Subgroups — #16015601

Order 3328.

1172 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718801	C₂	1	2	yes	maximal, complemented
2	#13116450	C₂	1	6	yes	maximal, complemented
3	#13116447	C₂	1	3	yes	maximal, complemented
4	C₂×C₂²⋊#6090	C₂	1	3	yes	maximal, complemented
5	C₂²×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#41	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#131	—	2	8	no	—
8	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
9	C₂×C₁₃×#131	C₄	1	3	yes	—
10	C₂×C₁₃×#131	C₂²	1	3	yes	—
11	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×M₁₆	C₄	1	3	yes	—
13	C₂³×C₈×C₁₃	C₂²	1	3	yes	—
14	#11718801	—	2	8	no	—
15	C₂×#88052	C₂²	1	6	yes	complemented
16	C₂²×#6090	C₂²	1	6	yes	complemented
17	C₂²⋊#6090	C₂²	1	1	yes	characteristic
18	C₂²⋊#6090	C₂²	1	3	yes	—
19	C₂²⋊#6090	C₂²	1	6	yes	complemented
20	C₂²⋊#6090	C₂²	1	6	yes	complemented
21	C₂×C₁₃×#41	—	2	4	no	—
22	C₂×C₁₃×#41	—	2	4	no	—
23	C₂×C₁₃×#41	C₈	1	1	yes	characteristic
24	C₂×C₁₃×#41	C₈	1	3	yes	—
25	C₂×C₁₃×#41	C₂×C₄	1	3	yes	—
26	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
27	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
28	C₂²×D₄×C₁₃	C₈	1	3	yes	—
29	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
30	C₁₃×#131	—	2	4	no	—
31	C₁₃×#131	—	2	12	no	—
32	C₁₃×#131	—	2	12	no	—
33	C₁₃×#131	D₄	1	8	yes	—
34	C₂×C₁₃×M₁₆	—	2	12	no	—
35	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
36	C₂×C₁₃×M₁₆	C₂×C₄	1	6	yes	—
37	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₁₃	—	2	12	no	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
41	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
45	#88052	—	2	12	no	—
46	C₂×#6090	—	2	12	no	—
47	C₂×#6090	C₂³	1	2	yes	—
48	C₂×#6090	C₂³	1	6	yes	—
49	#33334	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂×C₈	1	3	yes	—
51	C₁₃×#41	—	2	4	no	—
52	C₁₃×#41	—	2	12	no	—
53	C₁₃×#41	—	2	6	no	—
54	C₁₃×#41	—	2	6	no	—
55	C₁₃×#41	C₂²⋊C₄	1	4	yes	—
56	C₁₃×#41	M₁₆	1	4	yes	—
57	C₂×Q₈×C₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₃	C₂×C₈	1	3	yes	—
59	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂×D₄×C₁₃	—	2	12	no	—
61	C₂×D₄×C₁₃	C₂×C₈	1	6	yes	—
62	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
63	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
64	C₂²×C₄×C₁₃	—	2	6	no	—
65	C₂²×C₄×C₁₃	—	2	6	no	—
66	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
68	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
72	C₁₃×M₁₆	—	2	6	no	—
73	C₁₃×M₁₆	—	2	12	no	—
74	C₁₃×M₁₆	C₂²⋊C₄	1	12	yes	—
75	C₂×C₈×C₁₃	—	2	6	no	—
76	C₂×C₈×C₁₃	—	2	6	no	—
77	C₂×C₈×C₁₃	—	2	6	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	6	no	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	12	yes	—
84	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
85	#6090	—	2	2	no	—
86	#6090	—	2	6	no	—
87	C₂²×#131	D₁₃	1	1	yes	characteristic
88	C₂×#504	—	13	2	no	—
89	#2113	—	13	6	no	—
90	#2112	—	13	3	no	—
91	C₂×C₂²⋊C₁₆	—	13	3	no	—
92	C₂³×C₁₃	—	2	6	no	—
93	C₂³×C₁₃	—	2	6	no	—
94	C₂³×C₁₃	—	4	3	no	—
