Subgroups — #106×D₁₇₂ #25114697

Order 11008.

1126 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#106×D₄₃	C₂	1	2	yes	maximal, complemented
3	C₄⋊C₄×D₁₇₂	C₂	1	2	yes	maximal
4	#19135358	C₂	1	2	yes	maximal, complemented
5	#19134789	C₂	1	4	yes	maximal, complemented
6	#19134752	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134617	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₃	C₂²	1	2	yes	—
10	C₂×C₄×D₁₇₂	—	2	2	no	—
11	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
13	C₈×D₁₇₂	C₄	1	8	yes	complemented
14	C₂×#13063257	C₂²	1	2	yes	—
15	C₂×#13063257	C₂²	1	2	yes	—
16	C₂×#13063256	C₂²	1	4	yes	—
17	#106×D₄₃	—	2	8	no	—
18	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	—
20	#14582797	—	2	2	no	—
21	#14582773	—	2	4	no	—
22	#14582769	C₂²	1	2	yes	—
23	#14582756	C₂²	1	4	yes	—
24	#14582661	C₂²	1	4	yes	—
25	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
29	C₄₃×#340	C₂²	1	2	yes	—
30	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	—	2	4	no	—
32	C₂²×C₄×D₄₃	C₂³	1	2	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
34	C₂×C₈×D₄₃	—	2	4	no	—
35	C₂×C₈×D₄₃	—	2	4	no	—
36	C₂×C₈×D₄₃	C₂×C₄	1	8	yes	complemented
37	C₄×D₁₇₂	—	4	8	no	—
38	C₄×D₁₇₂	C₂×C₄	1	4	yes	—
39	C₄×D₁₇₂	Q₈	1	4	yes	—
40	C₂×C₄⋊Dic₄₃	—	2	2	no	—
41	C₂×C₄⋊Dic₄₃	—	2	2	no	—
42	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
44	C₂×Dic₄₃⋊C₄	—	2	4	no	—
45	C₄⋊C₄×D₄₃	—	2	8	no	—
46	C₄⋊C₄×D₄₃	—	4	4	no	—
47	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
48	C₂×(C₂×D₄₃)⋊C₄	D₄	1	2	yes	—
49	#13063257	—	2	4	no	—
50	#13063257	—	4	2	no	—
51	#13063256	—	2	8	no	—
52	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	complemented
53	(C₂×D₄₃)⋊C₈	C₂×C₄	1	8	yes	complemented
54	C₄₃⋊#95	—	2	4	no	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₃	—	2	2	no	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	#106×C₄₃	—	2	2	no	—
62	#106×C₄₃	D₄	1	4	yes	complemented
63	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
64	#95×C₄₃	—	2	2	no	—
65	C₂³×D₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₃	—	2	4	no	—
72	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
73	C₂×D₁₇₂	C₄⋊C₄	1	4	yes	—
74	C₂×D₁₇₂	SD₁₆	1	8	yes	—
75	C₂×C₄×D₄₃	—	2	2	no	—
76	C₂×C₄×D₄₃	—	2	2	no	—
77	C₂×C₄×D₄₃	—	2	2	no	—
78	C₂×C₄×D₄₃	—	2	2	no	—
79	C₂×C₄×D₄₃	—	4	8	no	—
80	C₂×C₄×D₄₃	—	4	8	no	—
81	C₂×C₄×D₄₃	—	8	4	no	—
82	C₂×C₄×D₄₃	C₂²×C₄	1	4	yes	—
83	C₂×C₄×D₄₃	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	1	no	—
86	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₃	#41	1	2	yes	—
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	C₂×D₄	1	4	yes	—
94	C₄²×C₄₃	—	4	4	no	—
95	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
97	(C₂×D₄₃)⋊C₄	—	4	8	no	—
98	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	8	yes	—
99	C₄⋊Dic₄₃	—	2	4	no	—
100	C₄⋊Dic₄₃	—	2	4	no	—
101	C₄⋊Dic₄₃	—	4	4	no	—
102	C₄⋊Dic₄₃	—	8	2	no	—
103	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
104	Dic₄₃⋊C₄	—	4	8	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂×C₄₃⋊C₈	—	2	2	no	—
107	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
108	C₈×D₄₃	—	2	16	no	—
109	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²×D₄₃	—	2	2	no	—
111	C₂²×D₄₃	—	2	2	no	—
112	C₂²×D₄₃	—	2	4	no	—
113	C₂²×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂²×D₄₃	C₂×SD₁₆	1	8	yes	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
125	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×Dic₄₃	—	2	2	no	—
132	C₂×Dic₄₃	—	2	2	no	—
133	C₂×Dic₄₃	—	4	4	no	—
134	C₂×Dic₄₃	—	4	4	no	—
135	C₂×Dic₄₃	—	4	4	no	—
