Subgroups — #16807218

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131616	C₂	1	1	yes	maximal, characteristic, complemented
2	#13131612	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131238	C₂	1	4	yes	maximal, complemented
4	#13131236	C₂	1	4	yes	maximal, complemented
5	D₁₃⋊#368	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131152	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131151	C₂	1	1	yes	maximal, characteristic, complemented
8	#13131044	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11718577	C₂	1	1	yes	maximal, characteristic
10	#11719625	—	2	1	no	—
11	#11719621	—	2	1	no	—
12	C₂×#88173	C₂²	1	1	yes	characteristic
13	C₂×#88166	C₂²	1	1	yes	characteristic
14	#11719155	—	2	1	no	—
15	#11719140	—	2	1	no	—
16	#11719112	—	2	2	no	—
17	#11719111	C₂²	1	4	yes	complemented
18	#11719104	—	2	2	no	—
19	#11719103	C₂²	1	4	yes	complemented
20	D₁₃⋊#108	C₂²	1	1	yes	characteristic
21	D₁₃⋊#108	C₂²	1	1	yes	characteristic
22	#11719089	C₂²	1	1	yes	characteristic
23	D₁₃⋊M₃₂	C₂²	1	1	yes	characteristic
24	C₂×C₁₃×SD₃₂	C₄	1	1	yes	characteristic, complemented
25	C₂×#88069	C₄	1	1	yes	characteristic, complemented
26	C₂×#88066	C₄	1	1	yes	characteristic, complemented
27	#11718577	C₄	1	2	yes	complemented
28	#11718577	C₄	1	2	yes	complemented
29	#11718577	C₂²	1	2	yes	—
30	#11718577	C₂²	1	2	yes	—
31	C₂×#88040	C₄	1	1	yes	characteristic, complemented
32	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
33	#11718281	C₂²	1	4	yes	complemented
34	#11718277	C₂²	1	4	yes	complemented
35	#11718224	C₂²	1	2	yes	complemented
36	#11718223	C₂²	1	2	yes	complemented
37	#11718216	C₂²	1	2	yes	complemented
38	C₁₆×#261	C₂²	1	2	yes	complemented
39	C₂×#6131	—	2	1	no	—
40	C₂×#6129	—	2	1	no	—
41	C₂×#6122	—	4	2	no	—
42	#88240	—	2	4	no	—
43	#88240	—	4	2	no	—
44	#88238	—	2	4	no	—
45	#88238	—	4	2	no	—
46	#88237	D₄	1	1	yes	characteristic
47	D₁₃⋊M₁₆	—	2	1	no	—
48	D₁₃⋊M₁₆	—	2	1	no	—
49	D₁₃⋊M₁₆	—	4	2	no	—
50	D₁₃⋊M₁₆	D₄	1	1	yes	characteristic
51	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
53	C₂×#6106	—	2	1	no	—
54	C₂×#6105	—	2	1	no	—
55	#88173	—	2	1	no	—
56	#88173	—	2	1	no	—
57	#88173	C₂×C₄	1	2	yes	—
58	#88173	C₂³	1	2	yes	—
59	#88166	—	2	1	no	—
60	#88166	—	2	1	no	—
61	#88166	C₂×C₄	1	2	yes	—
62	#88166	C₂³	1	2	yes	—
63	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
64	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
65	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
66	#88118	—	2	4	no	—
67	#88116	—	2	4	no	—
68	#88106	—	2	1	no	—
69	#88104	C₂³	1	2	yes	—
