Subgroups — #16016091

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 324 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	1	yes	maximal, characteristic, complemented
2	C₂³×#6090	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116662	C₂	1	4	yes	maximal, complemented
4	D₄×#6090	C₂	1	4	yes	maximal, complemented
5	#13116595	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116450	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116447	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116433	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2929	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#525	C₄	1	1	yes	characteristic
13	C₁₃×#435	C₄	1	4	yes	—
14	C₈×D₄×C₁₃	C₂²	1	4	yes	—
15	C₁₃×#424	C₄	1	1	yes	characteristic
16	C₁₃×#408	C₄	1	1	yes	characteristic
17	C₁₃×#407	C₄	1	1	yes	characteristic
18	C₁₃×#405	C₂²	1	1	yes	characteristic
19	#11718801	—	2	4	no	—
20	C₂×#88052	C₂²	1	1	yes	characteristic
21	C₂×#88052	C₂²	1	1	yes	characteristic
22	C₂×#88052	C₂²	1	2	yes	complemented
23	C₂²×#6090	—	2	4	no	—
24	C₂²×#6090	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	2	yes	—
26	C₂²×#6090	C₂²	1	2	yes	—
27	C₂²×#6090	C₂²	1	1	yes	characteristic
28	C₂²×#6090	C₂²	1	2	yes	complemented
29	C₂²⋊#6090	C₂²	1	4	yes	—
30	C₂²⋊#6090	C₂²	1	4	yes	complemented
31	C₄⋊#6090	C₂²	1	4	yes	complemented
32	#11718053	C₂²	1	2	yes	complemented
33	C₄×#6090	C₂²	1	2	yes	complemented
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₁₃×#131	—	2	2	no	—
37	C₁₃×#131	—	2	2	no	—
38	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₃	—	2	2	no	—
42	C₂²×C₈×C₁₃	—	2	2	no	—
43	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×#123	C₈	1	2	yes	—
49	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
50	C₁₃×#121	C₈	1	2	yes	—
51	C₁₃×#120	C₈	1	2	yes	—
52	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
53	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
54	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
55	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
56	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
57	C₁₃×#97	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	#88052	—	2	2	no	—
60	#88052	—	2	2	no	—
61	#88052	—	2	4	no	—
62	C₂×#6090	—	2	2	no	—
63	C₂×#6090	—	2	4	no	—
64	C₂×#6090	—	2	4	no	—
65	C₂×#6090	—	2	4	no	—
66	C₂×#6090	D₄	1	4	yes	complemented
67	C₂×#6090	C₂³	1	2	yes	—
68	C₂×#6090	C₂³	1	2	yes	—
69	C₂×#6090	C₂³	1	2	yes	—
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#33432	—	13	1	no	maximal, Sylow
73	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₁₃×#41	—	2	2	no	—
75	C₁₃×#41	—	2	2	no	—
76	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
78	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
79	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₁₃×M₁₆	—	2	2	no	—
88	C₁₃×M₁₆	—	2	2	no	—
89	C₁₃×M₁₆	—	2	4	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	4	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₁₃	#41	1	2	yes	—
102	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
104	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
105	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
106	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
107	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
108	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
109	#6090	—	2	2	no	—
110	#6090	—	2	2	no	—
111	#6090	—	2	2	no	—
112	#6090	C₂×D₄	1	2	yes	—
113	#6090	C₂×D₄	1	2	yes	—
114	C₂×#504	—	13	1	no	—
115	C₂³×C₁₆	—	13	1	no	—
116	#2929	D₁₃	1	1	yes	characteristic
117	#2168	—	13	4	no	—
118	D₄×C₁₆	—	13	4	no	—
119	#2151	—	13	1	no	—
120	#2113	—	13	1	no	—
121	#2112	—	13	1	no	—
122	#2108	—	13	1	no	—
123	C₂³×C₁₃	—	2	2	no	—
124	C₂³×C₁₃	—	2	4	no	—
125	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
126	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
128	Q₈×C₁₃	—	2	2	no	—
129	D₄×C₁₃	—	2	4	no	—
130	D₄×C₁₃	—	2	8	no	—
131	D₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
140	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
142	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#131	1	2	yes	—
144	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	C₄×D₄	1	2	yes	—
149	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
152	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
153	#525	Dic₁₃	1	1	yes	characteristic
154	#504	—	26	4	no	—
155	C₂×M₃₂	—	13	1	no	—
156	C₂×M₃₂	—	13	1	no	—
157	C₂×M₃₂	—	13	2	no	—
158	C₂²×C₁₆	—	13	1	no	—
159	C₂²×C₁₆	—	13	1	no	—
160	C₂²×C₁₆	—	13	2	no	—
161	C₂²×C₁₆	—	13	2	no	—
162	C₂²×C₁₆	—	13	2	no	—
163	C₂²×C₁₆	—	26	4	no	—
164	#435	Dic₁₃	1	4	yes	—
165	C₈×D₄	C₂×D₁₃	1	4	yes	—
166	#424	Dic₁₃	1	1	yes	characteristic
167	#408	Dic₁₃	1	1	yes	characteristic
168	#407	Dic₁₃	1	1	yes	characteristic
169	#405	C₂×D₁₃	1	1	yes	characteristic
170	C₄⋊C₁₆	—	13	4	no	—
171	C₂²⋊C₁₆	—	13	4	no	—
172	C₂²⋊C₁₆	—	13	4	no	—
173	#346	—	13	2	no	—