136	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
137	D₁₇₂	#106	1	16	yes	direct product factor
138	C₄×D₄₃	—	2	4	no	—
139	C₄×D₄₃	—	2	4	no	—
140	C₄×D₄₃	—	2	4	no	—
141	C₄×D₄₃	—	2	4	no	—
142	C₄×D₄₃	—	8	16	no	—
143	C₈×C₄₃	—	2	2	no	—
144	C₈×C₄₃	C₄×D₄	1	4	yes	complemented
145	C₄₃⋊C₈	—	2	4	no	—
146	D₄×#106	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₃	D₄²	1	1	yes	characteristic
149	C₂²×C₄₃	#547	1	1	yes	characteristic
150	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
151	C₂²×C₄₃	#577	1	2	yes	—
152	C₂×D₄₃	—	2	8	no	—
153	C₂×D₄₃	—	2	4	no	—
154	C₂×D₄₃	—	2	4	no	—
155	C₂×D₄₃	—	2	8	no	—
156	C₂×D₄₃	C₂×#106	1	8	yes	complemented
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	4	4	no	—
159	C₄×C₄₃	—	8	2	no	—
160	C₄×C₄₃	C₂×#106	1	4	yes	complemented
161	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₄₃	#520	1	1	yes	characteristic
163	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
164	C₄×C₄₃	#554	1	1	yes	characteristic
165	Dic₄₃	—	2	4	no	—
166	Dic₄₃	—	8	4	no	—
167	C₂×C₈×D₄	—	43	1	no	—
168	C₂²×#106	—	43	2	no	—
169	D₄×C₄⋊C₄	—	43	2	no	—
170	#1948	—	43	2	no	—
171	#1853	—	43	4	no	—
172	#1844	—	43	1	no	—
173	#1807	—	43	2	no	—
174	C₄×#106	D₄₃	1	1	yes	characteristic, complemented
175	C₂×C₄₃	D₄×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄₃	C₂²×#106	1	1	yes	characteristic
177	C₂×C₄₃	#2912	1	1	yes	characteristic
178	C₂×C₄₃	D₄×SD₁₆	1	2	yes	—
179	C₂×C₄₃	#3335	1	2	yes	—
180	D₄₃	—	2	8	no	—
181	C₂³×C₈	—	43	2	no	—
182	C₂×C₄×D₄	—	43	1	no	—
183	C₂×C₄×D₄	—	86	2	no	—
184	C₂²×C₄⋊C₄	—	43	4	no	—
185	C₈×D₄	—	43	8	no	—
186	C₂×#106	—	43	2	no	—
187	C₂×#106	—	43	4	no	—
188	C₂×#106	—	43	2	no	—
189	C₂×#106	—	86	8	no	—
190	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
191	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
192	C₂×C₄⋊C₈	—	43	1	no	—
193	C₂×C₂²⋊C₈	—	43	2	no	—
194	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
195	#389	—	86	2	no	—
196	#385	—	86	4	no	—
197	#384	—	43	2	no	—
198	#380	—	43	4	no	—
199	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
200	#340	—	43	4	no	—
201	#340	C₂×D₄₃	1	2	yes	—
202	C₄₃	D₄×#106	1	1	yes	characteristic, complemented, Sylow
203	C₂²×D₄	—	43	1	no	—
204	C₂³×C₄	—	43	2	no	—
205	C₂³×C₄	—	86	4	no	—
206	C₂²×C₈	—	43	2	no	—
207	C₂²×C₈	—	43	8	no	—
208	C₂²×C₈	—	86	4	no	—
209	C₂²×C₈	—	86	4	no	—
210	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
212	C₄×D₄	—	43	4	no	—
213	C₄×D₄	—	43	4	no	—
214	C₄×D₄	—	172	8	no	—
215	C₂×C₄⋊C₄	—	43	1	no	—
216	C₂×C₄⋊C₄	—	43	4	no	—
217	C₂×C₄⋊C₄	—	86	8	no	—
218	C₂×C₄⋊C₄	—	86	2	no	—
219	C₂×C₄⋊C₄	—	86	4	no	—
220	C₂×C₄⋊C₄	—	86	2	no	—
221	C₂×C₄⋊C₄	—	172	4	no	—
222	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
223	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
224	C₂×C₂²⋊C₄	—	43	2	no	—
225	C₂×C₂²⋊C₄	—	86	4	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
228	#106	—	2	2	no	—
229	#106	—	86	4	no	—
230	#106	—	86	8	no	—
231	#106	—	172	2	no	—
232	#106	D₁₇₂	1	4	yes	direct product factor
233	C₄⋊C₈	—	43	4	no	—
234	C₂²⋊C₈	—	43	8	no	—
235	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
236	#95	—	2	2	no	—
237	#95	—	86	4	no	—
238	C₂⁴	—	43	2	no	—
239	C₂×D₄	—	43	4	no	—
240	C₂×D₄	—	43	8	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	43	2	no	—
244	C₂²×C₄	—	43	4	no	—
245	C₂²×C₄	—	43	4	no	—
246	C₂²×C₄	—	86	2	no	—
247	C₂²×C₄	—	86	2	no	—
248	C₂²×C₄	—	86	2	no	—
249	C₂²×C₄	—	86	2	no	—
250	C₂²×C₄	—	86	4	no	—
251	C₂²×C₄	—	344	4	no	—
252	C₂²×C₄	—	172	8	no	—
253	C₂²×C₄	—	172	8	no	—
254	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	43	4	no	—
