes	—
98	Q₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
99	D₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
100	C₂×C₄×C₁₃	—	4	24	no	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
103	C₂×C₄×C₁₃	C₂³×C₄	1	16	yes	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
108	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
110	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
111	#1055	—	2	2	no	—
112	#1055	—	2	6	no	—
113	#1055	—	2	8	no	—
114	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
115	#568	—	26	8	no	—
116	C₂×#131	—	13	8	no	—
117	C₂×#131	—	13	8	no	—
118	C₂×#131	—	26	8	no	—
119	C₂²×M₁₆	—	13	6	no	—
120	C₂²×M₁₆	—	13	3	no	—
121	C₂³×C₈	—	13	3	no	—
122	#520	Dic₁₃	1	8	yes	—
123	#519	Dic₁₃	1	2	yes	—
124	#518	Dic₁₃	1	6	yes	—
125	C₄×#41	C₂×D₁₃	1	8	yes	—
126	C₂×#117	C₂×D₁₃	1	6	yes	—
127	#424	—	13	4	no	—
128	#424	—	13	12	no	—
129	#424	—	13	24	no	—
130	#424	—	13	24	no	—
131	#423	—	13	24	no	—
132	C₂×C₄⋊C₈	—	13	24	no	—
133	C₂²×C₁₃	—	2	12	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	4	12	no	—
136	C₂²×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
137	C₂²×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
138	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
139	C₄×C₁₃	—	2	16	no	—
140	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
141	C₄×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
142	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
143	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
144	C₂×#41	Dic₂₆	1	8	yes	—
145	C₂×#41	D₅₂	1	8	yes	—
146	C₂×#41	C₂×Dic₁₃	1	4	yes	—
147	C₂×#41	C₂²×D₁₃	1	4	yes	—
148	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
149	C₂²×D₄	C₂×Dic₁₃	1	3	yes	—
150	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
151	#131	—	26	32	no	—
152	#131	—	26	32	no	—
153	C₂×M₁₆	—	13	4	no	—
154	C₂×M₁₆	—	13	12	no	—
155	C₂×M₁₆	—	13	12	no	—
156	C₂×M₁₆	—	26	12	no	—
157	C₂×M₁₆	—	26	24	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	3	no	—
160	C₂²×C₈	—	13	12	no	—
161	C₂²×C₈	—	13	12	no	—
162	C₂²×C₈	—	26	12	no	—
163	C₄×Q₈	C₂×Dic₁₃	1	16	yes	—
164	C₄×D₄	C₂×Dic₁₃	1	48	yes	—
165	#117	C₂×Dic₁₃	1	12	yes	—
166	#117	C₂×Dic₁₃	1	12	yes	—
167	#117	C₂²×D₁₃	1	4	yes	—
168	#117	C₂²×D₁₃	1	12	yes	—
169	C₂×C₄⋊C₄	C₂×Dic₁₃	1	12	yes	—
170	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	12	yes	—
171	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
172	C₄⋊C₈	—	13	16	no	—
173	C₄⋊C₈	—	13	48	no	—
174	C₂×C₁₃	—	2	2	no	—
175	C₂×C₁₃	—	2	12	no	—
176	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₁₃	#3573	1	2	yes	—
178	C₂⁴	C₂²×Dic₁₃	1	3	yes	—
179	#41	C₄⋊Dic₁₃	1	32	yes	—
180	#41	C₂×Dic₂₆	1	16	yes	—
181	#41	C₂×D₅₂	1	16	yes	—
182	C₂×Q₈	C₄⋊Dic₁₃	1	8	yes	—
183	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
184	C₂×D₄	C₄⋊Dic₁₃	1	24	yes	—
185	C₂×D₄	C₂²×Dic₁₃	1	12	yes	—
186	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
187	C₂²×C₄	C₂×D₅₂	1	12	yes	—
188	C₂²×C₄	C₂²×Dic₁₃	1	24	yes	—
189	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
190	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
191	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
192	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
193	M₁₆	—	26	8	no	—
194	M₁₆	—	26	24	no	—
195	M₁₆	—	26	24	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	13	8	no	—
198	C₂×C₈	—	13	6	no	—
199	C₂×C₈	—	26	2	no	—
200	C₂×C₈	—	26	6	no	—
201	C₂×C₈	—	26	24	no	—
202	C₂×C₈	—	26	24	no	—
203	C₄⋊C₄	C₂²×Dic₁₃	1	8	yes	—
204	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
205	C₂²⋊C₄	C₂²×Dic₁₃	1	8	yes	—
206	C₂²⋊C₄	C₂²×Dic₁₃	1	24	yes	—
207	C₄²	C₂²×Dic₁₃	1	24	yes	—
208	C₄²	C₂³×D₁₃	1	8	yes	—
209	C₁₃	#60100	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	4	6	no	—
211	C₂³	C₂×C₄⋊Dic₁₃	1	12	yes	—
212	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
213	C₂³	C₂³×Dic₁₃	1	6	yes	—
214	Q₈	C₂×C₄⋊Dic₁₃	1	16	yes	—
215	D₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
216	C₂×C₄	—	4	24	no	—
217	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₁₃	1	24	yes	—
219	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
220	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
221	C₂×C₄	C₂²×D₅₂	1	2	yes	—
222	C₂×C₄	C₂²×D₅₂	1	12	yes	—
223	C₂×C₄	C₂³×Dic₁₃	1	16	yes	—
224	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
225	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
227	C₈	—	26	2	no	—
228	C₈	—	26	6	no	—
229	C₈	—	26	8	no	—
230	C₂²	—	2	12	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	12	no	—
233	C₂²	C₂²×C₄⋊Dic₁₃	1	2	yes	—
234	C₂²	C₂²×C₄⋊Dic₁₃	1	12	yes	—
235	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
236	C₄	—	2	16	no	—
237	C₄	C₂²×C₄⋊Dic₁₃	1	2	yes	—
238	C₄	C₂²×C₄⋊Dic₁₃	1	12	yes	—
239	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
240	C₄	C₂³×D₅₂	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	12	no	—
243	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic
244	C₂	#13129548	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₁₃
commutator, socle	C₂²×C₁₃	C₂⁴×C₄
Fitting	C₁₃×#3428	C₂
prime radical	C₂²×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	#3428	D₁₃
hypocenter	C₁₃	#60100
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #3428

Complements

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

N	H ≅ G/N	type
3. #13123404	C₂	semidirect
4. #13123400	C₂	semidirect
5. C₂×#11718660	C₂	semidirect
22. #11718660	C₂²	semidirect
23. #11718660	C₂²	semidirect
24. #11718585	C₂²	semidirect
25. C₂×C₄⋊#1055	C₂²	semidirect
209. C₁₃	#60100	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719524	2
C₂²×#88202	2
#13123404	2
#13123400	2
C₂×#11718660	2
C₁₃×#3428	2
#60100	13