Subgroups — C₄×#19153848 #37542541

Order 23168.

484 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×#19153848	C₂	1	1	yes	maximal, characteristic
2	C₄×#15668450	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041386	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041383	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041382	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#15668439	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15668438	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#15668436	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041320	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041286	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041279	C₂	1	1	yes	maximal, characteristic
14	C₄×#15668432	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15668450	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₈₁	C₂²	1	1	yes	characteristic
18	#19153848	C₄	1	8	yes	direct product factor
19	C₂×#15668439	C₂²	1	1	yes	characteristic
20	C₂×#15668438	C₂²	1	1	yes	characteristic
21	C₂×#15668436	C₂²	1	1	yes	characteristic
22	C₂×#15668432	C₂²	1	1	yes	characteristic
23	C₂×#15668432	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
26	C₄×Dic₃₆₂	—	2	1	no	—
27	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
28	#19153781	—	2	1	no	—
29	#19153781	C₂²	1	1	yes	characteristic
30	C₄×D₇₂₄	—	2	1	no	—
31	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
32	#19153778	—	2	1	no	—
33	#19153778	C₂²	1	1	yes	characteristic
34	#19153774	—	2	1	no	—
35	#19153774	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₈₁	—	2	1	no	—
37	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
38	C₄²×D₁₈₁	—	2	1	no	—
39	C₄²×D₁₈₁	—	2	1	no	—
40	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
41	#19153749	C₂²	1	1	yes	characteristic
42	#19153748	C₂²	1	1	yes	characteristic
43	#19153747	C₂²	1	1	yes	characteristic
44	#19153741	C₂²	1	1	yes	characteristic
45	#19153740	C₂²	1	1	yes	characteristic
46	#19153740	C₂²	1	1	yes	characteristic
47	#19153739	C₂²	1	1	yes	characteristic
48	#19153738	C₂²	1	1	yes	characteristic
49	#19153736	C₂²	1	1	yes	characteristic
50	#19153726	C₂²	1	1	yes	characteristic
51	#19153726	C₂²	1	1	yes	characteristic
52	C₄×#13089678	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
56	#107×C₁₈₁	C₂²	1	1	yes	characteristic
57	#103×C₁₈₁	C₂²	1	1	yes	characteristic
58	#103×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
60	#15668450	—	2	2	no	—
61	#15668450	C₂×C₄	1	4	yes	—
62	C₂×Dic₃₆₂	—	2	1	no	—
63	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
64	Q₈×D₁₈₁	—	2	2	no	—
65	Q₈×D₁₈₁	C₂×C₄	1	4	yes	complemented
66	C₂×D₇₂₄	—	2	1	no	—
67	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₈₁	—	2	1	no	—
69	C₂×C₄×D₁₈₁	—	2	1	no	—
70	C₂×C₄×D₁₈₁	—	2	1	no	—
71	C₂×C₄×D₁₈₁	—	2	1	no	—
72	C₂×C₄×D₁₈₁	—	2	1	no	—