70	#88104	C₂³	1	2	yes	—
71	#88103	—	2	1	no	—
72	#88103	—	2	1	no	—
73	#88100	D₄	1	2	yes	—
74	C₈×#261	C₂³	1	2	yes	—
75	#88098	—	2	1	no	—
76	D₁₃⋊C₁₆	C₂³	1	2	yes	—
77	C₁₃×SD₃₂	C₂×C₄	1	2	yes	—
78	C₁₃×SD₃₂	C₂×C₄	1	2	yes	—
79	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
80	#88069	C₂×C₄	1	2	yes	—
81	#88069	C₂×C₄	1	2	yes	—
82	#88066	C₂×C₄	1	2	yes	—
83	#88066	C₂×C₄	1	2	yes	—
84	C₂×#6090	C₂×C₄	1	1	yes	characteristic
85	#88040	C₂×C₄	1	2	yes	—
86	#88040	C₂×C₄	1	2	yes	—
87	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
88	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
89	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
90	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
91	#33465	—	13	1	no	maximal, Sylow
92	C₂×Q₈×C₁₃	—	2	1	no	—
93	C₂×D₄×C₁₃	—	2	1	no	—
94	C₂×C₂²⋊D₁₃	—	4	1	no	—
95	C₂²×Dic₁₃	—	4	1	no	—
96	#6131	—	2	2	no	—
97	#6131	—	2	2	no	—
98	#6131	—	4	1	no	—
99	#6131	—	4	1	no	—
100	#6129	—	2	2	no	—
101	#6129	—	2	2	no	—
102	#6129	—	4	1	no	—
103	#6129	—	4	1	no	—
104	C₂×D₅₂	—	2	1	no	—
105	C₂×D₅₂	—	4	1	no	—
106	C₂×C₄×D₁₃	—	4	1	no	—
107	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×Dic₂₆	—	2	1	no	—
109	C₂²⋊#261	—	2	1	no	—
110	#6122	—	4	1	no	—
111	#6122	—	4	1	no	—
112	#6122	—	4	4	no	—
113	#6122	D₈	1	2	yes	—
114	C₂×#1057	—	4	4	no	—
115	C₄⋊#261	—	2	1	no	—
116	C₄×#261	C₂×D₄	1	2	yes	—
117	#6118	—	2	2	no	—
118	#6118	—	2	2	no	—
119	#6118	—	4	4	no	—
120	#6118	D₈	1	2	yes	—
121	D₁₃⋊C₈	—	4	1	no	—
122	D₁₃⋊C₈	—	4	1	no	—
123	D₁₃⋊C₈	—	4	4	no	—
124	D₁₃⋊C₈	C₂×D₄	1	2	yes	—
125	C₁₃×Q₁₆	—	2	1	no	—
126	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
127	C₁₃×D₈	—	2	1	no	—
128	C₁₃×D₈	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
130	#6106	—	2	2	no	—
131	#6106	—	2	2	no	—
132	#6105	—	2	2	no	—
133	#6105	—	2	2	no	—
134	C₂×#1055	#41	1	1	yes	characteristic
135	Dic₅₂	—	2	1	no	—
136	Dic₅₂	C₂²×C₄	1	2	yes	—
137	D₁₀₄	—	2	1	no	—
138	D₁₀₄	C₂²×C₄	1	2	yes	—
139	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
140	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
141	C₈×D₁₃	#41	1	1	yes	characteristic
142	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
143	#6092	—	2	2	no	—
144	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
145	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
146	#6090	C₂²×C₄	1	1	yes	characteristic
147	#6090	C₂²×C₄	1	1	yes	characteristic
148	C₂×#508	—	13	1	no	—
149	#2957	—	13	1	no	—
150	#2957	—	13	1	no	—
151	#2179	—	13	4	no	—
152	#2178	—	13	4	no	—
153	C₂×#368	—	13	1	no	—
154	#2143	—	13	1	no	—
