Subgroups — C₄×#14583272 #25105018

Order 11008.

1254 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583272	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10621873	C₂	1	4	yes	maximal, complemented
3	C₄×#13063325	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
5	#19135449	C₂	1	4	yes	maximal, complemented
6	#19135401	C₂	1	1	yes	maximal, characteristic
7	#19135397	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134752	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10621873	C₂²	1	4	yes	—
11	C₂×#13063325	C₂²	1	1	yes	characteristic
12	C₂×#13063322	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
14	C₂×C₄×D₁₇₂	—	2	2	no	—
15	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
16	#14583272	C₄	1	8	yes	direct product factor
17	C₄×#10621873	—	2	16	no	—
18	C₂×#13063257	C₂²	1	4	yes	—
19	C₂×#13063250	C₂²	1	4	yes	—
20	C₂×#13063247	C₂²	1	4	yes	—
21	#14582805	C₂²	1	2	yes	—
22	#14582801	—	2	2	no	—
23	#14582801	C₂²	1	2	yes	—
24	#14582797	—	2	2	no	—
25	#14582769	C₂²	1	1	yes	characteristic
26	#14582769	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
32	C₄³×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₈₆	C₂³	1	2	yes	—
34	C₂²×D₁₇₂	—	2	2	no	—
35	C₂²×D₁₇₂	C₂³	1	2	yes	—
36	C₂²×C₄×D₄₃	—	2	2	no	—
37	C₂×#10621873	—	2	8	no	—
38	C₂×#10621873	—	2	8	no	—
39	C₂×#10621873	C₂×C₄	1	16	yes	complemented
40	#13063325	—	2	2	no	—
41	#13063325	C₂×C₄	1	4	yes	complemented
42	#13063322	—	2	2	no	—
43	#13063322	C₂×C₄	1	4	yes	—
44	C₄×Dic₈₆	—	2	8	no	—
45	C₄×D₁₇₂	—	2	8	no	—
46	C₄×D₁₇₂	—	4	8	no	—
47	C₂×C₄⋊Dic₄₃	—	2	2	no	—
48	C₂×C₄⋊Dic₄₃	—	2	2	no	—
49	C₂×C₄⋊Dic₄₃	—	2	2	no	—
50	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
52	C₂×Dic₄₃⋊C₄	—	2	2	no	—
53	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
54	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
55	C₂×C₄×Dic₄₃	—	2	2	no	—
56	#13063257	—	2	8	no	—
57	#13063250	—	2	8	no	—
58	#13063247	—	2	8	no	—
59	C₄₃⋊#95	—	2	4	no	—
60	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₄₃	D₄	1	2	yes	—
63	C₂×C₄²×C₄₃	D₄	1	2	yes	—
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₃	—	2	4	no	—
68	C₄×C₈×C₄₃	—	2	2	no	—
69	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
70	C₄×C₈×C₄₃	D₄	1	8	yes	complemented
71	C₂³×D₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₃	—	2	2	no	—
78	C₂²×Dic₄₃	—	2	2	no	—
79	C₂²×Dic₄₃	—	2	2	no	—
80	C₂×D₁₇₂	—	2	8	no	—
81	C₂×D₁₇₂	—	2	2	no	—
82	C₂×D₁₇₂	—	2	4	no	—
83	C₂×D₁₇₂	—	2	4	no	—
84	C₂×D₁₇₂	—	4	4	no	—
85	C₂×D₁₇₂	—	4	4	no	—
86	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
87	C₂×C₄×D₄₃	—	4	4	no	—
88	C₂×C₄×D₄₃	—	4	8	no	—
89	C₂×Dic₈₆	—	2	4	no	—
90	C₂×Dic₈₆	—	2	4	no	—
91	C₂×Dic₈₆	—	2	2	no	—
