Subgroups — #16018912

Order 3328.

688 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117103	C₂	1	2	yes	maximal, complemented
3	#13117079	C₂	1	2	yes	maximal, complemented
4	#13116876	C₂	1	2	yes	maximal, complemented
5	#13116871	C₂	1	2	yes	maximal, complemented
6	#13116764	C₂	1	2	yes	maximal, complemented
7	#13116747	C₂	1	2	yes	maximal, complemented
8	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3168	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#529	—	2	1	no	—
11	C₂²×D₁₀₄	—	2	1	no	—
12	C₂²×D₁₀₄	C₂²	1	2	yes	—
13	C₂×#88128	—	2	1	no	—
14	C₁₃×#499	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#495	C₂²	1	1	yes	characteristic, complemented
16	C₁₃×#488	—	2	1	no	—
17	C₁₃×#467	C₂²	1	1	yes	characteristic
18	C₁₃×#466	C₂²	1	1	yes	characteristic
19	C₁₃×#465	—	2	1	no	—
20	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₄×C₁₃×SD₁₆	—	2	1	no	—
22	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₂×#88068	C₂²	1	4	yes	complemented
25	C₂×#88066	C₂²	1	4	yes	complemented
26	#11718356	—	2	1	no	—
27	#11718356	—	2	1	no	—
28	#11718356	C₂²	1	2	yes	—
29	#11718356	C₂²	1	2	yes	—
30	C₂²⋊#6090	C₂²	1	2	yes	—
31	#11718083	C₂²	1	4	yes	complemented
32	#11718083	C₂²	1	4	yes	complemented
33	C₄⋊#6090	C₂²	1	2	yes	complemented
34	#11718053	C₂²	1	2	yes	complemented
35	C₂²×D₅₂	—	2	2	no	—
36	C₂²×D₅₂	—	4	1	no	—
37	C₂²×D₅₂	—	4	1	no	—
38	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₁₃×SD₁₆	—	2	1	no	—
40	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#126	—	4	1	no	—
44	C₁₃×#125	—	2	1	no	—
45	C₁₃×#124	—	2	1	no	—
46	C₁₃×#123	—	4	1	no	—
47	C₁₃×C₂²⋊Q₈	—	2	1	no	—
48	C₁₃×#121	—	2	1	no	—
49	C₄×Q₈×C₁₃	—	2	1	no	—
50	C₄×D₄×C₁₃	—	2	1	no	—
51	C₄×D₄×C₁₃	—	4	1	no	—
52	C₁₃×#117	—	4	1	no	—
53	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
54	C₂×D₁₀₄	—	2	2	no	—
55	C₂×D₁₀₄	—	2	2	no	—
56	C₂×D₁₀₄	—	2	2	no	—
57	C₂×D₁₀₄	—	4	4	no	—
58	C₂×D₁₀₄	—	4	4	no	—
59	C₂×D₁₀₄	C₂³	1	4	yes	—
60	#88128	—	2	2	no	—
61	#88128	—	2	2	no	—
62	#88128	—	4	1	no	—
63	#88128	—	4	1	no	—
64	C₁₃×#107	C₂³	1	1	yes	characteristic
65	C₁₃×#107	C₂³	1	1	yes	characteristic
66	C₁₃×#106	—	2	1	no	—
67	C₁₃×#103	—	2	1	no	—
68	C₁₃×#103	—	2	1	no	—
69	C₁₃×#102	—	2	1	no	—
70	C₁₃×#102	—	2	1	no	—
71	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#88068	—	4	4	no	—
76	#88066	—	4	4	no	—
77	C₂×#6090	C₂³	1	4	yes	—
78	#33783	—	13	1	no	maximal, Sylow
79	C₂³×D₁₃	—	4	2	no	—
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×D₅₂	—	2	4	no	—
88	C₂×D₅₂	—	4	4	no	—
89	C₂×D₅₂	—	4	4	no	—
90	C₂×D₅₂	—	4	2	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×D₅₂	—	4	2	no	—
93	C₂×D₅₂	—	4	2	no	—
94	C₂×D₅₂	—	4	2	no	—
95	C₂×D₅₂	—	8	4	no	—
96	C₂×D₅₂	—	8	4	no	—
97	C₁₃×Q₁₆	—	2	2	no	—
98	C₁₃×SD₁₆	—	4	2	no	—
99	C₁₃×D₈	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	—	2	1	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	1	no	—
114	C₁₃×C₂²⋊C₄	—	4	1	no	—
115	C₁₃×C₂²⋊C₄	—	4	1	no	—
116	C₁₃×C₂²⋊C₄	—	4	1	no	—
117	C₄²×C₁₃	—	4	1	no	—
118	C₄²×C₁₃	D₈	1	1	yes	characteristic
119	C₄²×C₁₃	D₈	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
122	D₁₀₄	—	2	4	no	—
123	D₁₀₄	—	2	4	no	—
124	D₁₀₄	—	4	4	no	—
125	D₁₀₄	—	4	4	no	—
126	D₁₀₄	—	4	4	no	—
127	#6090	—	2	4	no	—
128	#3168	D₁₃	1	1	yes	characteristic, complemented
129	C₂×#493	—	13	1	no	—
130	#2250	—	13	2	no	—
131	#2244	—	13	2	no	—
132	#2207	—	13	2	no	—
133	#2206	—	13	2	no	—
134	#2187	—	13	2	no	—
135	#2184	—	13	2	no	—
136	#2163	—	13	1	no	—
137	C₂³×C₁₃	—	4	1	no	—
138	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₁₃	—	4	4	no	—
140	C₂²×D₁₃	—	8	4	no	—
141	C₂²×D₁₃	—	8	4	no	—
142	C₂²×D₁₃	—	8	4	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	4	1	no	—
145	D₄×C₁₃	—	4	1	no	—
146	D₄×C₁₃	—	4	1	no	—
147	D₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	1	no	—
150	C₂×C₄×C₁₃	—	2	1	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	—	8	2	no	—
156	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	D₅₂	—	4	4	no	—
163	D₅₂	—	4	4	no	—
164	D₅₂	—	8	4	no	—
165	D₅₂	—	8	4	no	—
166	D₅₂	—	8	4	no	—
167	D₅₂	—	8	4	no	—
168	D₅₂	—	8	4	no	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
172	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
173	C₂²×D₈	—	13	2	no	—
174	C₂²×D₈	—	26	1	no	—
175	C₂×#127	—	26	1	no	—
176	#529	—	2	1	no	—
177	C₂×SD₃₂	—	13	4	no	—
178	C₂×D₁₆	—	13	4	no	—
179	#499	C₂×D₁₃	1	1	yes	characteristic, complemented
180	#495	C₂×D₁₃	1	1	yes	characteristic, complemented
181	#493	—	13	2	no	—
182	#493	—	13	2	no	—
183	#493	—	26	1	no	—
184	#493	—	26	1	no	—
185	#488	—	2	1	no	—
186	#467	C₂×D₁₃	1	1	yes	characteristic
187	#466	C₂×D₁₃	1	1	yes	characteristic