260	C₂×C₈	—	86	2	no	—
261	C₂×C₈	—	86	2	no	—
262	C₂×C₈	—	86	16	no	—
263	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
265	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
266	C₂×C₈	#10621898	1	2	yes	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	43	4	no	—
271	C₄⋊C₄	—	86	4	no	—
272	C₄⋊C₄	—	86	4	no	—
273	C₄⋊C₄	—	344	2	no	—
274	C₄⋊C₄	—	172	8	no	—
275	C₄⋊C₄	—	172	4	no	—
276	C₄⋊C₄	C₂×D₁₇₂	1	4	yes	—
277	C₂²⋊C₄	—	43	8	no	—
278	C₂²⋊C₄	—	172	8	no	—
279	C₄²	—	4	4	no	—
280	C₄²	C₂×C₄×D₄₃	1	2	yes	—
281	C₄²	Q₈×D₄₃	1	2	yes	—
282	C₂³	—	43	4	no	—
283	C₂³	—	43	8	no	—
284	C₂³	—	86	2	no	—
285	C₂³	—	86	4	no	—
286	C₂³	—	86	2	no	—
287	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
288	D₄	—	43	16	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	43	4	no	—
297	C₂×C₄	—	86	2	no	—
298	C₂×C₄	—	86	2	no	—
299	C₂×C₄	—	86	4	no	—
300	C₂×C₄	—	86	4	no	—
301	C₂×C₄	—	86	4	no	—
302	C₂×C₄	—	86	4	no	—
303	C₂×C₄	—	344	16	no	—
304	C₂×C₄	—	172	4	no	—
305	C₂×C₄	—	172	4	no	—
306	C₂×C₄	—	172	4	no	—
307	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
308	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
309	C₂×C₄	SD₁₆×D₄₃	1	4	yes	—
310	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
312	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
313	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
314	C₂×C₄	#13063371	1	1	yes	characteristic
315	C₂×C₄	#13063372	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	86	4	no	—
318	C₈	C₄×D₁₇₂	1	4	yes	complemented
319	C₂²	—	43	8	no	—
320	C₂²	—	86	4	no	—
321	C₂²	—	86	8	no	—
322	C₂²	—	86	8	no	—
323	C₂²	—	86	4	no	—
324	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
325	C₂²	D₄×D₁₇₂	1	1	yes	characteristic
326	C₂²	#14583554	1	1	yes	characteristic
327	C₂²	C₂×SD₁₆×D₄₃	1	2	yes	—
328	C₂²	#14583706	1	2	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	4	4	no	—
331	C₄	—	8	2	no	—
332	C₄	—	86	4	no	—
333	C₄	—	344	4	no	—
334	C₄	#106×D₄₃	1	4	yes	complemented
335	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
336	C₄	#14583379	1	1	yes	characteristic
337	C₄	Q₈×D₁₇₂	1	1	yes	characteristic
338	C₄	#14583609	1	1	yes	characteristic
339	C₂	—	86	8	no	—
340	C₂	C₄⋊C₄×D₁₇₂	1	1	yes	characteristic
341	C₂	C₂×#106×D₄₃	1	1	yes	characteristic
342	C₂	#19143187	1	1	yes	characteristic
343	C₂	SD₁₆×D₁₇₂	1	2	yes	—
344	C₂	#19146954	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#106×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₁₇₂	D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#106	D₄₃
hypocenter	C₄₃	D₄×#106
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#106×D₄₃	C₂	semidirect
4. #19135358	C₂	semidirect
5. #19134789	C₂	semidirect
6. #19134752	C₂	semidirect
7. #19134617	C₂	semidirect
8. C₄×#106×C₄₃	C₂	semidirect
13. C₈×D₁₇₂	C₄	semidirect
36. C₂×C₈×D₄₃	C₂×C₄	semidirect
52. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
53. (C₂×D₄₃)⋊C₈	C₂×C₄	semidirect
62. #106×C₄₃	D₄	semidirect
63. C₄×C₈×C₄₃	C₂×C₄	semidirect
137. D₁₇₂	#106	direct (N × H)
144. C₈×C₄₃	C₄×D₄	semidirect
156. C₂×D₄₃	C₂×#106	semidirect
160. C₄×C₄₃	C₂×#106	semidirect
174. C₄×#106	D₄₃	semidirect
202. C₄₃	D₄×#106	semidirect
232. #106	D₁₇₂	direct (N × H)
235. C₄×C₈	C₄×D₄₃	semidirect
318. C₈	C₄×D₁₇₂	semidirect
334. C₄	#106×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₇₂	2
C₂×#106×D₄₃	2
C₄⋊C₄×D₁₇₂	2
#19135358	2
#19134789	2
#19134752	2
#19134617	2
C₄×#106×C₄₃	2
D₄×#106	43

Subgroups — #106×D172 #25114697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #106×D₁₇₂ #25114697