155	#2143	—	13	1	no	—
156	#2108	—	13	1	no	—
157	C₂³×C₁₃	—	4	1	no	—
158	C₂²×D₁₃	—	4	1	no	—
159	C₂²×D₁₃	#135	1	1	yes	characteristic
160	C₂×#261	—	2	2	no	—
161	Q₈×C₁₃	—	2	2	no	—
162	Q₈×C₁₃	—	4	1	no	—
163	D₄×C₁₃	—	2	2	no	—
164	D₄×C₁₃	—	4	1	no	—
165	C₂×C₄×C₁₃	—	4	1	no	—
166	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
167	C₂²⋊D₁₃	—	4	2	no	—
168	C₂²⋊D₁₃	—	4	2	no	—
169	C₂×Dic₁₃	—	4	1	no	—
170	C₂×Dic₁₃	—	4	2	no	—
171	C₂×Dic₁₃	—	4	2	no	—
172	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
173	D₅₂	—	2	2	no	—
174	D₅₂	—	4	2	no	—
175	D₅₂	—	4	2	no	—
176	D₅₂	—	4	1	no	—
177	C₄×D₁₃	—	4	2	no	—
178	C₄×D₁₃	—	4	2	no	—
179	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
180	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
181	C₄×D₁₃	#135	1	1	yes	characteristic
182	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
183	Dic₂₆	—	2	2	no	—
184	Dic₂₆	—	4	1	no	—
185	#1057	—	4	2	no	—
186	#1057	—	4	2	no	—
187	#1057	C₂×D₈	1	4	yes	—
188	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
189	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
190	#1055	C₄×D₄	1	1	yes	characteristic
191	#1055	C₂×#41	1	1	yes	characteristic
192	C₂×#135	—	13	1	no	—
193	C₂×#135	—	13	1	no	—
194	#568	—	26	1	no	—
195	#568	—	26	1	no	—
196	#508	—	13	2	no	—
197	#508	—	13	2	no	—
198	#508	—	13	2	no	—
199	#508	—	13	2	no	—
200	C₂×Q₃₂	—	13	1	no	—
201	C₂×SD₃₂	—	13	1	no	—
202	C₂×SD₃₂	#261	1	1	yes	characteristic, complemented
203	C₂×D₁₆	—	13	1	no	—
204	C₂×M₃₂	—	13	1	no	—
205	C₂²×C₁₆	—	13	1	no	—
206	#444	—	26	2	no	—
207	#444	—	26	2	no	—
208	#443	—	13	4	no	—
209	#443	—	13	4	no	—
210	C₂×#108	—	13	1	no	—
211	C₂×#108	—	13	1	no	—
212	#421	—	26	1	no	—
213	#421	—	26	1	no	—
214	#405	—	13	1	no	—
215	#368	—	13	2	no	—
216	#368	—	13	2	no	—
217	#359	—	13	4	no	—
218	#359	—	13	4	no	—
219	#346	—	13	2	no	—
220	C₄×C₁₆	—	13	2	no	—
221	C₂²×C₁₃	—	4	2	no	—
222	C₂²×C₁₃	—	16	1	no	—
223	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
224	C₂×D₁₃	—	4	1	no	—
225	C₂×D₁₃	—	4	2	no	—
226	C₂×D₁₃	—	16	1	no	—
227	C₂×D₁₃	C₄×D₈	1	1	yes	characteristic
228	C₂×D₁₃	C₂×#135	1	1	yes	characteristic
229	#261	—	4	2	no	—
230	C₄×C₁₃	—	4	2	no	—
231	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
232	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
233	Dic₁₃	—	4	2	no	—
234	Dic₁₃	C₄×D₈	1	1	yes	characteristic
235	Dic₁₃	C₂²×D₈	1	1	yes	characteristic
236	C₂×#41	—	26	1	no	—
237	C₂×#41	—	26	1	no	—
238	#135	—	13	2	no	—
239	#135	—	13	2	no	—
240	#135	—	13	2	no	—
241	#135	—	13	2	no	—
242	#135	—	26	1	no	—
243	#135	—	26	1	no	—