188	#465	—	2	1	no	—
189	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
190	C₄×SD₁₆	—	2	1	no	—
191	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic, complemented
192	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
193	C₄⋊C₁₆	—	13	2	no	—
194	#357	—	13	4	no	—
195	#357	—	13	4	no	—
196	C₂²⋊C₁₆	—	13	2	no	—
197	#346	—	13	2	no	—
198	C₂²×C₁₃	—	2	1	no	—
199	C₂²×C₁₃	—	2	1	no	—
200	C₂²×C₁₃	—	8	1	no	—
201	C₂²×C₁₃	—	8	1	no	—
202	C₂²×C₁₃	—	8	1	no	—
203	C₂²×C₁₃	#509	1	2	yes	—
204	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
205	C₂×D₁₃	—	8	4	no	—
206	C₂×D₁₃	—	8	4	no	—
207	C₂×D₁₃	—	8	4	no	—
208	C₂×D₁₃	—	16	8	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	—	8	1	no	—
214	C₄×C₁₃	—	8	1	no	—
215	C₄×C₁₃	—	8	1	no	—
216	C₄×C₁₃	#509	1	2	yes	—
217	C₄×C₁₃	#535	1	1	yes	characteristic
218	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂²×D₄	—	26	2	no	—
220	C₂²×D₄	—	52	1	no	—
221	C₂²×D₄	—	52	1	no	—
222	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
223	C₂×SD₁₆	—	2	1	no	—
224	C₂×D₈	—	13	4	no	—
225	C₂×D₈	—	26	2	no	—
226	C₂×D₈	—	26	2	no	—
227	C₂×D₈	—	26	2	no	—
228	C₂×D₈	—	52	4	no	—
229	C₂×D₈	—	52	4	no	—
230	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	#127	—	26	2	no	—
234	#127	—	26	2	no	—
235	#127	—	52	1	no	—
236	#127	—	52	1	no	—
237	#126	—	4	1	no	—
238	#125	—	2	1	no	—
239	#124	—	2	1	no	—
240	#123	—	4	1	no	—
241	C₂²⋊Q₈	—	2	1	no	—
242	#121	—	2	1	no	—
243	C₄×Q₈	—	2	1	no	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	—	4	1	no	—
246	#117	—	4	1	no	—
247	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
248	SD₃₂	—	52	4	no	—
249	D₁₆	—	52	4	no	—
250	C₂×C₁₆	—	13	4	no	—
251	#107	C₂²×D₁₃	1	1	yes	characteristic
252	#107	C₂²×D₁₃	1	1	yes	characteristic
253	#106	—	2	1	no	—
254	#103	—	2	1	no	—
255	#103	—	2	1	no	—
256	#102	—	2	1	no	—
257	#102	—	2	1	no	—
258	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
262	C₂×C₁₃	—	2	2	no	—
263	C₂×C₁₃	—	16	1	no	—
264	C₂×C₁₃	#3205	1	1	yes	characteristic
265	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
266	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
267	D₁₃	—	16	4	no	—
268	C₂⁴	—	52	2	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	26	4	no	—
273	C₂×D₄	—	52	4	no	—
274	C₂×D₄	—	52	4	no	—
275	C₂×D₄	—	52	2	no	—
276	C₂×D₄	—	52	2	no	—
277	C₂×D₄	—	52	2	no	—
278	C₂×D₄	—	52	2	no	—
279	C₂×D₄	—	52	2	no	—
280	C₂×D₄	—	104	4	no	—
281	C₂×D₄	—	104	4	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	#6104	1	1	yes	characteristic
284	C₂²×C₄	#6104	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
286	Q₁₆	—	2	2	no	—
287	SD₁₆	—	4	2	no	—
288	D₈	—	2	2	no	—
289	D₈	—	26	4	no	—
290	D₈	—	26	4	no	—
291	D₈	—	52	4	no	—
292	D₈	—	52	4	no	—
293	D₈	—	52	4	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	4	1	no	—
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	4	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	#6104	1	1	yes	characteristic
313	C₄²	#6104	1	1	yes	characteristic
314	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
316	C₁₆	—	26	4	no	—
317	C₁₃	#33783	1	1	yes	characteristic, complemented, Sylow
318	C₂³	—	4	1	no	—
319	C₂³	—	52	4	no	—
320	C₂³	—	104	4	no	—
321	C₂³	—	104	4	no	—
322	C₂³	—	104	4	no	—
323	C₂³	C₂×#6104	1	1	yes	characteristic
324	Q₈	—	4	1	no	—
325	Q₈	—	4	1	no	—
326	D₄	—	4	1	no	—
327	D₄	—	4	1	no	—
328	D₄	—	8	2	no	—
329	D₄	—	52	4	no	—
330	D₄	—	52	4	no	—
331	D₄	—	104	4	no	—
332	D₄	—	104	4	no	—
333	D₄	—	104	4	no	—
334	D₄	—	104	4	no	—
335	D₄	—	104	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	8	2	no	—
344	C₂×C₄	C₂×#6104	1	1	yes	characteristic
345	C₂×C₄	C₂×#6104	1	1	yes	characteristic
346	C₂×C₄	C₂×#6104	1	1	yes	characteristic
347	C₂×C₄	C₂×#6104	1	1	yes	characteristic
348	C₂×C₄	C₂×#6104	1	1	yes	characteristic
349	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	#88203	1	2	yes	—
353	C₈	#88256	1	2	yes	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	104	4	no	—
360	C₂²	—	104	4	no	—
361	C₂²	—	104	4	no	—
362	C₂²	—	208	8	no	—
363	C₂²	#11718854	1	2	yes	—
364	C₂²	C₂²×#6104	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	8	1	no	—
370	C₄	—	8	1	no	—
371	C₄	—	8	1	no	—
372	C₄	#11718854	1	2	yes	—
373	C₄	C₂×#88203	1	1	yes	characteristic
374	C₄	#11719535	1	1	yes	characteristic
375	C₂	—	2	2	no	—
376	C₂	—	16	1	no	—
377	C₂	—	208	4	no	—
378	C₂	#13125751	1	1	yes	characteristic
379	C₂	C₂×#11718854	1	1	yes	characteristic
380	C₂	C₂×#11718854	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3168	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3168	D₁₃
hypocenter	C₁₃	#33783
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3168

