Subgroups — #16732583

Order 3328.

1890 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719175	C₂	1	2	yes	maximal
2	#13125388	C₂	1	8	yes	maximal, complemented
3	#13125382	C₂	1	8	yes	maximal, complemented
4	#13125360	C₂	1	4	yes	maximal, complemented
5	#13125350	C₂	1	4	yes	maximal, complemented
6	#13123426	C₂	1	4	yes	maximal
7	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	4	yes	—
11	C₂×#88178	—	2	4	no	—
12	#11719175	—	2	4	no	—
13	#11719175	—	2	4	no	—
14	#11719175	C₂²	1	4	yes	—
15	#11719175	C₂²	1	4	yes	—
16	#11719175	C₂²	1	4	yes	—
17	#11719175	C₂²	1	4	yes	—
18	C₂×#88124	C₂²	1	4	yes	—
19	C₂×#88123	C₂²	1	4	yes	—
20	C₂×C₄×Dic₂₆	—	2	4	no	—
21	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
22	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
23	C₂×C₁₃×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
25	#11718782	C₂²	1	32	yes	—
26	#11718767	C₂²	1	16	yes	—
27	#11718363	C₂²	1	32	yes	complemented
28	#11718344	C₂²	1	16	yes	complemented
29	#11718343	C₂²	1	8	yes	complemented
30	#11718341	C₂²	1	8	yes	complemented
31	C₂³×C₄×C₁₃	D₄	1	2	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₂₆	—	2	4	no	—
34	C₂×C₁₃×M₁₆	Q₈	1	16	yes	complemented
35	C₂×C₁₃×M₁₆	C₂³	1	8	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₁₃	D₄	1	4	yes	—
38	C₂×C₄²×C₁₃	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
44	#88180	—	2	4	no	—
45	#88180	—	2	4	no	—
46	#88180	C₂³	1	8	yes	—
47	#88180	C₂³	1	8	yes	—
48	C₂×C₄⋊Dic₁₃	—	2	4	no	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
50	#88178	—	2	8	no	—
51	#88178	—	2	8	no	—
52	#88178	—	2	8	no	—
53	#88178	—	2	8	no	—
54	C₂×#6101	—	2	8	no	—
55	C₂×C₄×Dic₁₃	—	2	4	no	—
56	#88124	—	2	8	no	—
57	#88124	C₂³	1	8	yes	—
58	#88124	C₂³	1	8	yes	—
59	#88123	—	2	4	no	—
60	#88123	—	2	4	no	—
61	#88123	C₂³	1	8	yes	—
62	#88123	C₂³	1	8	yes	—
63	C₄×Dic₂₆	—	2	8	no	—
64	C₄×Dic₂₆	—	2	8	no	—
65	C₄×Dic₂₆	—	2	8	no	—
66	C₄×Dic₂₆	—	2	8	no	—
67	C₁₃×#97	C₂³	1	8	yes	—
68	C₄×C₈×C₁₃	C₂³	1	4	yes	—
69	C₄×C₈×C₁₃	C₂³	1	4	yes	—
70	#88058	C₂³	1	32	yes	—
71	#88057	C₂³	1	32	yes	—
72	#60663	—	13	1	no	maximal, Sylow
73	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	—	4	4	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
82	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₁₃	—	2	8	no	—
85	C₂×Dic₂₆	—	2	8	no	—
86	C₂×Dic₂₆	—	2	8	no	—
87	C₂×Dic₂₆	—	4	16	no	—
88	C₁₃×M₁₆	C₂×Q₈	1	32	yes	complemented
89	C₂×C₈×C₁₃	C₂×Q₈	1	16	yes	complemented
90	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
91	C₄²×C₁₃	C₂×D₄	1	2	yes	—
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃	C₂×D₄	1	4	yes	—
94	C₄²×C₁₃	C₂×D₄	1	4	yes	—
95	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₄²×C₁₃	C₂⁴	1	2	yes	—
98	C₂²⋊Dic₁₃	—	2	8	no	—
99	C₂²⋊Dic₁₃	—	2	8	no	—
100	C₄⋊Dic₁₃	—	2	8	no	—
101	C₄⋊Dic₁₃	—	2	8	no	—
102	C₄⋊Dic₁₃	—	2	8	no	—
103	C₄⋊Dic₁₃	C₂⁴	1	16	yes	—
104	#6101	—	2	16	no	—
105	#6101	—	2	16	no	—
106	C₄×Dic₁₃	—	2	8	no	—
107	C₄×Dic₁₃	—	2	8	no	—