244	#135	—	26	1	no	—
245	#135	—	26	1	no	—
246	C₂×Q₁₆	—	13	1	no	—
247	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
248	C₂×SD₁₆	—	26	1	no	—
249	C₂×SD₁₆	—	26	1	no	—
250	C₂×D₈	—	13	1	no	—
251	C₂×D₈	C₂×#261	1	1	yes	characteristic
252	#131	—	26	4	no	—
253	#131	—	26	4	no	—
254	#131	—	52	2	no	—
255	#131	—	52	2	no	—
256	C₂×M₁₆	—	13	1	no	—
257	C₂×M₁₆	—	26	1	no	—
258	C₂×M₁₆	—	26	1	no	—
259	C₂×M₁₆	—	52	2	no	—
260	C₂×M₁₆	—	52	2	no	—
261	C₂²×C₈	—	13	1	no	—
262	#117	—	13	1	no	—
263	Q₃₂	—	13	2	no	—
264	Q₃₂	—	13	2	no	—
265	SD₃₂	—	13	2	no	—
266	SD₃₂	—	13	2	no	—
267	SD₃₂	C₂×#261	1	2	yes	—
268	SD₃₂	C₂×#261	1	2	yes	—
269	D₁₆	—	13	2	no	—
270	D₁₆	—	13	2	no	—
271	M₃₂	—	26	1	no	—
272	M₃₂	—	26	1	no	—
273	C₂×C₁₆	—	13	1	no	—
274	C₂×C₁₆	—	13	1	no	—
275	C₂×C₁₆	—	13	1	no	—
276	C₂×C₁₆	—	13	1	no	—
277	C₂×C₁₆	—	13	1	no	—
278	C₂×C₁₆	—	13	2	no	—
279	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
280	#108	—	13	2	no	—
281	#108	—	13	2	no	—
282	#108	—	26	1	no	—
283	#108	—	26	1	no	—
284	#104	—	26	4	no	—
285	#104	—	26	4	no	—
286	#97	—	13	2	no	—
287	C₄×C₈	—	13	2	no	—
288	C₂×C₁₃	—	2	1	no	—
289	C₂×C₁₃	—	8	2	no	—
290	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
291	D₁₃	—	2	1	no	—
292	D₁₃	—	2	1	no	—
293	D₁₃	—	8	2	no	—
294	#41	—	26	2	no	—
295	#41	—	26	2	no	—
296	#41	—	26	2	no	—
297	#41	—	26	2	no	—
298	#41	—	52	1	no	—
299	#41	—	52	1	no	—
300	#41	—	52	1	no	—
301	#41	—	52	1	no	—
302	C₂×Q₈	—	2	1	no	—
303	C₂×Q₈	—	26	1	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	26	1	no	—
306	C₂×D₄	—	52	1	no	—
307	C₂×D₄	—	52	1	no	—
308	C₂²×C₄	—	13	1	no	—
309	C₂²×C₄	—	52	1	no	—
310	C₂²×C₄	—	52	1	no	—
311	Q₁₆	—	2	1	no	—
312	Q₁₆	—	13	2	no	—
313	Q₁₆	—	26	1	no	—
314	Q₁₆	C₂²×#261	1	2	yes	—
315	SD₁₆	—	26	2	no	—
316	SD₁₆	—	26	2	no	—
317	SD₁₆	—	26	2	no	—
318	SD₁₆	—	26	2	no	—
319	D₈	—	2	1	no	—
320	D₈	—	13	2	no	—
321	D₈	—	26	1	no	—
322	D₈	C₂²×#261	1	2	yes	—
323	M₁₆	—	13	2	no	—
324	M₁₆	—	13	2	no	—
325	M₁₆	—	26	2	no	—
326	M₁₆	—	26	2	no	—
327	M₁₆	—	52	4	no	—
328	M₁₆	—	52	4	no	—
329	M₁₆	—	52	1	no	—
330	M₁₆	—	52	1	no	—
331	C₂×C₈	—	13	2	no	—
332	C₂×C₈	—	13	1	no	—
333	C₂×C₈	—	13	1	no	—
334	C₂×C₈	—	13	1	no	—
335	C₂×C₈	—	13	1	no	—
336	C₂×C₈	—	13	1	no	—
337	C₂×C₈	—	52	4	no	—
338	C₂×C₈	—	52	4	no	—
339	C₂×C₈	—	52	1	no	—
340	C₂×C₈	—	52	1	no	—
341	C₂×C₈	C₂²×#261	1	1	yes	characteristic
342	C₄⋊C₄	—	26	1	no	—
343	C₂²⋊C₄	—	26	1	no	—
344	C₄²	—	13	2	no	—
345	C₁₆	—	13	1	no	—
346	C₁₆	—	13	1	no	—
347	C₁₆	—	26	2	no	—
348	C₁₆	C₂²×#261	1	1	yes	characteristic
349	C₁₆	C₂²×#261	1	1	yes	characteristic