92	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
97	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₄₃	#41	1	4	yes	—
99	C₄²×C₄₃	—	2	4	no	—
100	C₄²×C₄₃	—	2	2	no	—
101	C₄²×C₄₃	SD₁₆	1	8	yes	complemented
102	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₃	C₂×D₄	1	4	yes	—
104	C₄²×C₄₃	#41	1	2	yes	—
105	(C₂×D₄₃)⋊C₄	—	4	4	no	—
106	(C₂×D₄₃)⋊C₄	—	4	8	no	—
107	C₄⋊Dic₄₃	—	2	8	no	—
108	C₄⋊Dic₄₃	—	2	4	no	—
109	C₄⋊Dic₄₃	—	2	4	no	—
110	C₄⋊Dic₄₃	—	4	4	no	—
111	C₄⋊Dic₄₃	—	4	4	no	—
112	Dic₄₃⋊C₄	—	4	4	no	—
113	C₄×Dic₄₃	—	4	4	no	—
114	#10621873	—	2	32	no	—
115	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×D₄₃	—	2	4	no	—
117	C₂²×D₄₃	—	4	2	no	—
118	C₂²×D₄₃	—	4	2	no	—
119	C₂²×D₄₃	—	4	4	no	—
120	C₂²×D₄₃	—	4	4	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
125	C₂×C₄×C₄₃	#127	1	2	yes	—
126	C₂×C₄×C₄₃	#127	1	2	yes	—
127	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
128	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
130	C₂×C₄×C₄₃	#135	1	4	yes	—
131	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
134	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₄₃	—	2	4	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	—	4	4	no	—
141	C₂×Dic₄₃	—	4	4	no	—
142	D₁₇₂	—	2	8	no	—
143	D₁₇₂	—	4	4	no	—
144	D₁₇₂	—	4	16	no	—
145	C₄×D₄₃	—	8	8	no	—
146	Dic₈₆	—	2	8	no	—
147	Dic₈₆	—	4	4	no	—
148	C₈×C₄₃	—	2	4	no	—
149	C₈×C₄₃	C₄×D₄	1	8	yes	complemented
150	C₄×#492	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
153	C₂²×C₄₃	#532	1	1	yes	characteristic
154	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
155	C₂²×C₄₃	C₂×#135	1	2	yes	—
156	C₂×D₄₃	—	4	4	no	—
157	C₂×D₄₃	—	4	8	no	—
158	C₂×D₄₃	—	8	2	no	—
159	C₂×D₄₃	—	8	2	no	—
160	C₂×D₄₃	—	8	4	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	4	no	—
163	C₄×C₄₃	C₄×SD₁₆	1	8	yes	complemented
164	C₄×C₄₃	#492	1	4	yes	complemented
165	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
166	C₄×C₄₃	#532	1	2	yes	—
167	Dic₄₃	—	4	4	no	—
168	Dic₄₃	—	8	2	no	—
169	Dic₄₃	—	8	2	no	—
170	C₂×#492	—	43	1	no	—
171	C₂×C₄×SD₁₆	—	43	4	no	—
172	C₄×C₄⋊Q₈	—	43	1	no	—
173	C₄×#127	—	43	1	no	—
174	#1968	—	43	4	no	—
175	#1957	—	43	1	no	—
176	#1956	—	43	1	no	—
177	#1844	—	43	1	no	—
178	C₄²×C₈	D₄₃	1	1	yes	characteristic, complemented
179	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
180	C₂×C₄₃	C₂×C₄×SD₁₆	1	2	yes	—
181	C₂×C₄₃	C₂×#492	1	1	yes	characteristic
182	C₂×C₄₃	#3147	1	1	yes	characteristic
183	C₂×C₄₃	#3167	1	2	yes	—
184	D₄₃	—	8	4	no	—
185	C₂²×SD₁₆	—	43	4	no	—
186	C₂×C₄⋊Q₈	—	43	1	no	—
187	C₂×#127	—	43	1	no	—
188	C₂×C₄×Q₈	—	43	2	no	—
189	C₂×C₄×D₄	—	43	2	no	—
190	C₂×C₄×D₄	—	86	2	no	—
191	#492	—	43	8	no	—
192	C₄×SD₁₆	—	86	16	no	—