95	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
96	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
97	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
98	Q₈×C₁₃	—	2	6	no	—
99	Q₈×C₁₃	C₂²⋊C₈	1	4	yes	—
100	D₄×C₁₃	—	2	12	no	—
101	D₄×C₁₃	—	2	6	no	—
102	D₄×C₁₃	C₂²⋊C₈	1	12	yes	—
103	C₂×C₄×C₁₃	—	2	6	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	3	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	3	no	—
108	C₂×C₄×C₁₃	—	2	6	no	—
109	C₂×C₄×C₁₃	—	2	6	no	—
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
111	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	3	no	—
118	C₈×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
119	C₈×C₁₃	C₂²×D₄	1	2	yes	—
120	C₂²×#41	Dic₁₃	1	1	yes	characteristic
121	C₂×#131	—	2	8	no	—
122	C₂×#131	Dic₁₃	1	1	yes	characteristic
123	C₂×#131	Dic₁₃	1	3	yes	—
124	C₂×#131	C₂×D₁₃	1	3	yes	—
125	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
126	C₂²×M₁₆	Dic₁₃	1	3	yes	—
127	C₂³×C₈	C₂×D₁₃	1	3	yes	—
128	#504	—	26	8	no	—
129	C₂×M₃₂	—	13	6	no	—
130	C₂²×C₁₆	—	13	6	no	—
131	C₂²⋊C₁₆	—	13	1	no	—
132	C₂²⋊C₁₆	—	13	3	no	—
133	C₂²⋊C₁₆	—	13	6	no	—
134	C₂²⋊C₁₆	—	13	6	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	3	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	3	no	—
139	C₂²×C₁₃	—	2	3	no	—
140	C₂²×C₁₃	—	2	3	no	—
141	C₂²×C₁₃	—	2	3	no	—
142	C₂²×C₁₃	—	4	12	no	—
143	C₂²×C₁₃	C₂×C₂²⋊C₈	1	6	yes	—
144	C₂²×C₁₃	#504	1	2	yes	complemented
145	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	3	no	—
148	C₄×C₁₃	C₂×C₂²⋊C₈	1	6	yes	—
149	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
151	C₂×#41	—	2	4	no	—
152	C₂×#41	—	2	4	no	—
153	C₂×#41	#1055	1	1	yes	characteristic
154	C₂×#41	#1055	1	3	yes	—
155	C₂×#41	C₂×Dic₁₃	1	3	yes	—
156	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂²×Q₈	#1055	1	1	yes	characteristic
158	C₂²×D₄	#1055	1	3	yes	—
159	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
160	#131	—	2	4	no	—
161	#131	—	2	12	no	—
162	#131	—	2	12	no	—
163	#131	C₂²⋊D₁₃	1	8	yes	—
164	C₂×M₁₆	—	2	12	no	—
165	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
166	C₂×M₁₆	C₂×Dic₁₃	1	6	yes	—
167	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
168	C₂²×C₈	—	2	12	no	—
169	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
170	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
171	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
172	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
174	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
175	M₃₂	—	26	12	no	—
176	C₂×C₁₆	—	13	2	no	—
177	C₂×C₁₆	—	13	6	no	—
178	C₂×C₁₆	—	26	12	no	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	—	2	6	no	—
181	C₂×C₁₃	—	4	3	no	—
182	C₂×C₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
183	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
184	C₂×C₁₃	#3407	1	1	yes	characteristic
185	C₂⁴	C₂×#1055	1	3	yes	—
186	#41	—	2	4	no	—
187	#41	—	2	12	no	—
188	#41	—	2	6	no	—
189	#41	—	2	6	no	—
190	#41	#6099	1	4	yes	—
191	#41	C₂²⋊Dic₁₃	1	4	yes	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	C₂×#1055	1	3	yes	—
