Subgroups — #15871722

Order 3328.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718344	C₂	1	1	yes	maximal, characteristic
2	C₄×#88124	C₂	1	2	yes	maximal
3	#13115801	C₂	1	2	yes	maximal
4	#13115579	C₂	1	2	yes	maximal
5	#13115574	C₂	1	2	yes	maximal
6	#13115000	C₂	1	1	yes	maximal, characteristic
7	#13114733	C₂	1	2	yes	maximal
8	#13114733	C₂	1	2	yes	maximal
9	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
10	C₂×#88124	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88058	C₂²	1	2	yes	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88057	C₂²	1	2	yes	—
21	C₂×#88057	C₂²	1	2	yes	—
22	#11718597	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
24	#11718594	—	2	4	no	—
25	C₄×#6101	—	2	4	no	—
26	#11718589	C₂²	1	4	yes	—
27	#11718344	C₄	1	8	yes	complemented
28	#11718143	C₂²	1	4	yes	—
29	#11718143	C₂²	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	4	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
40	C₂×#6101	—	2	4	no	—
41	C₂×#6101	—	2	4	no	—
42	C₂×C₄×Dic₁₃	—	2	4	no	—
43	#88124	—	2	4	no	—
44	#88124	C₂×C₄	1	8	yes	—
45	C₁₃×#97	—	2	2	no	—
46	C₁₃×#97	—	2	2	no	—
47	C₁₃×#97	Q₈	1	4	yes	—
48	C₁₃×#97	Q₈	1	4	yes	—
49	C₄×C₈×C₁₃	—	2	2	no	—
50	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
51	#88071	—	2	4	no	—
52	#88071	—	2	4	no	—
53	#88071	—	2	4	no	—
54	#88058	—	2	4	no	—
55	#88058	C₂×C₄	1	8	yes	—
56	#88057	—	2	4	no	—
57	#88057	C₂×C₄	1	8	yes	—
58	#21728	—	13	1	no	maximal, Sylow
59	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	2	4	no	—
64	C₂²×Dic₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	—	2	1	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₁₃	#41	1	2	yes	—
76	C₂×C₈×C₁₃	#41	1	2	yes	—
77	C₄²×C₁₃	—	2	1	no	—
78	C₄²×C₁₃	—	2	1	no	—
79	C₄²×C₁₃	—	4	4	no	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
82	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₁₃	#41	1	2	yes	—
84	C₄⋊Dic₁₃	—	2	4	no	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
88	#6101	—	2	16	no	—
89	#6101	—	4	8	no	—
90	C₄×Dic₁₃	—	4	8	no	—
91	C₂×#499	—	13	1	no	—
92	C₄×#125	—	13	2	no	—
93	#1991	—	13	2	no	—
94	#1944	—	13	2	no	—
95	#1943	—	13	2	no	—
96	#1846	—	13	1	no	—
97	#1776	—	13	2	no	—
98	#1776	—	13	2	no	—
99	C₄×#97	D₁₃	1	1	yes	characteristic
100	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
107	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₁₃	—	2	8	no	—
114	C₂×Dic₁₃	—	4	4	no	—
115	C₂×Dic₁₃	—	4	4	no	—
116	C₂×Dic₁₃	—	4	4	no	—
117	C₂×Dic₁₃	—	4	4	no	—
118	C₂×Dic₁₃	—	4	4	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	C₄×Q₈	1	4	yes	—
122	C₈×C₁₃	C₄×Q₈	1	4	yes	—
123	C₂×#125	—	13	2	no	—
124	#499	—	13	8	no	—
125	C₂×#107	—	13	2	no	—
126	C₂×#107	—	13	2	no	—
127	C₂×#106	—	13	2	no	—
128	C₂×#106	—	13	2	no	—
129	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
130	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
131	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
132	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
135	#389	—	13	4	no	—
136	#387	—	26	4	no	—
137	#382	—	13	2	no	—
138	C₄×C₄⋊C₄	—	13	4	no	—
139	C₄×C₄⋊C₄	—	26	4	no	—
140	C₄³	C₂×D₁₃	1	1	yes	characteristic
141	#340	—	13	4	no	—
142	#340	—	13	4	no	—
143	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₁₃	#532	1	1	yes	characteristic
146	C₂²×C₁₃	#575	1	4	yes	—
