Subgroups — #16018365

Order 3328.

834 conjugacy classes of subgroups.

The subgroup lattice has 356 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, complemented
2	C₂×#11718854	C₂	1	2	yes	maximal, complemented
3	C₂×#11718357	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117105	C₂	1	2	yes	maximal, complemented
5	#13117100	C₂	1	2	yes	maximal, complemented
6	#13117095	C₂	1	4	yes	maximal, complemented
7	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	—	2	1	no	—
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₁₃×#532	—	2	1	no	—
12	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
13	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
14	C₂²×#6095	—	2	1	no	—
15	C₂×#88129	—	2	1	no	—
16	C₁₃×#495	—	2	1	no	—
17	C₁₃×#495	C₂²	1	2	yes	complemented
18	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
20	C₁₃×#492	—	2	1	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	#11718856	—	2	8	no	—
23	#11718854	—	2	8	no	—
24	C₂×#88069	C₂²	1	4	yes	complemented
25	C₂×#88068	C₂²	1	4	yes	—
26	C₂×#88067	C₂²	1	4	yes	complemented
27	C₂×#88066	C₂²	1	4	yes	—
28	C₂×#88052	C₂²	1	2	yes	—
29	#11718357	—	2	1	no	—
30	#11718357	—	2	1	no	—
31	#11718357	C₂²	1	2	yes	—
32	#11718357	C₂²	1	2	yes	—
33	#11718053	C₂²	1	2	yes	complemented
34	C₄×#6090	C₂²	1	2	yes	complemented
35	C₂×C₁₃×#41	—	2	2	no	—
36	C₂²×D₅₂	—	2	1	no	—
37	C₂²×Dic₂₆	—	2	1	no	—
38	C₁₃×#135	—	2	4	no	—
39	C₁₃×#135	—	2	4	no	—
40	C₁₃×#135	—	4	4	no	—
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×SD₁₆	—	2	2	no	—
45	C₂×C₁₃×D₈	—	2	1	no	—
46	C₂×C₁₃×D₈	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#127	—	2	1	no	—
51	C₁₃×#124	—	2	2	no	—
52	C₁₃×#121	—	4	2	no	—
53	C₄×D₄×C₁₃	—	4	2	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₂×#6103	—	4	2	no	—
56	C₂×Dic₅₂	—	2	4	no	—
57	C₂×Dic₅₂	—	2	1	no	—
58	C₂×Dic₅₂	C₂³	1	2	yes	—
59	C₂×D₁₀₄	—	2	4	no	—
60	C₂×D₁₀₄	—	2	1	no	—
61	C₂×D₁₀₄	C₂³	1	2	yes	—
62	C₂×#6095	—	2	2	no	—
63	C₂×#6095	—	2	2	no	—
64	C₂×#6095	—	4	4	no	—
65	#88129	—	2	2	no	—
66	#88129	—	2	2	no	—
67	#88129	—	4	2	no	—
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	#88069	—	2	8	no	—
73	#88068	—	2	8	no	—
74	#88067	—	2	8	no	—
75	#88066	—	2	8	no	—
76	#88052	D₄	1	8	yes	complemented
77	C₂×#6090	C₂³	1	4	yes	—
78	#33719	—	13	1	no	maximal, Sylow
79	C₂³×D₁₃	—	4	1	no	—
80	C₁₃×#41	—	4	4	no	—
81	C₁₃×#41	—	4	4	no	—
82	C₂×Q₈×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	2	2	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	2	no	—
86	C₂²×C₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	D₈	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₁₃	—	4	1	no	—
90	C₂×D₅₂	—	2	2	no	—
91	C₂×D₅₂	—	4	2	no	—
92	C₂×D₅₂	—	4	2	no	—
93	C₂×D₅₂	—	4	1	no	—
94	C₂×Dic₂₆	—	2	2	no	—
95	C₂×Dic₂₆	—	4	2	no	—
96	C₂×Dic₂₆	—	4	2	no	—
97	C₂×Dic₂₆	—	4	1	no	—
98	C₁₃×Q₁₆	—	2	2	no	—
99	C₁₃×Q₁₆	—	2	2	no	—
100	C₁₃×Q₁₆	—	4	2	no	—
101	C₁₃×SD₁₆	—	4	4	no	—
102	C₁₃×SD₁₆	—	4	4	no	—
103	C₁₃×D₈	—	2	2	no	—
104	C₁₃×D₈	—	2	2	no	—
105	C₁₃×D₈	—	4	2	no	—
106	C₂×C₈×C₁₃	D₈	1	4	yes	—
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	—	4	2	no	—
113	C₁₃×C₂²⋊C₄	—	4	4	no	—
114	C₄²×C₁₃	D₈	1	2	yes	—
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	#6103	—	4	4	no	—
118	#6103	—	4	4	no	—
119	Dic₅₂	—	2	4	no	—
120	Dic₅₂	—	4	2	no	—
121	D₁₀₄	—	2	4	no	—
122	D₁₀₄	—	4	2	no	—
123	#6095	—	4	4	no	—
124	#6095	—	4	4	no	—
