Subgroups — #16789588

Order 3328.

350 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	#13131169	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718217	C₂	1	2	yes	maximal
3	C₄×#88100	C₂	1	1	yes	maximal, characteristic
4	#13129876	C₂	1	2	yes	maximal
5	#13116590	C₂	1	1	yes	maximal, characteristic
6	C₂×#88100	C₄	1	1	yes	characteristic
7	C₂×#88100	C₄	1	1	yes	characteristic
8	C₂×#88100	C₂²	1	1	yes	characteristic
9	C₂×#88100	C₂²	1	1	yes	characteristic
10	C₂×#88098	C₂²	1	2	yes	—
11	C₄²×#261	C₄	1	1	yes	characteristic
12	C₄×D₁₃⋊C₈	C₄	1	1	yes	characteristic
13	C₂×#88038	C₂²	1	2	yes	—
14	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
15	#11718260	C₄	1	2	yes	complemented
16	C₄⋊#6092	C₄	1	2	yes	complemented
17	#11718217	—	2	4	no	—
18	C₁₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
19	#11718093	C₄	1	2	yes	complemented
20	C₄⋊#6090	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×#261	—	2	1	no	—
22	C₂×C₄×#261	—	2	1	no	—
23	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
25	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
29	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×#1057	—	2	1	no	—
31	C₄×#1057	—	2	1	no	—
32	C₂×#6092	C₂×C₄	1	4	yes	—
33	#88100	—	2	2	no	—
34	#88100	—	2	2	no	—
35	#88100	—	2	1	no	—
36	#88100	—	2	1	no	—
37	#88100	D₄	1	2	yes	—
38	#88100	Q₈	1	2	yes	—
39	#88098	—	2	2	no	—
40	#88098	—	2	2	no	—
41	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×#6090	C₂×C₄	1	2	yes	—
46	C₄×#1055	C₂×C₄	1	1	yes	characteristic
47	#88038	—	2	2	no	—
48	#88038	—	2	2	no	—
49	#12658	—	13	1	no	maximal, Sylow
50	C₂²×#261	—	2	1	no	—
51	C₂²×#261	—	2	1	no	—
52	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
53	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×#1057	—	2	1	no	—
56	C₂×#1057	—	2	1	no	—
57	C₄×#261	—	2	1	no	—
58	C₄×#261	—	2	1	no	—
59	C₄×#261	—	4	2	no	—
60	C₄×#261	—	4	2	no	—
61	C₄×#261	C₄⋊C₄	1	2	yes	—
62	C₄×#261	M₁₆	1	2	yes	—
63	D₁₃⋊C₈	—	2	1	no	—
64	D₁₃⋊C₈	—	2	1	no	—
65	D₁₃⋊C₈	C₄⋊C₄	1	2	yes	—
66	D₁₃⋊C₈	M₁₆	1	2	yes	—
67	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₄²	1	1	yes	characteristic
70	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
71	C₄×Dic₁₃	M₁₆	1	1	yes	characteristic
72	C₄×Dic₁₃	M₁₆	1	1	yes	characteristic
73	C₂×#1055	C₄²	1	1	yes	characteristic
74	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
75	C₈×D₁₃	—	2	1	no	—
76	C₈×D₁₃	—	2	1	no	—
77	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
78	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
79	C₈×D₁₃	#41	1	1	yes	characteristic
80	C₈×D₁₃	#41	1	1	yes	characteristic
81	#6092	—	2	4	no	—
82	C₁₃×C₁₆	—	2	2	no	—
83	#6090	—	2	2	no	—
84	#2150	—	13	1	no	—
85	#2150	—	13	1	no	—
