Subgroups — #16017218

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718856	C₂	1	2	yes	maximal
2	C₂×#11718360	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116886	C₂	1	4	yes	maximal, complemented
4	#13116881	C₂	1	4	yes	maximal, complemented
5	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116555	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×#41	—	2	1	no	—
10	C₂²×Dic₅₂	—	2	1	no	—
11	C₂²×Dic₅₂	C₂²	1	2	yes	—
12	C₂×#88123	—	2	1	no	—
13	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
14	C₄×C₁₃×SD₁₆	—	2	2	no	—
15	C₄×C₁₃×D₈	C₂²	1	2	yes	—
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#416	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718856	—	2	8	no	—
20	C₂×#88069	C₂²	1	4	yes	—
21	C₂×#88067	C₂²	1	4	yes	—
22	C₂×#88052	C₂²	1	2	yes	—
23	#11718360	—	2	2	no	—
24	#11718360	C₂²	1	4	yes	—
25	#11718084	C₂²	1	8	yes	—
26	C₄⋊#6090	C₂²	1	4	yes	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×Dic₂₆	—	2	2	no	—
29	C₁₃×#135	—	2	2	no	—
30	C₁₃×#135	D₄	1	4	yes	—
31	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×SD₁₆	—	2	1	no	—
33	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₃	—	2	2	no	—
37	C₄×D₄×C₁₃	—	2	2	no	—
38	C₄×D₄×C₁₃	—	4	2	no	—
39	C₁₃×#117	—	2	1	no	—
40	C₁₃×#117	—	4	1	no	—
41	C₂×C₄²×C₁₃	—	4	1	no	—
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₃	—	4	2	no	—
44	C₂×Dic₅₂	—	2	4	no	—
45	C₂×Dic₅₂	—	2	4	no	—
46	C₂×Dic₅₂	—	2	4	no	—
47	C₂×Dic₅₂	—	2	2	no	—
48	C₂×Dic₅₂	—	4	4	no	—
49	C₂×Dic₅₂	C₂³	1	4	yes	—
50	#88123	—	2	4	no	—
51	#88123	—	4	1	no	—
52	#88123	—	4	1	no	—
53	C₁₃×#107	C₂³	1	2	yes	—
54	C₁₃×#106	—	2	1	no	—
55	C₁₃×#103	—	2	2	no	—
56	C₁₃×#102	—	2	2	no	—
57	C₄×C₈×C₁₃	D₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	#88069	—	2	8	no	—
60	#88067	—	2	8	no	—
61	#88052	—	2	4	no	—
62	C₂×#6090	C₂³	1	4	yes	—
63	#33583	—	13	1	no	maximal, Sylow
64	C₁₃×#41	—	2	4	no	—
65	C₁₃×#41	—	4	2	no	—
66	C₂×Q₈×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	—	2	1	no	—
68	C₂×D₄×C₁₃	—	4	1	no	—
69	C₂²×C₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	D₈	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	4	2	no	—
74	C₂×Dic₂₆	—	2	4	no	—
75	C₂×Dic₂₆	—	4	4	no	—
76	C₂×Dic₂₆	—	4	4	no	—
77	C₂×Dic₂₆	—	4	2	no	—
78	C₁₃×Q₁₆	—	2	1	no	—
79	C₁₃×Q₁₆	C₂×D₄	1	2	yes	—
80	C₁₃×SD₁₆	—	2	4	no	—
81	C₁₃×D₈	—	2	1	no	—
82	C₁₃×D₈	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₁₃×C₂²⋊C₄	—	4	1	no	—
92	C₁₃×C₂²⋊C₄	—	4	2	no	—
93	C₄²×C₁₃	—	4	1	no	—
94	C₄²×C₁₃	—	4	2	no	—
95	C₄²×C₁₃	D₈	1	1	yes	characteristic
96	C₄²×C₁₃	D₈	1	1	yes	characteristic
97	C₄²×C₁₃	C₂×D₄	1	2	yes	—
98	C₄⋊Dic₁₃	—	4	4	no	—
99	C₄⋊Dic₁₃	—	4	4	no	—
100	Dic₅₂	—	2	4	no	—
101	Dic₅₂	—	2	4	no	—
102	Dic₅₂	—	4	8	no	—
103	Dic₅₂	—	4	4	no	—
104	#6090	—	2	4	no	—
105	C₂×#510	—	13	2	no	—
106	C₂×#494	—	13	1	no	—
107	C₄×#135	D₁₃	1	1	yes	characteristic
108	#2209	—	13	4	no	—
109	#2208	—	13	4	no	—
110	#2150	—	13	1	no	—
111	#2141	—	13	2	no	—
112	C₂³×C₁₃	—	4	1	no	—
113	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₁₃	—	2	2	no	—
115	Q₈×C₁₃	—	4	1	no	—
116	D₄×C₁₃	—	2	2	no	—
117	D₄×C₁₃	—	4	4	no	—
118	D₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	4	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	8	4	no	—
124	C₂×C₄×C₁₃	#121	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	8	4	no	—
132	Dic₂₆	—	4	4	no	—
133	Dic₂₆	—	4	4	no	—
134	Dic₂₆	—	8	4	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	#121	1	2	yes	—
137	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
140	C₂²×Q₁₆	—	13	2	no	—
141	C₂²×Q₁₆	—	26	1	no	—
142	C₂×C₄⋊Q₈	—	26	1	no	—
143	C₄×#41	—	2	1	no	—
144	#510	—	26	8	no	—
145	C₂×Q₃₂	—	13	4	no	—
146	C₂×SD₃₂	—	13	4	no	—
147	C₂×M₃₂	—	13	2	no	—
148	#494	—	13	4	no	—
149	#494	—	26	2	no	—
150	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
151	C₄×SD₁₆	—	2	2	no	—
152	C₄×D₈	C₂×D₁₃	1	2	yes	—
153	#427	C₂×D₁₃	1	1	yes	characteristic
154	#416	—	2	1	no	—