193	C₂×#106	—	43	4	no	—
194	C₂×#103	—	43	4	no	—
195	C₂×#102	—	43	4	no	—
196	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
198	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
200	#391	—	43	2	no	—
201	#390	—	43	2	no	—
202	#390	—	86	2	no	—
203	#389	—	86	2	no	—
204	#384	—	43	1	no	—
205	#384	—	43	1	no	—
206	C₄×C₄⋊C₄	—	86	2	no	—
207	C₄³	C₂×D₄₃	1	1	yes	characteristic
208	C₄₃	C₄×#492	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	—	43	2	no	—
210	C₂²×D₄	—	43	2	no	—
211	C₂²×D₄	—	86	2	no	—
212	C₂³×C₄	—	86	2	no	—
213	C₂×SD₁₆	—	43	16	no	—
214	C₂×SD₁₆	—	86	8	no	—
215	C₂×SD₁₆	—	86	8	no	—
216	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
217	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
218	C₄⋊Q₈	—	43	4	no	—
219	C₄⋊Q₈	—	86	2	no	—
220	#127	—	43	4	no	—
221	#127	—	86	2	no	—
222	C₄×Q₈	—	86	8	no	—
223	C₄×D₄	—	86	8	no	—
224	C₄×D₄	—	172	8	no	—
225	C₂×C₄⋊C₄	—	43	2	no	—
226	C₂×C₄⋊C₄	—	43	2	no	—
227	C₂×C₄⋊C₄	—	86	2	no	—
228	C₂×C₄⋊C₄	—	86	2	no	—
229	C₂×C₄⋊C₄	—	86	2	no	—
230	C₂×C₄⋊C₄	—	86	2	no	—
231	C₂×C₂²⋊C₄	—	86	2	no	—
232	C₂×C₂²⋊C₄	—	86	4	no	—
233	C₂×C₄²	—	86	2	no	—
234	C₂×C₄²	D₁₇₂	1	2	yes	—
235	C₂×C₄²	D₁₇₂	1	2	yes	—
236	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
239	#106	—	86	8	no	—
240	#103	—	86	8	no	—
241	#102	—	86	8	no	—
242	C₄×C₈	—	2	4	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
245	C₄×C₈	D₁₇₂	1	8	yes	complemented
246	#95	—	86	4	no	—
247	C₂⁴	—	86	2	no	—
248	C₂×Q₈	—	43	4	no	—
249	C₂×Q₈	—	86	2	no	—
250	C₂×Q₈	—	86	4	no	—
251	C₂×Q₈	—	86	4	no	—
252	C₂×D₄	—	43	4	no	—
253	C₂×D₄	—	86	8	no	—
254	C₂×D₄	—	86	4	no	—
255	C₂×D₄	—	86	4	no	—
256	C₂×D₄	—	86	2	no	—
257	C₂×D₄	—	172	4	no	—
258	C₂×D₄	—	172	4	no	—
259	C₂²×C₄	—	86	2	no	—
260	C₂²×C₄	—	86	2	no	—
261	C₂²×C₄	—	86	2	no	—
262	C₂²×C₄	—	172	4	no	—
263	C₂²×C₄	—	172	8	no	—
264	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
266	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
267	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
269	SD₁₆	—	86	32	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
273	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
274	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
275	C₂×C₈	#10621898	1	4	yes	—
276	C₄⋊C₄	—	86	8	no	—
277	C₄⋊C₄	—	86	4	no	—
278	C₄⋊C₄	—	86	4	no	—
279	C₄⋊C₄	—	172	4	no	—
280	C₄⋊C₄	—	172	4	no	—
281	C₄⋊C₄	—	172	4	no	—
282	C₂²⋊C₄	—	172	4	no	—
283	C₂²⋊C₄	—	172	8	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	172	4	no	—
287	C₄²	#10621873	1	8	yes	complemented
288	C₄²	C₂×C₄×D₄₃	1	2	yes	—
289	C₄²	C₂×D₁₇₂	1	4	yes	—
290	C₄²	#10621898	1	2	yes	—
291	C₂³	—	86	4	no	—
292	C₂³	—	172	4	no	—
293	C₂³	—	172	4	no	—
294	C₂³	—	172	2	no	—
295	C₂³	—	172	2	no	—