147	C₄×C₁₃	—	2	4	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₁₃	#533	1	1	yes	characteristic
153	C₄×C₁₃	#533	1	1	yes	characteristic
154	Dic₁₃	—	4	8	no	—
155	Dic₁₃	—	8	4	no	—
156	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
160	#125	—	13	8	no	—
161	#125	—	26	4	no	—
162	C₂×C₄⋊C₄	—	13	4	no	—
163	C₂×C₄⋊C₄	—	13	4	no	—
164	C₂×C₄⋊C₄	—	26	4	no	—
165	C₂×C₄⋊C₄	—	26	4	no	—
166	C₂×C₄²	—	26	4	no	—
167	C₂×C₄²	D₅₂	1	4	yes	—
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
171	#107	—	13	8	no	—
172	#107	—	26	4	no	—
173	#106	—	13	8	no	—
174	#106	—	26	4	no	—
175	#97	—	2	2	no	—
176	#97	—	2	2	no	—
177	#97	Dic₂₆	1	4	yes	—
178	#97	Dic₂₆	1	4	yes	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	C₄×D₁₃	1	4	yes	—
181	#95	—	26	4	no	—
182	#95	—	26	4	no	—
183	#95	—	26	4	no	—
184	C₂×C₁₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
185	C₂×C₁₃	#2943	1	2	yes	—
186	C₂×C₁₃	#3166	1	2	yes	—
187	C₂×C₁₃	#3183	1	2	yes	—
188	C₂²×C₄	—	26	4	no	—
189	C₂²×C₄	—	26	4	no	—
190	C₂²×C₄	C₂×D₅₂	1	2	yes	—
191	C₂²×C₄	C₂×D₅₂	1	2	yes	—
192	C₂²×C₄	C₂×D₅₂	1	2	yes	—
193	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
199	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
200	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
201	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
204	C₂×C₈	#6127	1	2	yes	—
205	C₂×C₈	#6127	1	2	yes	—
206	C₄⋊C₄	—	13	8	no	—
207	C₄⋊C₄	—	26	16	no	—
208	C₄⋊C₄	—	26	4	no	—
209	C₄⋊C₄	—	26	4	no	—
210	C₄⋊C₄	—	26	4	no	—
211	C₄⋊C₄	—	52	8	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	4	no	—
215	C₄²	—	52	8	no	—
216	C₄²	C₂×Dic₂₆	1	2	yes	—
217	C₄²	C₂×Dic₂₆	1	2	yes	—
218	C₄²	C₂×C₄×D₁₃	1	2	yes	—
219	C₄²	#6127	1	2	yes	—
220	C₁₃	#21728	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₂²×D₅₂	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	26	8	no	—
227	C₂×C₄	—	52	4	no	—
228	C₂×C₄	—	52	4	no	—
229	C₂×C₄	—	52	4	no	—
230	C₂×C₄	—	52	4	no	—
231	C₂×C₄	—	52	4	no	—
232	C₂×C₄	#88123	1	4	yes	—
233	C₂×C₄	C₄×D₅₂	1	4	yes	—
234	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
235	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂×#6127	1	1	yes	characteristic
238	C₂×C₄	C₂×#6127	1	1	yes	characteristic
239	C₂×C₄	C₂×#6127	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₄×Dic₂₆	1	4	yes	—
243	C₈	C₄×Dic₂₆	1	4	yes	—
244	C₂²	C₂×#88123	1	1	yes	characteristic
245	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
246	C₂²	#11719182	1	1	yes	characteristic
247	C₂²	#11719453	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	—	4	2	no	—
251	C₄	—	52	8	no	—
252	C₄	—	104	4	no	—
253	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
254	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
255	C₄	#11719175	1	1	yes	characteristic
256	C₄	#11719175	1	1	yes	characteristic
257	C₂	C₄×#88123	1	1	yes	characteristic
258	C₂	#13123638	1	2	yes	—
259	C₂	#13125388	1	2	yes	—
260	C₂	#13125565	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#97	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	#21728
LCS	G ▷ C₂×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
27. #11718344	C₄	semidirect
220. C₁₃	#21728	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718344	2
C₄×#88124	2
#13115801	2
#13115579	2
#13115574	2
#13115000	2
#13114733	2
C₄×C₁₃×#97	2
#21728	13