86	C₂×#347	—	13	2	no	—
87	C₄×#97	—	13	1	no	—
88	#1375	—	13	2	no	—
89	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×#261	—	2	2	no	—
91	C₂×#261	—	2	2	no	—
92	C₂×#261	—	4	1	no	—
93	C₂×#261	—	4	1	no	—
94	C₂×#261	—	4	1	no	—
95	C₂×#261	—	4	1	no	—
96	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
99	C₂×Dic₁₃	#97	1	1	yes	characteristic
100	C₂×Dic₁₃	#97	1	1	yes	characteristic
101	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
102	C₄×D₁₃	—	2	2	no	—
103	C₄×D₁₃	—	2	2	no	—
104	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₁₃	#117	1	1	yes	characteristic
106	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
108	#1057	—	2	2	no	—
109	#1057	—	2	2	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
113	#1055	—	2	1	no	—
114	#1055	C₄×D₄	1	1	yes	characteristic
115	#1055	C₄×Q₈	1	1	yes	characteristic
116	C₂×M₃₂	—	13	2	no	—
117	C₂×M₃₂	—	13	2	no	—
118	C₂×#97	—	13	1	no	—
119	C₂×#97	—	13	1	no	—
120	C₂×#97	—	13	1	no	—
121	C₂×#97	—	13	1	no	—
122	C₂×C₄×C₈	—	13	1	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₄³	—	13	1	no	—
125	C₄⋊C₁₆	—	13	2	no	—
126	C₄⋊C₁₆	—	13	2	no	—
127	C₄⋊C₁₆	—	13	2	no	—
128	C₄⋊C₁₆	—	13	1	no	—
129	C₄⋊C₁₆	#261	1	1	yes	characteristic, complemented
130	#347	—	26	4	no	—
131	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
132	C₂×D₁₃	—	2	1	no	—
133	C₂×D₁₃	—	2	1	no	—
134	C₂×D₁₃	#423	1	1	yes	characteristic
135	C₂×D₁₃	#432	1	1	yes	characteristic
136	#261	—	4	2	no	—
137	#261	—	4	2	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	#347	1	2	yes	complemented
140	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
142	Dic₁₃	—	2	1	no	—
143	Dic₁₃	#347	1	2	yes	complemented
144	Dic₁₃	#423	1	1	yes	characteristic
145	Dic₁₃	#432	1	1	yes	characteristic
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	1	no	—
148	C₂²×C₈	—	13	1	no	—
149	C₂²×C₈	—	13	1	no	—
150	C₂×C₄²	—	13	1	no	—
151	C₂×C₄²	—	13	1	no	—
152	C₂×C₄²	—	13	1	no	—
153	C₂×C₄²	—	26	1	no	—
154	C₂×C₄²	—	26	1	no	—
155	M₃₂	—	26	2	no	—
156	M₃₂	—	26	2	no	—
157	M₃₂	—	26	2	no	—
158	M₃₂	—	26	2	no	—
159	C₂×C₁₆	—	13	4	no	—
160	C₂×C₁₆	—	13	2	no	—
161	C₂×C₁₆	C₂×#261	1	2	yes	—
162	#97	—	13	2	no	—
163	#97	—	13	2	no	—
164	#97	—	26	2	no	—
165	#97	—	26	1	no	—
166	#97	—	26	1	no	—
167	#97	—	26	2	no	—
168	C₄×C₈	—	13	1	no	—
169	C₄×C₈	—	13	1	no	—
170	C₄×C₈	—	13	1	no	—
171	C₄×C₈	—	26	1	no	—
172	C₄×C₈	—	26	1	no	—
173	C₄×C₈	C₂×#261	1	1	yes	characteristic
174	C₂×C₁₃	#1767	1	1	yes	characteristic
175	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
176	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
177	D₁₃	—	2	1	no	—
178	D₁₃	—	2	1	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	—	13	1	no	—
181	C₂²×C₄	—	13	1	no	—