296	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
297	Q₈	—	86	8	no	—
298	Q₈	—	172	4	no	—
299	D₄	—	86	8	no	—
300	D₄	—	172	4	no	—
301	D₄	—	172	16	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	86	4	no	—
306	C₂×C₄	—	344	8	no	—
307	C₂×C₄	—	172	4	no	—
308	C₂×C₄	—	172	4	no	—
309	C₂×C₄	—	172	2	no	—
310	C₂×C₄	—	172	2	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
314	C₂×C₄	#13063322	1	2	yes	—
315	C₂×C₄	#13063322	1	2	yes	—
316	C₂×C₄	C₂×#10621873	1	4	yes	—
317	C₂×C₄	C₂×#10621873	1	4	yes	—
318	C₂×C₄	C₂×#10621873	1	4	yes	—
319	C₂×C₄	#13063347	1	4	yes	—
320	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
322	C₂×C₄	C₂×#10621898	1	2	yes	—
323	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	C₄×D₁₇₂	1	8	yes	complemented
326	C₂²	—	344	2	no	—
327	C₂²	—	344	2	no	—
328	C₂²	—	344	4	no	—
329	C₂²	—	172	4	no	—
330	C₂²	—	172	8	no	—
331	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
332	C₂²	C₂×#13063322	1	1	yes	characteristic
333	C₂²	#14583449	1	1	yes	characteristic
334	C₂²	C₂²×#10621873	1	2	yes	—
335	C₂²	C₂×#13063347	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	344	2	no	—
339	C₄	—	344	2	no	—
340	C₄	—	172	4	no	—
341	C₄	C₄×#10621873	1	8	yes	complemented
342	C₄	#14583272	1	4	yes	direct product factor
343	C₄	C₂×C₄×D₁₇₂	1	2	yes	—
344	C₄	#14583449	1	2	yes	—
345	C₂	—	344	4	no	—
346	C₂	C₄×#13063322	1	1	yes	characteristic
347	C₂	C₂×C₄×#10621873	1	2	yes	—
348	C₂	C₂×#14583272	1	1	yes	characteristic
349	C₂	#19145023	1	1	yes	characteristic
350	C₂	#19145154	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂×#13063322	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	C₄×#492
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#10621873	C₂	semidirect
3. C₄×#13063325	C₂	semidirect
5. #19135449	C₂	semidirect
7. #19135397	C₂	semidirect
8. #19134752	C₂	semidirect
9. C₄²×C₈×C₄₃	C₂	semidirect
16. #14583272	C₄	direct (N × H)
39. C₂×#10621873	C₂×C₄	semidirect
41. #13063325	C₂×C₄	semidirect
69. C₄×C₈×C₄₃	C₂×C₄	semidirect
70. C₄×C₈×C₄₃	D₄	semidirect
101. C₄²×C₄₃	SD₁₆	semidirect
149. C₈×C₄₃	C₄×D₄	semidirect
163. C₄×C₄₃	C₄×SD₁₆	semidirect
164. C₄×C₄₃	#492	semidirect
178. C₄²×C₈	D₄₃	semidirect
208. C₄₃	C₄×#492	semidirect
244. C₄×C₈	C₄×D₄₃	semidirect
245. C₄×C₈	D₁₇₂	semidirect
287. C₄²	#10621873	semidirect
325. C₈	C₄×D₁₇₂	semidirect
341. C₄	C₄×#10621873	semidirect
342. C₄	#14583272	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×#14583272	2
C₂×C₄×#10621873	2
C₄×#13063325	2
C₄×#13063322	2
#19135449	2
#19135401	2
#19135397	2
#19134752	2
C₄²×C₈×C₄₃	2
C₄×#492	43

Subgroups — C4×#14583272 #25105018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14583272 #25105018