Subgroups — #16017624

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123713	C₂	1	1	yes	maximal, characteristic
2	#13116997	C₂	1	2	yes	maximal, complemented
3	#13116990	C₂	1	2	yes	maximal, complemented
4	#13116985	C₂	1	2	yes	maximal, complemented
5	#13116970	C₂	1	2	yes	maximal, complemented
6	#13116625	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718081	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116595	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116546	C₂	1	1	yes	maximal, characteristic
10	#13116541	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#544	—	2	1	no	—
13	C₂×#88158	C₂²	1	1	yes	characteristic
14	#11719228	—	2	1	no	—
15	C₁₃×#501	—	2	2	no	—
16	C₁₃×#500	C₂²	1	2	yes	—
17	C₁₃×#499	C₂²	1	2	yes	—
18	C₁₃×#428	—	2	1	no	—
19	C₁₃×#427	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
21	C₁₃×#405	C₂²	1	1	yes	characteristic
22	#11718794	—	2	1	no	—
23	C₂×#88058	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718371	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	2	yes	—
28	C₄×D₁₀₄	C₂²	1	2	yes	—
29	#11718084	C₂²	1	1	yes	characteristic
30	#11718084	C₂²	1	2	yes	—
31	#11718084	C₂²	1	1	yes	characteristic
32	#11718083	C₂²	1	1	yes	characteristic
33	#11718083	C₂²	1	2	yes	—
34	#11718083	C₂²	1	1	yes	characteristic
35	#11718082	C₂²	1	2	yes	—
36	#11718081	C₂²	1	2	yes	—
37	#11718081	C₂²	1	2	yes	—
38	#11718081	C₂²	1	2	yes	—
39	C₄⋊#6090	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	2	yes	—
41	C₂×#6127	—	2	1	no	—
42	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	—	2	2	no	—
45	C₁₃×C₄⋊Q₈	—	4	1	no	—
46	C₁₃×#125	—	2	2	no	—
47	C₁₃×#125	—	4	1	no	—
48	C₁₃×C₂²⋊Q₈	—	4	2	no	—
49	C₁₃×#117	—	2	1	no	—
50	C₁₃×#117	D₄	1	1	yes	characteristic
51	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
52	C₂×C₄⋊Dic₁₃	—	2	1	no	—
53	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
54	#88158	—	2	2	no	—
55	#88158	Q₈	1	4	yes	complemented
56	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
57	C₂×#6095	—	2	1	no	—
58	#88146	—	4	1	no	—
59	C₄⋊C₄×D₁₃	—	4	1	no	—
60	#88135	—	4	2	no	—
61	C₄×D₅₂	—	2	2	no	—
62	C₄×Dic₂₆	—	2	2	no	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#107	—	2	2	no	—
65	C₁₃×#107	C₂³	1	2	yes	—
66	C₁₃×#107	C₂³	1	1	yes	characteristic
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#106	—	2	1	no	—
69	C₁₃×#106	—	2	2	no	—
70	C₁₃×#97	D₄	1	2	yes	—
71	C₄×C₈×C₁₃	C₂³	1	2	yes	—
72	#88061	—	2	2	no	—
73	#88058	—	2	1	no	—
74	#88058	C₂³	1	2	yes	—
75	#88056	—	2	2	no	—
76	#88052	—	2	2	no	—
77	C₂×#6090	—	2	2	no	—
78	C₂×#6090	C₂³	1	1	yes	characteristic
79	C₂×#6090	C₂³	1	1	yes	characteristic
80	C₂×#6090	C₂³	1	2	yes	—
81	#33633	—	13	1	no	maximal, Sylow
82	C₂×Q₈×C₁₃	—	4	2	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₁₃	—	4	1	no	—
86	C₂²×Dic₁₃	—	4	1	no	—
87	#6127	—	2	4	no	—
88	#6127	—	4	2	no	—
89	C₂×D₅₂	—	2	1	no	—
90	C₂×C₄×D₁₃	—	4	1	no	—
91	C₂×C₄×D₁₃	—	4	2	no	—
92	C₂×Dic₂₆	—	2	1	no	—
93	C₁₃×M₁₆	—	2	2	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₁₃	#41	1	2	yes	—
98	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	—	4	2	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
110	C₄²×C₁₃	—	4	1	no	—
111	C₄²×C₁₃	C₂×D₄	1	2	yes	—
112	#6103	—	4	2	no	—
113	C₄⋊Dic₁₃	—	2	2	no	—