108	C₂×#533	—	13	2	no	—
109	#3166	—	13	8	no	—
110	#3165	—	13	8	no	—
111	#3162	—	13	4	no	—
112	#3160	—	13	4	no	—
113	#2911	—	13	4	no	—
114	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
117	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
124	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
131	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
132	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	16	no	—
134	C₂×Dic₁₃	—	4	16	no	—
135	Dic₂₆	—	4	16	no	—
136	Dic₂₆	—	4	16	no	—
137	C₈×C₁₃	C₂²×Q₈	1	16	yes	—
138	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
139	#533	—	13	4	no	—
140	#533	—	13	4	no	—
141	#533	—	13	4	no	—
142	#533	—	13	4	no	—
143	#533	—	26	4	no	—
144	#533	—	26	4	no	—
145	C₂×C₄⋊Q₈	—	13	4	no	—
146	C₂×#125	—	13	4	no	—
147	C₂×C₂²⋊Q₈	—	26	4	no	—
148	C₂×C₄×Q₈	—	26	4	no	—
149	C₂×#117	—	13	4	no	—
150	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
151	#501	—	13	32	no	—
152	#500	—	13	8	no	—
153	#499	—	13	16	no	—
154	#498	—	13	8	no	—
155	#428	—	13	32	no	—
156	#427	—	13	16	no	—
157	C₄×M₁₆	C₂×D₁₃	1	4	yes	—
158	C₄×M₁₆	C₂×D₁₃	1	4	yes	—
159	C₂×#97	C₂×D₁₃	1	2	yes	—
160	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	4	4	no	—
164	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
165	C₂²×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
166	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
167	C₄×C₁₃	—	2	4	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
170	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
171	C₄×C₁₃	#575	1	4	yes	—
172	C₄×C₁₃	C₂³×Q₈	1	2	yes	—
173	Dic₁₃	—	4	16	no	—
174	C₂²×Q₈	—	26	4	no	—
175	C₂³×C₄	D₅₂	1	2	yes	—
176	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×M₁₆	Dic₂₆	1	16	yes	complemented
178	C₂×M₁₆	C₂²×D₁₃	1	8	yes	—
179	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
180	C₄⋊Q₈	—	13	8	no	—
181	C₄⋊Q₈	—	13	8	no	—
182	C₄⋊Q₈	—	26	4	no	—
183	C₄⋊Q₈	—	26	4	no	—
184	#125	—	13	8	no	—
185	#125	—	13	8	no	—
186	#125	—	26	8	no	—
187	C₂²⋊Q₈	—	26	8	no	—
188	C₂²⋊Q₈	—	26	8	no	—
189	C₂²⋊Q₈	—	26	8	no	—
190	C₂²⋊Q₈	—	26	8	no	—
191	C₄×Q₈	—	26	8	no	—
192	C₄×Q₈	—	26	8	no	—
193	C₄×Q₈	—	26	8	no	—
194	C₄×Q₈	—	26	8	no	—
195	#117	—	13	8	no	—
196	#117	—	13	8	no	—
197	#117	—	26	4	no	—
198	#117	—	26	4	no	—
199	C₂×C₄⋊C₄	—	13	8	no	—
200	C₂×C₄⋊C₄	—	26	8	no	—
201	C₂×C₄⋊C₄	—	26	4	no	—
202	C₂×C₂²⋊C₄	—	26	4	no	—
203	C₂×C₄²	—	26	4	no	—
204	C₂×C₄²	D₅₂	1	4	yes	—
205	C₂×C₄²	D₅₂	1	2	yes	—
206	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
207	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
208	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
210	#107	—	13	32	no	—
211	#106	—	13	32	no	—
212	#97	C₂²×D₁₃	1	8	yes	—
213	C₄×C₈	C₂²×D₁₃	1	4	yes	—
214	C₄×C₈	C₂²×D₁₃	1	4	yes	—
215	C₂×C₁₃	—	2	4	no	—
216	C₂×C₁₃	—	2	2	no	—
217	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
218	C₂×C₁₃	C₂×#575	1	2	yes	—
219	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