125	#6090	C₂×D₄	1	8	yes	—
126	C₂×#510	—	13	2	no	—
127	C₂×#509	—	13	2	no	—
128	#3147	D₁₃	1	1	yes	characteristic, complemented
129	C₂×#495	—	13	1	no	—
130	#2251	—	13	2	no	—
131	#2250	—	13	2	no	—
132	#2249	—	13	4	no	—
133	C₄×M₃₂	—	13	1	no	—
134	C₂³×C₁₃	—	4	2	no	—
135	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₁₃	—	4	2	no	—
137	C₂²×D₁₃	—	8	4	no	—
138	C₂²×D₁₃	—	8	2	no	—
139	Q₈×C₁₃	—	4	2	no	—
140	Q₈×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	2	no	—
142	D₄×C₁₃	—	4	2	no	—
143	D₄×C₁₃	—	8	4	no	—
144	D₄×C₁₃	—	8	4	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	8	4	no	—
148	C₂×C₄×C₁₃	#127	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×Dic₁₃	—	4	2	no	—
155	C₂×Dic₁₃	—	8	2	no	—
156	D₅₂	—	4	2	no	—
157	D₅₂	—	4	2	no	—
158	D₅₂	—	8	2	no	—
159	Dic₂₆	—	4	2	no	—
160	Dic₂₆	—	4	2	no	—
161	Dic₂₆	—	8	2	no	—
162	C₈×C₁₃	#127	1	2	yes	—
163	C₈×C₁₃	C₂×D₈	1	2	yes	—
164	C₈×C₁₃	C₂×D₈	1	2	yes	—
165	C₈×C₁₃	C₂²×D₄	1	2	yes	—
166	C₂×#135	—	2	1	no	—
167	C₂×#135	C₂×D₁₃	1	2	yes	—
168	C₂²×Q₁₆	—	13	1	no	—
169	C₂²×SD₁₆	—	26	1	no	—
170	C₂²×D₈	—	13	1	no	—
171	#532	—	2	1	no	—
172	C₂×#124	—	26	1	no	—
173	#510	—	26	8	no	—
174	#509	—	26	8	no	—
175	C₂×Q₃₂	—	13	4	no	—
176	C₂×SD₃₂	—	13	4	no	—
177	C₂×SD₃₂	—	13	4	no	—
178	C₂×D₁₆	—	13	4	no	—
179	C₂×M₃₂	—	13	2	no	—
180	#495	—	2	1	no	—
181	#495	—	13	2	no	—
182	#495	—	13	2	no	—
183	#495	—	26	1	no	—
184	#495	—	26	1	no	—
185	#495	C₂×D₁₃	1	2	yes	complemented
186	#494	C₂×D₁₃	1	1	yes	characteristic, complemented
187	#493	C₂×D₁₃	1	1	yes	characteristic, complemented
188	#492	—	2	1	no	—
189	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
190	#346	—	13	2	no	—
191	C₄×C₁₆	—	13	2	no	—
192	C₂²×C₁₃	—	2	2	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	—	8	4	no	—
195	C₂²×C₁₃	#493	1	2	yes	—
196	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
197	C₂×D₁₃	—	8	2	no	—
198	C₂×D₁₃	—	8	4	no	—
199	C₂×D₁₃	—	16	4	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	2	2	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	#493	1	2	yes	—
204	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
205	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
206	Dic₁₃	—	8	2	no	—
207	C₂×#41	—	2	2	no	—
208	C₂²×Q₈	—	26	1	no	—
209	C₂²×D₄	—	26	1	no	—
210	#135	—	2	4	no	—
211	#135	—	2	4	no	—
212	#135	—	4	4	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	13	2	no	—
215	C₂×Q₁₆	—	26	4	no	—
216	C₂×Q₁₆	—	26	1	no	—
217	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×SD₁₆	—	26	2	no	—
221	C₂×SD₁₆	—	26	2	no	—
222	C₂×SD₁₆	—	52	4	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	—	13	2	no	—
225	C₂×D₈	—	26	4	no	—
226	C₂×D₈	—	26	1	no	—
227	C₂×D₈	C₂²×D₁₃	1	2	yes	—
228	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
230	C₄⋊Q₈	—	2	1	no	—
231	#127	—	2	1	no	—
232	#124	—	2	2	no	—
233	#124	—	26	2	no	—
234	#124	—	26	2	no	—
235	#124	—	52	2	no	—
236	#121	—	4	2	no	—
237	C₄×D₄	—	4	2	no	—
238	C₂×C₂²⋊C₄	—	52	2	no	—
239	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
240	Q₃₂	—	26	8	no	—
241	SD₃₂	—	26	8	no	—
242	SD₃₂	—	26	8	no	—
243	D₁₆	—	26	8	no	—
244	M₃₂	—	13	8	no	—
245	C₂×C₁₆	—	13	4	no	—
246	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
247	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
250	C₂×C₁₃	—	2	2	no	—
251	C₂×C₁₃	—	16	2	no	—
252	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