114	C₄⋊Dic₁₃	—	4	1	no	—
115	#6101	—	4	2	no	—
116	C₄×Dic₁₃	—	4	2	no	—
117	Dic₅₂	—	2	1	no	—
118	Dic₅₂	C₂×Q₈	1	2	yes	—
119	D₁₀₄	—	2	1	no	—
120	D₁₀₄	C₂×Q₈	1	2	yes	—
121	#6095	—	2	4	no	—
122	#6090	—	2	1	no	—
123	#6090	—	2	1	no	—
124	#6090	—	2	2	no	—
125	#3164	D₁₃	1	1	yes	characteristic, complemented
126	#2951	—	13	1	no	—
127	#2229	—	13	2	no	—
128	#2228	—	13	2	no	—
129	#2227	—	13	2	no	—
130	#2224	—	13	2	no	—
131	#2159	—	13	1	no	—
132	C₂×#366	—	13	1	no	—
133	#2151	—	13	1	no	—
134	#2139	—	13	1	no	—
135	#2138	—	13	1	no	—
136	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂²×D₁₃	—	4	1	no	—
138	Q₈×C₁₃	—	16	2	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	16	1	no	—
143	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#137	1	2	yes	—
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₁₃	—	4	4	no	—
150	C₂×Dic₁₃	—	4	1	no	—
151	C₂×Dic₁₃	—	4	2	no	—
152	C₂×Dic₁₃	—	16	1	no	—
153	D₅₂	—	2	2	no	—
154	D₅₂	—	4	1	no	—
155	C₄×D₁₃	—	4	4	no	—
156	C₄×D₁₃	—	16	2	no	—
157	Dic₂₆	—	2	2	no	—
158	Dic₂₆	—	4	1	no	—
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
161	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
162	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
163	C₂×#135	—	13	1	no	—
164	#544	—	2	1	no	—
165	#520	—	26	1	no	—
166	C₂×M₃₂	—	13	1	no	—
167	C₂²×C₁₆	—	13	1	no	—
168	#501	—	2	2	no	—
169	#500	C₂×D₁₃	1	2	yes	—
170	#499	C₂×D₁₃	1	2	yes	—
171	#440	—	26	2	no	—
172	C₄×Q₁₆	—	13	2	no	—
173	C₄×D₈	—	13	2	no	—
174	#428	—	2	1	no	—
175	#427	C₂×D₁₃	1	1	yes	characteristic
176	C₂×#107	—	13	1	no	—
177	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
178	#417	—	26	1	no	—
179	#405	C₂×D₁₃	1	1	yes	characteristic
180	#367	—	13	2	no	—
181	#366	—	13	2	no	—
182	#366	—	13	2	no	—
183	#366	—	13	2	no	—
184	C₄⋊C₁₆	—	13	2	no	—
185	C₄⋊C₁₆	—	13	2	no	—
186	#358	—	13	1	no	—
187	#358	—	13	1	no	—
188	#358	—	13	2	no	—
189	#357	—	13	1	no	—
190	#357	—	13	1	no	—
191	#357	—	13	2	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	#474	1	2	yes	—
195	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
196	C₂×D₁₃	—	4	2	no	—
197	C₂×D₁₃	—	8	1	no	—
198	C₂×D₁₃	—	8	1	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	—	8	1	no	—
203	C₄×C₁₃	—	8	1	no	—
204	C₄×C₁₃	#474	1	2	yes	—
205	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
206	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
207	Dic₁₃	—	4	2	no	—
208	Dic₁₃	—	8	2	no	—
209	C₂×#41	—	26	1	no	—
210	#135	—	13	4	no	—
211	#135	—	26	2	no	—
212	C₂×Q₁₆	—	13	1	no	—
213	C₂×SD₁₆	—	26	1	no	—
214	C₂×D₈	—	13	1	no	—
215	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	—	4	1	no	—
219	#125	—	2	2	no	—
220	#125	—	4	1	no	—
221	C₂²⋊Q₈	—	4	2	no	—
222	C₄×Q₈	—	26	2	no	—
223	C₄×D₄	—	26	2	no	—
224	C₄×D₄	—	52	2	no	—
225	#117	—	2	1	no	—
226	#117	—	52	1	no	—
227	#117	C₂²⋊D₁₃	1	1	yes	characteristic
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	26	1	no	—
230	C₂×C₄⋊C₄	—	52	1	no	—
231	M₃₂	—	26	2	no	—
232	C₂×C₁₆	—	13	1	no	—
233	C₂×C₁₆	—	13	1	no	—
234	C₂×C₁₆	—	13	2	no	—
235	C₂×C₁₆	—	26	2	no	—
236	#107	—	2	1	no	—
237	#107	—	2	2	no	—
238	#107	—	13	2	no	—
239	#107	—	26	1	no	—
240	#107	C₂²×D₁₃	1	2	yes	—
241	#107	C₂²×D₁₃	1	1	yes	characteristic
242	#107	C₂²×D₁₃	1	1	yes	characteristic
243	#106	—	2	1	no	—