220	C₂×Q₈	—	26	8	no	—
221	C₂×Q₈	—	26	8	no	—
222	C₂×Q₈	—	52	16	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	26	8	no	—
225	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
226	C₂²×C₄	C₂×D₅₂	1	4	yes	—
227	C₂²×C₄	C₂×D₅₂	1	2	yes	—
228	C₂²×C₄	C₂×D₅₂	1	4	yes	—
229	C₂²×C₄	C₂×D₅₂	1	2	yes	—
230	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₅₂	1	2	yes	—
232	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
233	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
234	M₁₆	C₂×Dic₂₆	1	32	yes	complemented
235	C₂×C₈	C₂×Dic₂₆	1	16	yes	complemented
236	C₂×C₈	C₂³×D₁₃	1	8	yes	—
237	C₄⋊C₄	—	13	16	no	—
238	C₄⋊C₄	—	26	8	no	—
239	C₄⋊C₄	—	26	16	no	—
240	C₄⋊C₄	—	26	16	no	—
241	C₄⋊C₄	—	26	8	no	—
242	C₄⋊C₄	—	26	8	no	—
243	C₂²⋊C₄	—	26	8	no	—
244	C₂²⋊C₄	—	26	8	no	—
245	C₄²	—	26	8	no	—
246	C₄²	—	26	8	no	—
247	C₄²	C₂×D₅₂	1	4	yes	—
248	C₄²	C₂×D₅₂	1	4	yes	—
249	C₄²	C₂×D₅₂	1	2	yes	—
250	C₄²	C₂×D₅₂	1	2	yes	—
251	C₄²	C₂³×D₁₃	1	1	yes	characteristic
252	C₄²	C₂³×D₁₃	1	1	yes	characteristic
253	C₄²	C₂³×D₁₃	1	2	yes	—
254	C₁₃	#60663	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	4	2	no	—
256	C₂³	#88123	1	4	yes	—
257	C₂³	C₂²×D₅₂	1	1	yes	characteristic
258	C₂³	C₂²×D₅₂	1	2	yes	—
259	Q₈	—	52	16	no	—
260	Q₈	—	52	16	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	26	16	no	—
266	C₂×C₄	—	52	16	no	—
267	C₂×C₄	#88123	1	4	yes	—
268	C₂×C₄	#88123	1	8	yes	—
269	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
270	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
271	C₂×C₄	C₂²×D₅₂	1	2	yes	—
272	C₂×C₄	C₂²×D₅₂	1	4	yes	—
273	C₂×C₄	C₂²×D₅₂	1	2	yes	—
274	C₂×C₄	C₂²×D₅₂	1	2	yes	—
275	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
276	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
277	C₈	C₂²×Dic₂₆	1	16	yes	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	C₂×#88123	1	2	yes	—
282	C₂²	C₂×#88123	1	4	yes	—
283	C₂²	C₂³×D₅₂	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	52	16	no	—
287	C₄	C₂×#88123	1	4	yes	—
288	C₄	C₂×#88123	1	2	yes	—
289	C₄	#11719453	1	4	yes	—
290	C₄	C₂³×Dic₂₆	1	2	yes	—
291	C₂	—	2	4	no	—
292	C₂	—	2	2	no	—
293	C₂	C₂²×#88123	1	1	yes	characteristic
294	C₂	C₂×#11719453	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂³×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#60663
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13125388	C₂	semidirect
3. #13125382	C₂	semidirect
4. #13125360	C₂	semidirect
5. #13125350	C₂	semidirect
27. #11718363	C₂²	semidirect
28. #11718344	C₂²	semidirect
29. #11718343	C₂²	semidirect
30. #11718341	C₂²	semidirect
34. C₂×C₁₃×M₁₆	Q₈	semidirect
88. C₁₃×M₁₆	C₂×Q₈	semidirect
89. C₂×C₈×C₁₃	C₂×Q₈	semidirect
177. C₂×M₁₆	Dic₂₆	semidirect
234. M₁₆	C₂×Dic₂₆	semidirect
235. C₂×C₈	C₂×Dic₂₆	semidirect
254. C₁₃	#60663	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719175	2
#13125388	2
#13125382	2
#13125360	2
#13125350	2
#13123426	2
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