253	C₂×C₁₃	#3414	1	2	yes	—
254	D₁₃	—	16	2	no	—
255	C₂⁴	—	52	1	no	—
256	#41	—	4	4	no	—
257	#41	—	4	4	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	26	2	no	—
260	C₂×Q₈	—	52	1	no	—
261	C₂×Q₈	—	52	2	no	—
262	C₂×Q₈	—	52	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	26	2	no	—
267	C₂×D₄	—	52	2	no	—
268	C₂×D₄	—	52	1	no	—
269	C₂×D₄	—	52	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	52	1	no	—
272	C₂²×C₄	#6104	1	2	yes	—
273	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	4	2	no	—
277	Q₁₆	—	26	4	no	—
278	Q₁₆	—	52	2	no	—
279	SD₁₆	—	4	4	no	—
280	SD₁₆	—	4	4	no	—
281	SD₁₆	—	52	4	no	—
282	SD₁₆	—	52	4	no	—
283	D₈	—	2	2	no	—
284	D₈	—	2	2	no	—
285	D₈	—	4	2	no	—
286	D₈	—	26	4	no	—
287	D₈	—	52	2	no	—
288	C₂×C₈	#6104	1	4	yes	—
289	C₂×C₈	D₄×D₁₃	1	4	yes	—
290	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
294	C₄⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	4	no	—
296	C₂²⋊C₄	—	52	4	no	—
297	C₂²⋊C₄	—	52	4	no	—
298	C₄²	#6104	1	2	yes	—
299	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
301	C₁₆	—	13	8	no	—
302	C₁₃	#33719	1	1	yes	characteristic, complemented, Sylow
303	C₂³	—	4	2	no	—
304	C₂³	—	52	2	no	—
305	C₂³	—	104	4	no	—
306	C₂³	—	104	2	no	—
307	C₂³	C₂×#6104	1	1	yes	characteristic
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	52	2	no	—
311	Q₈	—	52	2	no	—
312	Q₈	—	104	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	4	no	—
316	D₄	—	8	4	no	—
317	D₄	—	52	2	no	—
318	D₄	—	52	2	no	—
319	D₄	—	104	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	4	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	—	104	2	no	—
325	C₂×C₄	C₂×#6104	1	2	yes	—
326	C₂×C₄	C₂×#6104	1	2	yes	—
327	C₂×C₄	C₂×#6104	1	2	yes	—
328	C₂×C₄	C₂×#6104	1	1	yes	characteristic
329	C₂×C₄	#88194	1	2	yes	—
330	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
331	C₈	C₂×#6104	1	2	yes	—
332	C₈	C₂×#6104	1	2	yes	—
333	C₈	#88194	1	2	yes	—
334	C₈	C₂×D₄×D₁₃	1	2	yes	—
335	C₂²	—	2	2	no	—
336	C₂²	—	8	4	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	104	4	no	—
339	C₂²	—	104	2	no	—
340	C₂²	—	208	4	no	—
341	C₂²	#11718734	1	2	yes	—
342	C₂²	C₂²×#6104	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	104	2	no	—
347	C₄	#11718734	1	2	yes	—
348	C₄	C₂²×#6104	1	1	yes	characteristic
349	C₄	C₂×#88194	1	1	yes	characteristic
350	C₂	—	2	2	no	—
351	C₂	—	16	2	no	—
352	C₂	—	208	2	no	—
353	C₂	C₂×#11718734	1	1	yes	characteristic
354	C₂	#13127970	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3147	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3147	D₁₃
hypocenter	C₁₃	#33719
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#11718856	C₂	semidirect
2. C₂×#11718854	C₂	semidirect
3. C₂×#11718357	C₂	semidirect
4. #13117105	C₂	semidirect
5. #13117100	C₂	semidirect
6. #13117095	C₂	semidirect
7. C₄×#88052	C₂	semidirect
8. C₁₃×#3147	C₂	semidirect
17. C₁₃×#495	C₂²	semidirect
18. C₁₃×#494	C₂²	semidirect
19. C₁₃×#493	C₂²	semidirect
24. C₂×#88069	C₂²	semidirect
26. C₂×#88067	C₂²	semidirect
33. #11718053	C₂²	semidirect
34. C₄×#6090	C₂²	semidirect
76. #88052	D₄	semidirect
128. #3147	D₁₃	semidirect
185. #495	C₂×D₁₃	semidirect
186. #494	C₂×D₁₃	semidirect
187. #493	C₂×D₁₃	semidirect
302. C₁₃	#33719	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×#11718854	2
C₂×#11718357	2
#13117105	2
#13117100	2
#13117095	2
C₄×#88052	2
C₁₃×#3147	2
#33719	13