194	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
195	C₂×D₄	—	2	12	no	—
196	C₂×D₄	C₂×#1055	1	6	yes	—
197	C₂×D₄	C₂×#1055	1	3	yes	—
198	C₂×D₄	C₂×#1055	1	3	yes	—
199	C₂²×C₄	—	2	6	no	—
200	C₂²×C₄	—	2	6	no	—
201	C₂²×C₄	C₂×#1055	1	3	yes	—
202	C₂²×C₄	C₂×#1055	1	3	yes	—
203	C₂²×C₄	C₂×#1055	1	3	yes	—
204	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
206	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
207	M₁₆	—	2	6	no	—
208	M₁₆	—	2	12	no	—
209	M₁₆	C₂²⋊Dic₁₃	1	12	yes	—
210	C₂×C₈	—	2	6	no	—
211	C₂×C₈	—	2	6	no	—
212	C₂×C₈	—	2	6	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	6	no	—
215	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
217	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
218	C₂×C₈	C₂×C₂²⋊D₁₃	1	12	yes	—
219	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
220	C₁₆	—	26	2	no	—
221	C₁₆	—	26	6	no	—
222	C₁₃	#33334	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	6	no	—
224	C₂³	—	2	6	no	—
225	C₂³	—	4	3	no	—
226	C₂³	C₂²×#1055	1	3	yes	—
227	C₂³	C₂²×#1055	1	1	yes	characteristic
228	C₂³	C₂²×#1055	1	3	yes	—
229	Q₈	—	2	6	no	—
230	Q₈	C₂²⋊#1055	1	4	yes	—
231	D₄	—	2	12	no	—
232	D₄	—	2	6	no	—
233	D₄	C₂²⋊#1055	1	12	yes	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	3	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	3	no	—
239	C₂×C₄	—	2	6	no	—
240	C₂×C₄	—	2	6	no	—
241	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
242	C₂×C₄	C₂²×#1055	1	3	yes	—
243	C₂×C₄	C₂²×#1055	1	3	yes	—
244	C₂×C₄	C₂×#6099	1	6	yes	—
245	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
246	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
247	C₈	—	2	1	no	—
248	C₈	—	2	3	no	—
249	C₈	C₂×C₂²⋊Dic₁₃	1	6	yes	—
250	C₈	C₂²×C₂²⋊D₁₃	1	2	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	3	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	2	3	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	2	3	no	—
258	C₂²	—	4	12	no	—
259	C₂²	#11718801	1	2	yes	complemented
260	C₂²	C₂×C₂²⋊#1055	1	6	yes	—
261	C₂²	C₂³×#1055	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	3	no	—
264	C₄	C₂×C₂²⋊#1055	1	6	yes	—
265	C₄	C₂²×#6099	1	1	yes	characteristic
266	C₄	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	6	no	—
269	C₂	—	4	3	no	—
270	C₂	C₂²×C₂²⋊#1055	1	1	yes	characteristic
271	C₂	C₂×#11718801	1	1	yes	characteristic
272	C₂	#13127939	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂²×C₁₃×#131	C₂
prime radical	C₂²×C₁₃×#41	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×#131	D₁₃
hypocenter	C₁₃	#33334
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#11718801	C₂	semidirect
2. #13116450	C₂	semidirect
3. #13116447	C₂	semidirect
4. C₂×C₂²⋊#6090	C₂	semidirect
15. C₂×#88052	C₂²	semidirect
16. C₂²×#6090	C₂²	semidirect
19. C₂²⋊#6090	C₂²	semidirect
20. C₂²⋊#6090	C₂²	semidirect
144. C₂²×C₁₃	#504	semidirect
222. C₁₃	#33334	semidirect
259. C₂²	#11718801	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718801	2
#13116450	2
#13116447	2
C₂×C₂²⋊#6090	2
C₂²×C₁₃×#131	2
#33334	13