73	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
76	#15668439	C₂×C₄	1	4	yes	complemented
77	#15668438	C₂×C₄	1	4	yes	complemented
78	#15668436	C₂×C₄	1	4	yes	complemented
79	#15668432	—	2	2	no	—
80	#15668432	C₂×C₄	1	4	yes	complemented
81	C₂×#13089678	C₂³	1	1	yes	characteristic
82	C₂×#13089678	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₈₁	—	2	1	no	—
84	C₄⋊Dic₁₈₁	—	2	1	no	—
85	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
87	#15668428	—	2	1	no	—
88	#15668428	—	2	1	no	—
89	#15668428	—	2	1	no	—
90	#15668426	—	2	1	no	—
91	#15668426	—	2	1	no	—
92	#15668426	—	2	1	no	—
93	C₄×Dic₁₈₁	—	2	1	no	—
94	C₄×Dic₁₈₁	—	2	1	no	—
95	C₄×Dic₁₈₁	—	2	1	no	—
96	C₄×Dic₁₈₁	—	2	1	no	—
97	C₄×Dic₁₈₁	—	2	1	no	—
98	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
99	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
100	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
103	Q₁₆×C₁₈₁	C₂×C₄	1	4	yes	complemented
104	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₈₁	—	2	1	no	—
107	C₄⋊C₄×C₁₈₁	—	2	1	no	—
108	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
110	C₄²×C₁₈₁	—	2	1	no	—
111	C₄²×C₁₈₁	—	2	1	no	—
112	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
113	C₂²×D₁₈₁	—	2	1	no	—
114	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
115	Dic₃₆₂	—	2	1	no	—
116	Dic₃₆₂	—	2	4	no	—
117	Dic₃₆₂	C₂²×C₄	1	2	yes	—
118	D₇₂₄	—	2	4	no	—
119	D₇₂₄	—	2	1	no	—
120	D₇₂₄	C₂²×C₄	1	2	yes	—
121	C₂×Dic₁₈₁	—	2	1	no	—
122	C₂×Dic₁₈₁	—	2	1	no	—
123	C₂×Dic₁₈₁	—	2	1	no	—
124	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₈₁	—	2	4	no	—
128	C₄×D₁₈₁	—	2	2	no	—
129	C₄×D₁₈₁	—	2	4	no	—
130	C₄×D₁₈₁	—	4	1	no	—
131	C₄×D₁₈₁	—	4	2	no	—
132	C₄×D₁₈₁	—	4	2	no	—
133	C₄×D₁₈₁	—	4	2	no	—
134	C₄×D₁₈₁	C₂²×C₄	1	2	yes	—
135	C₄×D₁₈₁	#41	1	2	yes	—
136	#13089678	—	2	1	no	—
137	#13089678	C₂²×C₄	1	2	yes	—
138	Q₈×C₁₈₁	—	2	1	no	—
139	Q₈×C₁₈₁	—	2	1	no	—
140	Q₈×C₁₈₁	C₂²×C₄	1	2	yes	—
141	Q₈×C₁₈₁	C₂²×C₄	1	2	yes	—
142	C₂×C₄×C₁₈₁	—	2	1	no	—
143	C₂×C₄×C₁₈₁	—	2	1	no	—
144	C₂×C₄×C₁₈₁	—	2	1	no	—
145	C₂×C₄×C₁₈₁	—	2	1	no	—
146	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
149	C₈×C₁₈₁	—	2	1	no	—
150	C₈×C₁₈₁	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₈₁	—	2	1	no	—
153	C₂×D₁₈₁	—	2	1	no	—
154	C₂×D₁₈₁	—	2	2	no	—
155	C₂×D₁₈₁	—	4	1	no	—
156	C₂×D₁₈₁	—	4	1	no	—
157	C₂×D₁₈₁	C₄×D₄	1	2	yes	—
158	C₄×C₁₈₁	—	2	2	no	—
159	C₄×C₁₈₁	—	2	2	no	—
160	C₄×C₁₈₁	—	2	1	no	—
161	C₄×C₁₈₁	—	4	1	no	—
162	C₄×C₁₈₁	—	4	1	no	—
163	C₄×C₁₈₁	#137	1	2	yes	complemented
164	C₄×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
165	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
166	Dic₁₈₁	—	2	2	no	—
167	Dic₁₈₁	—	2	1	no	—
168	Dic₁₈₁	—	2	1	no	—
169	Dic₁₈₁	—	4	1	no	—
170	Dic₁₈₁	—	4	1	no	—
171	Dic₁₈₁	C₄×D₄	1	2	yes	—
172	C₂×C₁₈₁	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×C₁₈₁	C₂×#137	1	1	yes	characteristic
174	C₂×C₁₈₁	#575	1	1	yes	characteristic