244	#106	—	2	2	no	—
245	#103	—	26	2	no	—
246	#102	—	26	2	no	—
247	#97	C₂²⋊D₁₃	1	2	yes	—
248	C₄×C₈	C₂²×D₁₃	1	2	yes	—
249	C₂×C₁₃	—	2	2	no	—
250	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
251	C₂×C₁₃	#3415	1	1	yes	characteristic
252	C₂×C₁₃	#3417	1	1	yes	characteristic
253	D₁₃	—	8	2	no	—
254	#41	—	26	4	no	—
255	#41	—	52	2	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	—	26	1	no	—
258	C₂×D₄	—	26	1	no	—
259	C₂×D₄	—	52	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	52	1	no	—
262	C₂²×C₄	—	52	1	no	—
263	C₂²×C₄	—	52	2	no	—
264	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
265	Q₁₆	—	13	2	no	—
266	Q₁₆	—	26	1	no	—
267	SD₁₆	—	26	4	no	—
268	D₈	—	13	2	no	—
269	D₈	—	26	1	no	—
270	M₁₆	—	2	2	no	—
271	C₂×C₈	Q₈×D₁₃	1	2	yes	—
272	C₂×C₈	#6131	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
274	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	26	2	no	—
284	C₄⋊C₄	—	52	1	no	—
285	C₄⋊C₄	—	52	2	no	—
286	C₄⋊C₄	#6104	1	1	yes	characteristic
287	C₄⋊C₄	#6104	1	1	yes	characteristic
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	52	2	no	—
290	C₂²⋊C₄	#6104	1	2	yes	—
291	C₄²	—	4	1	no	—
292	C₄²	—	52	2	no	—
293	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
294	C₁₆	—	26	1	no	—
295	C₁₆	—	26	1	no	—
296	C₁₆	—	26	2	no	—
297	C₁₃	#33633	1	1	yes	characteristic, complemented, Sylow
298	C₂³	—	52	1	no	—
299	C₂³	C₂×#6104	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	Q₈	—	26	2	no	—
302	Q₈	—	52	1	no	—
303	D₄	—	26	2	no	—
304	D₄	—	52	4	no	—
305	D₄	—	52	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	—	52	1	no	—
311	C₂×C₄	—	52	4	no	—
312	C₂×C₄	—	52	2	no	—
313	C₂×C₄	—	208	2	no	—
314	C₂×C₄	—	208	1	no	—
315	C₂×C₄	C₂×#6104	1	2	yes	—
316	C₂×C₄	C₂×#6104	1	2	yes	—
317	C₂×C₄	C₂×#6104	1	1	yes	characteristic
318	C₂×C₄	#88196	1	2	yes	—
319	C₂×C₄	#88200	1	2	yes	—
320	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#88200	1	2	yes	—
323	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
324	C₈	C₂×#6131	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	52	2	no	—
328	C₂²	—	104	1	no	—
329	C₂²	—	104	1	no	—
330	C₂²	#11718748	1	2	yes	—
331	C₂²	C₂²×#6104	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	—	52	2	no	—
338	C₄	—	104	2	no	—
339	C₄	#11718748	1	2	yes	—
340	C₄	C₂×#88196	1	1	yes	characteristic
341	C₄	C₂×#88200	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	104	2	no	—
344	C₂	C₂×#11718748	1	1	yes	characteristic
345	C₂	#13127975	1	1	yes	characteristic
346	C₂	#13127987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3164	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88158	C₂²
hypercenter	#3164	D₁₃
hypocenter	C₁₃	#33633
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #13116997	C₂	semidirect
3. #13116990	C₂	semidirect
4. #13116985	C₂	semidirect
5. #13116970	C₂	semidirect
6. #13116625	C₂	semidirect
7. C₂×#11718081	C₂	semidirect
8. #13116595	C₂	semidirect
11. C₁₃×#3164	C₂	semidirect
55. #88158	Q₈	semidirect
125. #3164	D₁₃	semidirect
297. C₁₃	#33633	semidirect

Maximal subgroups

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

subgroup	index
#13123713	2
#13116997	2
#13116990	2
#13116985	2
#13116970	2
#13116625	2
C₂×#11718081	2
#13116595	2
#13116546	2
#13116541	2
C₁₃×#3164	2
#33633	13