350	C₁₃	#33465	1	1	yes	characteristic, complemented, Sylow
351	C₂³	—	4	1	no	—
352	C₂³	—	13	1	no	—
353	C₂³	—	52	1	no	—
354	Q₈	—	2	2	no	—
355	Q₈	—	4	1	no	—
356	Q₈	—	26	2	no	—
357	Q₈	—	52	1	no	—
358	D₄	—	2	2	no	—
359	D₄	—	4	1	no	—
360	D₄	—	26	2	no	—
361	D₄	—	52	2	no	—
362	D₄	—	52	2	no	—
363	D₄	—	52	2	no	—
364	D₄	—	52	2	no	—
365	D₄	—	52	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	13	1	no	—
368	C₂×C₄	—	13	1	no	—
369	C₂×C₄	—	13	1	no	—
370	C₂×C₄	—	13	1	no	—
371	C₂×C₄	—	13	1	no	—
372	C₂×C₄	—	26	2	no	—
373	C₂×C₄	—	52	2	no	—
374	C₂×C₄	—	52	2	no	—
375	C₂×C₄	—	52	2	no	—
376	C₂×C₄	—	52	2	no	—
377	C₂×C₄	—	52	1	no	—
378	C₂×C₄	D₄×#261	1	1	yes	characteristic
379	C₈	—	13	4	no	—
380	C₈	—	13	1	no	—
381	C₈	—	13	1	no	—
382	C₈	—	52	2	no	—
383	C₈	—	52	2	no	—
384	C₈	D₄×#261	1	1	yes	characteristic
385	C₈	C₂³×#261	1	1	yes	characteristic
386	C₂²	—	4	2	no	—
387	C₂²	—	13	1	no	—
388	C₂²	—	13	1	no	—
389	C₂²	—	16	1	no	—
390	C₂²	—	52	1	no	—
391	C₂²	—	52	2	no	—
392	C₂²	—	208	1	no	—
393	C₂²	D₈×#261	1	1	yes	characteristic
394	C₄	—	4	2	no	—
395	C₄	—	13	1	no	—
396	C₄	—	13	1	no	—
397	C₄	—	52	2	no	—
398	C₄	—	52	2	no	—
399	C₄	D₈×#261	1	1	yes	characteristic
400	C₄	C₂×D₄×#261	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	8	2	no	—
403	C₂	—	26	1	no	—
404	C₂	—	26	1	no	—
405	C₂	—	104	2	no	—
406	C₂	C₂×D₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#261
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×SD₃₂	C₄
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	C₂×#11718577	C₂
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×SD₃₂	#261
hypocenter	C₁₃	#33465
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #13131616	C₂	semidirect
2. #13131612	C₂	semidirect
3. #13131238	C₂	semidirect
4. #13131236	C₂	semidirect
5. D₁₃⋊#368	C₂	semidirect
6. #13131152	C₂	semidirect
7. #13131151	C₂	semidirect
8. #13131044	C₂	semidirect
17. #11719111	C₂²	semidirect
19. #11719103	C₂²	semidirect
24. C₂×C₁₃×SD₃₂	C₄	semidirect
25. C₂×#88069	C₄	semidirect
26. C₂×#88066	C₄	semidirect
27. #11718577	C₄	semidirect
28. #11718577	C₄	semidirect
31. C₂×#88040	C₄	semidirect
33. #11718281	C₂²	semidirect
34. #11718277	C₂²	semidirect
35. #11718224	C₂²	semidirect
36. #11718223	C₂²	semidirect
37. #11718216	C₂²	semidirect
38. C₁₆×#261	C₂²	semidirect
202. C₂×SD₃₂	#261	semidirect
350. C₁₃	#33465	semidirect

Maximal subgroups

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

subgroup	index
#13131616	2
#13131612	2
#13131238	2
#13131236	2
D₁₃⋊#368	2
#13131152	2
#13131151	2
#13131044	2
C₂×#11718577	2
#33465	13