175	D₁₈₁	—	2	2	no	—
176	D₁₈₁	—	4	2	no	—
177	C₁₈₁	C₄×#137	1	1	yes	characteristic, complemented, Sylow
178	C₄×#137	—	181	1	no	maximal, Sylow
179	C₂×#137	—	181	1	no	—
180	C₄×#41	—	181	1	no	—
181	C₂×C₄×Q₈	—	181	1	no	—
182	#441	—	181	1	no	—
183	#441	—	181	1	no	—
184	#440	—	181	1	no	—
185	#440	—	181	1	no	—
186	C₄×Q₁₆	—	181	1	no	—
187	C₄×Q₁₆	D₁₈₁	1	1	yes	characteristic, complemented
188	C₄×SD₁₆	—	181	1	no	—
189	C₄×SD₁₆	—	181	1	no	—
190	#428	—	181	1	no	—
191	#419	—	181	1	no	—
192	#417	—	181	1	no	—
193	C₄×M₁₆	—	181	1	no	—
194	C₂×#41	—	181	1	no	—
195	C₂²×Q₈	—	181	1	no	—
196	#137	—	181	8	no	—
197	C₂×Q₁₆	—	181	1	no	—
198	C₂×Q₁₆	C₂×D₁₈₁	1	1	yes	characteristic
199	C₂×SD₁₆	—	181	1	no	—
200	C₂×SD₁₆	—	181	1	no	—
201	C₂×M₁₆	—	181	1	no	—
202	C₂×M₁₆	—	181	1	no	—
203	C₄×Q₈	—	362	1	no	—
204	C₄×Q₈	—	362	1	no	—
205	C₄×Q₈	—	181	1	no	—
206	C₄×Q₈	—	181	1	no	—
207	C₄×Q₈	—	181	1	no	—
208	C₄×Q₈	—	181	1	no	—
209	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
211	C₄×D₄	—	362	1	no	—
212	C₄×D₄	—	362	1	no	—
213	C₄×D₄	—	181	1	no	—
214	C₄×D₄	—	181	1	no	—
215	#117	—	362	1	no	—
216	#117	—	181	1	no	—
217	C₂×C₄⋊C₄	—	362	1	no	—
218	C₂×C₄⋊C₄	—	181	1	no	—
219	C₂×C₄²	—	362	1	no	—
220	C₂×C₄²	—	362	1	no	—
221	C₂×C₄²	—	181	1	no	—
222	#107	—	181	1	no	—
223	#107	C₂×D₁₈₁	1	1	yes	characteristic
224	#106	—	181	1	no	—
225	#106	—	181	1	no	—
226	#103	—	181	1	no	—
227	#103	—	181	1	no	—
228	#103	—	181	1	no	—
229	#103	—	181	1	no	—
230	#103	C₂×D₁₈₁	1	1	yes	characteristic
231	#103	C₂×D₁₈₁	1	1	yes	characteristic
232	#102	—	181	1	no	—
233	#102	—	181	1	no	—
234	#97	—	181	1	no	—
235	#97	—	181	1	no	—
236	C₄×C₈	—	181	1	no	—
237	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
238	#41	—	362	2	no	—
239	#41	—	181	4	no	—
240	C₂×Q₈	—	362	1	no	—
241	C₂×Q₈	—	362	2	no	—
242	C₂×Q₈	—	181	4	no	—
243	C₂×Q₈	—	181	1	no	—
244	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
246	C₂×D₄	—	362	1	no	—
247	C₂×D₄	—	181	1	no	—
248	C₂²×C₄	—	362	1	no	—
249	C₂²×C₄	—	362	1	no	—
250	C₂²×C₄	—	362	1	no	—
251	C₂²×C₄	—	362	1	no	—
252	C₂²×C₄	—	362	1	no	—
253	C₂²×C₄	—	181	1	no	—
254	C₂²×C₄	—	181	1	no	—
255	C₂²×C₄	—	181	1	no	—
256	Q₁₆	—	181	4	no	—
257	Q₁₆	C₄×D₁₈₁	1	4	yes	complemented
258	SD₁₆	—	181	4	no	—
259	SD₁₆	—	181	4	no	—
260	M₁₆	—	362	2	no	—
261	M₁₆	—	181	4	no	—
262	C₂×C₈	—	181	1	no	—
263	C₂×C₈	—	181	1	no	—
264	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	362	1	no	—
269	C₄⋊C₄	—	362	1	no	—
270	C₄⋊C₄	—	362	1	no	—
271	C₄⋊C₄	—	362	1	no	—
272	C₄⋊C₄	—	362	1	no	—
273	C₄⋊C₄	—	181	1	no	—
274	C₄⋊C₄	—	181	1	no	—
275	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
277	C₂²⋊C₄	—	362	1	no	—
278	C₂²⋊C₄	—	362	1	no	—
279	C₂²⋊C₄	—	362	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	362	1	no	—
283	C₄²	—	362	1	no	—
284	C₄²	—	362	1	no	—
285	C₄²	—	362	1	no	—
286	C₄²	—	362	1	no	—
287	C₄²	—	181	1	no	—
288	C₄²	—	181	1	no	—
289	C₄²	—	181	1	no	—