182	C₂²×C₄	—	26	1	no	—
183	C₂²×C₄	—	26	1	no	—
184	C₂×C₈	—	13	1	no	—
185	C₂×C₈	—	13	1	no	—
186	C₂×C₈	—	13	1	no	—
187	C₂×C₈	—	13	1	no	—
188	C₂×C₈	—	13	1	no	—
189	C₂×C₈	—	13	2	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	26	1	no	—
193	C₂×C₈	—	26	1	no	—
194	C₂×C₈	—	26	1	no	—
195	C₂×C₈	—	26	1	no	—
196	C₂×C₈	—	26	1	no	—
197	C₂×C₈	—	26	1	no	—
198	C₂×C₈	C₄×#261	1	1	yes	characteristic
199	C₂×C₈	C₂²×#261	1	1	yes	characteristic
200	C₄²	—	13	1	no	—
201	C₄²	—	13	1	no	—
202	C₄²	—	13	1	no	—
203	C₄²	—	13	2	no	—
204	C₄²	—	13	2	no	—
205	C₄²	—	26	1	no	—
206	C₄²	—	26	1	no	—
207	C₄²	—	52	2	no	—
208	C₄²	—	52	2	no	—
209	C₄²	C₄×#261	1	1	yes	characteristic
210	C₁₆	—	2	2	no	—
211	C₁₆	—	26	4	no	—
212	C₁₆	—	26	2	no	—
213	C₁₃	#12658	1	1	yes	characteristic, complemented, Sylow
214	C₂³	—	13	1	no	—
215	C₂×C₄	—	13	1	no	—
216	C₂×C₄	—	13	1	no	—
217	C₂×C₄	—	13	1	no	—
218	C₂×C₄	—	13	1	no	—
219	C₂×C₄	—	13	1	no	—
220	C₂×C₄	—	13	1	no	—
221	C₂×C₄	—	13	1	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	2	no	—
224	C₂×C₄	—	26	2	no	—
225	C₂×C₄	—	26	2	no	—
226	C₂×C₄	—	52	1	no	—
227	C₂×C₄	—	52	1	no	—
228	C₂×C₄	—	52	1	no	—
229	C₂×C₄	—	52	1	no	—
230	C₂×C₄	#88100	1	1	yes	characteristic
231	C₂×C₄	#88100	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	13	1	no	—
235	C₈	—	13	1	no	—
236	C₈	—	26	2	no	—
237	C₈	—	26	2	no	—
238	C₈	—	26	1	no	—
239	C₈	D₄×#261	1	1	yes	characteristic
240	C₈	Q₈×#261	1	1	yes	characteristic
241	C₂²	—	13	1	no	—
242	C₂²	—	13	1	no	—
243	C₂²	—	26	1	no	—
244	C₂²	—	26	1	no	—
245	C₂²	C₂×#88100	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	13	2	no	—
248	C₄	—	13	1	no	—
249	C₄	—	13	1	no	—
250	C₄	—	26	1	no	—
251	C₄	—	52	2	no	—
252	C₄	—	52	2	no	—
253	C₄	#11718217	1	2	yes	complemented
254	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
255	C₄	M₁₆×#261	1	1	yes	characteristic
256	C₂	—	26	1	no	—
257	C₂	—	26	1	no	—
258	C₂	#13130623	1	1	yes	characteristic
259	C₂	C₂×#11718217	1	1	yes	characteristic
260	C₂	#13131051	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×C₄⋊C₁₆	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂²×D₁₃	C₂×M₁₆
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₄⋊C₁₆	#261
hypocenter	C₁₃	#12658
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
15. #11718260	C₄	semidirect
16. C₄⋊#6092	C₄	semidirect
18. C₁₃×C₄⋊C₁₆	C₄	semidirect
19. #11718093	C₄	semidirect
20. C₄⋊#6090	C₄	semidirect
129. C₄⋊C₁₆	#261	semidirect
139. C₄×C₁₃	#347	semidirect
143. Dic₁₃	#347	semidirect
213. C₁₃	#12658	semidirect
253. C₄	#11718217	semidirect

Maximal subgroups

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

subgroup	index
#13131169	2
C₂×#11718217	2
C₄×#88100	2
#13129876	2
#13116590	2
#12658	13