174	C₄×C₁₆	—	13	2	no	—
175	C₂²×C₁₃	—	2	8	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	—	2	2	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	4	no	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	C₈×D₄	1	2	yes	—
182	C₂²×C₁₃	#504	1	4	yes	complemented
183	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
184	C₄×C₁₃	—	2	4	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	C₈×D₄	1	2	yes	—
188	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
190	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
191	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
192	#131	—	2	2	no	—
193	#131	—	2	2	no	—
194	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
195	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
196	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
201	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
202	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
203	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	#123	#1055	1	2	yes	—
205	C₂²⋊Q₈	#1055	1	2	yes	—
206	#121	#1055	1	2	yes	—
207	#120	#1055	1	2	yes	—
208	C₄×D₄	C₂×Dic₁₃	1	4	yes	—
209	#117	C₂×Dic₁₃	1	1	yes	characteristic
210	M₃₂	—	26	4	no	—
211	M₃₂	—	26	2	no	—
212	M₃₂	—	26	2	no	—
213	C₂×C₁₆	—	13	4	no	—
214	C₂×C₁₆	—	13	1	no	—
215	C₂×C₁₆	—	13	1	no	—
216	C₂×C₁₆	—	13	2	no	—
217	C₂×C₁₆	—	13	2	no	—
218	C₂×C₁₆	—	13	2	no	—
219	C₂×C₁₆	—	26	4	no	—
220	C₂×C₁₆	—	26	4	no	—
221	C₂×C₁₆	—	26	4	no	—
222	C₂×C₁₆	—	26	2	no	—
223	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
224	C₂²⋊C₈	C₂×Dic₁₃	1	4	yes	—
225	C₂²⋊C₈	C₂×Dic₁₃	1	4	yes	—
226	#97	C₂×Dic₁₃	1	2	yes	—
227	C₄×C₈	C₂²×D₁₃	1	2	yes	—
228	C₂×C₁₃	—	2	2	no	—
229	C₂×C₁₃	—	2	4	no	—
230	C₂×C₁₃	—	4	2	no	—
231	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
232	C₂×C₁₃	C₂×#504	1	2	yes	—
233	C₂⁴	C₂×#1055	1	1	yes	characteristic
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
237	C₂×D₄	C₂×#1055	1	4	yes	—
238	C₂×D₄	C₂×#1055	1	2	yes	—
239	C₂×D₄	C₂×#1055	1	1	yes	characteristic
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×#1055	1	2	yes	—
243	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
244	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
247	M₁₆	—	2	2	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	D₄×D₁₃	1	2	yes	—
257	C₂×C₈	#6129	1	2	yes	—
258	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
259	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
260	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
261	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂×#1055	1	2	yes	—
264	C₄⋊C₄	C₂×#1055	1	4	yes	—
265	C₂²⋊C₄	C₂×#1055	1	2	yes	—
266	C₂²⋊C₄	C₂×#1055	1	4	yes	—
267	C₂²⋊C₄	C₂×#1055	1	4	yes	—
268	C₄²	C₂²×Dic₁₃	1	2	yes	—
269	C₁₆	—	13	2	no	—
270	C₁₆	—	13	2	no	—
271	C₁₆	—	26	2	no	—
272	C₁₆	—	26	2	no	—
273	C₁₆	—	26	2	no	—
274	C₁₃	#33432	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	2	2	no	—
276	C₂³	—	2	4	no	—
277	C₂³	C₂²×#1055	1	2	yes	—
278	C₂³	C₂²×#1055	1	1	yes	characteristic
279	C₂³	C₂²×#1055	1	2	yes	—
280	Q₈	—	2	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	2	8	no	—
283	D₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	C₂²×#1055	1	4	yes	—
291	C₂×C₄	C₂²×#1055	1	2	yes	—
292	C₂×C₄	C₂²×#1055	1	2	yes	—
293	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
294	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
295	C₂×C₄	#88202	1	2	yes	—
296	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	D₄×Dic₁₃	1	2	yes	—
301	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
302	C₈	C₂×#6129	1	1	yes	characteristic
303	C₂²	—	2	8	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	D₄×#1055	1	2	yes	—
310	C₂²	#11718801	1	4	yes	complemented
311	C₂²	C₂³×#1055	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	D₄×#1055	1	2	yes	—
316	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
317	C₄	C₂×#88202	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	4	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×D₄×#1055	1	1	yes	characteristic
322	C₂	C₂×#11718801	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2929	C₂
prime radical	C₁₃×#525	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2929	D₁₃
hypocenter	C₁₃	#33432
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. C₂×#11718801	C₂	semidirect
2. C₂³×#6090	C₂	semidirect
3. #13116662	C₂	semidirect
4. D₄×#6090	C₂	semidirect
5. #13116595	C₂	semidirect
6. #13116450	C₂	semidirect
7. #13116447	C₂	semidirect
8. #13116433	C₂	semidirect
22. C₂×#88052	C₂²	semidirect
28. C₂²×#6090	C₂²	semidirect
30. C₂²⋊#6090	C₂²	semidirect
31. C₄⋊#6090	C₂²	semidirect
32. #11718053	C₂²	semidirect
33. C₄×#6090	C₂²	semidirect
66. C₂×#6090	D₄	semidirect
182. C₂²×C₁₃	#504	semidirect
274. C₁₃	#33432	semidirect
310. 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
C₂³×#6090	2
#13116662	2
D₄×#6090	2
#13116595	2
#13116450	2
#13116447	2
#13116433	2
C₁₃×#2929	2
#33432	13