290	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
291	C₂³	—	362	1	no	—
292	C₂³	—	181	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	362	4	no	—
296	Q₈	—	362	1	no	—
297	Q₈	—	181	2	no	—
298	Q₈	C₂×C₄×D₁₈₁	1	2	yes	—
299	Q₈	C₂×C₄×D₁₈₁	1	2	yes	—
300	D₄	—	362	4	no	—
301	D₄	—	362	1	no	—
302	D₄	—	181	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	362	4	no	—
308	C₂×C₄	—	362	4	no	—
309	C₂×C₄	—	362	1	no	—
310	C₂×C₄	—	362	2	no	—
311	C₂×C₄	—	362	1	no	—
312	C₂×C₄	—	362	1	no	—
313	C₂×C₄	—	181	1	no	—
314	C₂×C₄	—	181	1	no	—
315	C₂×C₄	—	181	1	no	—
316	C₂×C₄	—	181	2	no	—
317	C₂×C₄	—	181	2	no	—
318	C₂×C₄	—	724	1	no	—
319	C₂×C₄	—	724	2	no	—
320	C₂×C₄	—	724	2	no	—
321	C₂×C₄	—	724	2	no	—
322	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
323	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	362	1	no	—
327	C₈	—	181	2	no	—
328	C₈	C₂×C₄×D₁₈₁	1	2	yes	—
329	C₂²	—	362	1	no	—
330	C₂²	—	362	1	no	—
331	C₂²	—	362	2	no	—
332	C₂²	—	181	2	no	—
333	C₂²	—	724	1	no	—
334	C₂²	—	724	1	no	—
335	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	362	2	no	—
342	C₄	—	362	1	no	—
343	C₄	—	362	1	no	—
344	C₄	—	181	2	no	—
345	C₄	—	724	1	no	—
346	C₄	—	724	1	no	—
347	C₄	#19153848	1	2	yes	direct product factor
348	C₄	C₂²×C₄×D₁₈₁	1	1	yes	characteristic
349	C₄	#41×D₁₈₁	1	1	yes	characteristic
350	C₂	—	362	2	no	—
351	C₂	—	724	2	no	—
352	C₂	C₄×D₄×D₁₈₁	1	1	yes	characteristic
353	C₂	C₂×#19153848	1	1	yes	characteristic
354	C₂	#26042076	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₈₁
commutator	C₄×C₁₈₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	C₄×Q₁₆×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×#15668450	C₂²
cosocle	C₂×C₄×C₁₈₁	C₂⁴
hypercenter	C₄×Q₁₆	D₁₈₁
hypocenter	C₁₈₁	C₄×#137
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Q₈×D₁₈₁	C₂	semidirect
4. #26041386	C₂	semidirect
5. #26041383	C₂	semidirect
6. #26041382	C₂	semidirect
7. #26041378	C₂	semidirect
8. C₄×#15668439	C₂	semidirect
9. C₄×#15668438	C₂	semidirect
10. C₄×#15668436	C₂	semidirect
11. #26041320	C₂	semidirect
12. #26041286	C₂	semidirect
14. C₄×#15668432	C₂	semidirect
15. C₄×Q₁₆×C₁₈₁	C₂	semidirect
18. #19153848	C₄	direct (N × H)
65. Q₈×D₁₈₁	C₂×C₄	semidirect
76. #15668439	C₂×C₄	semidirect
77. #15668438	C₂×C₄	semidirect
78. #15668436	C₂×C₄	semidirect
80. #15668432	C₂×C₄	semidirect
103. Q₁₆×C₁₈₁	C₂×C₄	semidirect
163. C₄×C₁₈₁	#137	semidirect
177. C₁₈₁	C₄×#137	semidirect
187. C₄×Q₁₆	D₁₈₁	semidirect
257. Q₁₆	C₄×D₁₈₁	semidirect
347. C₄	#19153848	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19153848	2
C₄×#15668450	2
C₄×Q₈×D₁₈₁	2
#26041386	2
#26041383	2
#26041382	2
#26041378	2
C₄×#15668439	2
C₄×#15668438	2
C₄×#15668436	2
#26041320	2
#26041286	2
#26041279	2
C₄×#15668432	2
C₄×Q₁₆×C₁₈₁	2
C₄×#137	181

Subgroups — C4×#19153848 #37542541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19153848 #37542541