Complements

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

N	H ≅ G/N	type
1. C₂×#11718356	C₂	semidirect
2. #13117103	C₂	semidirect
3. #13117079	C₂	semidirect
4. #13116876	C₂	semidirect
5. #13116871	C₂	semidirect
6. #13116764	C₂	semidirect
7. #13116747	C₂	semidirect
8. #13116639	C₂	semidirect
9. C₁₃×#3168	C₂	semidirect
14. C₁₃×#499	C₂²	semidirect
15. C₁₃×#495	C₂²	semidirect
20. C₄×C₁₃×Q₁₆	C₂²	semidirect
22. C₄×C₁₃×D₈	C₂²	semidirect
24. C₂×#88068	C₂²	semidirect
25. C₂×#88066	C₂²	semidirect
31. #11718083	C₂²	semidirect
32. #11718083	C₂²	semidirect
33. C₄⋊#6090	C₂²	semidirect
34. #11718053	C₂²	semidirect
128. #3168	D₁₃	semidirect
179. #499	C₂×D₁₃	semidirect
180. #495	C₂×D₁₃	semidirect
189. C₄×Q₁₆	C₂×D₁₃	semidirect
191. C₄×D₈	C₂×D₁₃	semidirect
317. C₁₃	#33783	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718356	2
#13117103	2
#13117079	2
#13116876	2
#13116871	2
#13116764	2
#13116747	2
#13116639	2
C₁₃×#3168	2
#33783	13