155	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
156	C₄⋊C₁₆	—	13	4	no	—
157	#358	—	13	8	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	4	2	no	—
160	C₂²×C₁₃	—	8	2	no	—
161	C₂²×C₁₃	#459	1	2	yes	—
162	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	—	8	2	no	—
166	C₄×C₁₃	#459	1	2	yes	—
167	C₄×C₁₃	#510	1	4	yes	complemented
168	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
169	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	Dic₁₃	—	8	4	no	—
171	C₂×#41	—	2	1	no	—
172	C₂²×Q₈	—	26	2	no	—
173	#135	—	2	2	no	—
174	#135	D₅₂	1	4	yes	—
175	C₂×Q₁₆	—	13	4	no	—
176	C₂×Q₁₆	—	26	4	no	—
177	C₂×Q₁₆	—	26	4	no	—
178	C₂×Q₁₆	—	26	2	no	—
179	C₂×Q₁₆	—	26	4	no	—
180	C₂×Q₁₆	—	52	4	no	—
181	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×SD₁₆	—	2	1	no	—
183	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₄⋊Q₈	—	26	4	no	—
187	C₄⋊Q₈	—	52	1	no	—
188	C₄⋊Q₈	—	52	1	no	—
189	C₄×Q₈	—	2	2	no	—
190	C₄×D₄	—	2	2	no	—
191	C₄×D₄	—	4	2	no	—
192	#117	—	2	1	no	—
193	#117	—	4	1	no	—
194	C₂×C₄⋊C₄	—	52	2	no	—
195	C₂×C₄²	—	4	1	no	—
196	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
197	Q₃₂	—	26	8	no	—
198	SD₃₂	—	26	8	no	—
199	M₃₂	—	26	4	no	—
200	C₂×C₁₆	—	13	4	no	—
201	#107	C₂²×D₁₃	1	2	yes	—
202	#106	—	2	1	no	—
203	#103	—	2	2	no	—
204	#102	—	2	2	no	—
205	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
206	C₄×C₈	C₂²×D₁₃	1	2	yes	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	—	8	2	no	—
209	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
210	C₂×C₁₃	C₂×#510	1	2	yes	—
211	#41	—	2	4	no	—
212	#41	—	4	2	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	26	4	no	—
215	C₂×Q₈	—	52	2	no	—
216	C₂×Q₈	—	52	4	no	—
217	C₂×Q₈	—	52	4	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	52	2	no	—
223	C₂²×C₄	#6104	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	Q₁₆	—	2	1	no	—
226	Q₁₆	—	26	4	no	—
227	Q₁₆	—	26	4	no	—
228	Q₁₆	—	52	8	no	—
229	Q₁₆	—	52	4	no	—
230	Q₁₆	C₂×D₅₂	1	2	yes	—
231	SD₁₆	—	2	4	no	—
232	D₈	—	2	1	no	—
233	D₈	C₂×D₅₂	1	2	yes	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂×D₅₂	1	2	yes	—
236	C₂×C₈	#6127	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	52	4	no	—
243	C₄⋊C₄	—	52	4	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₄²	#6104	1	1	yes	characteristic
249	C₄²	#6104	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
251	C₁₆	—	26	4	no	—
252	C₁₃	#33583	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	4	1	no	—
254	C₂³	C₂×#6104	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	4	1	no	—
257	Q₈	—	52	4	no	—
258	Q₈	—	52	4	no	—
259	Q₈	—	104	4	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	52	4	no	—
269	C₂×C₄	—	104	4	no	—
270	C₂×C₄	#88184	1	2	yes	—
271	C₂×C₄	C₂×#6104	1	2	yes	—
272	C₂×C₄	C₂×#6104	1	1	yes	characteristic
273	C₂×C₄	C₂×#6104	1	2	yes	—
274	C₂×C₄	#88203	1	2	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	#88184	1	2	yes	—
278	C₈	C₂²×D₅₂	1	1	yes	characteristic
279	C₈	C₂×#6127	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	#11718676	1	2	yes	—
284	C₂²	C₂²×#6104	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	2	no	—
288	C₄	—	104	4	no	—
289	C₄	#11718676	1	2	yes	—
290	C₄	#11718856	1	4	yes	complemented
291	C₄	C₂×#88184	1	1	yes	characteristic
292	C₄	C₂×#88203	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	8	2	no	—
295	C₂	C₂×#11718676	1	1	yes	characteristic
296	C₂	C₂×#11718856	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×C₁₃×#135	C₂²
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	#33583
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#11718360	C₂	semidirect
3. #13116886	C₂	semidirect
4. #13116881	C₂	semidirect
5. #13116592	C₂	semidirect
6. #13116555	C₂	semidirect
167. C₄×C₁₃	#510	semidirect
252. C₁₃	#33583	semidirect
290. C₄	#11718856	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718856	2
C₂×#11718360	2
#13116886	2
#13116881	2
#13116592	2
#13116555	2
C₄×C₁₃×#135	2
#33583	13