Subgroups — #31890632

Order 18176.

432 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597306	C₂	1	1	yes	maximal, characteristic
2	#22588361	C₂	1	2	yes	maximal
3	#22587099	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586243	C₂	1	2	yes	maximal, complemented
5	C₄×#106×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11765865	C₂²	1	1	yes	characteristic
7	#17986129	C₄	1	1	yes	characteristic, complemented
8	#17986129	C₄	1	1	yes	characteristic, complemented
9	#17986078	C₄	1	1	yes	characteristic, complemented
10	#17986078	C₄	1	1	yes	characteristic, complemented
11	#17986070	—	2	1	no	—
12	#17986070	—	2	1	no	—
13	#17986052	C₂²	1	1	yes	characteristic
14	#17986050	—	2	1	no	—
15	#17986023	C₂²	1	2	yes	—
16	C₂×#14556056	—	2	2	no	—
17	C₄×#11765848	—	2	1	no	—
18	#17985759	—	2	2	no	—
19	#17985752	—	2	1	no	—
20	#17985752	—	2	1	no	—
21	C₂×#106×C₇₁	—	2	1	no	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
24	#340×C₇₁	—	2	1	no	—
25	C₂×#11765865	D₄	1	1	yes	characteristic
26	C₂×#11765849	—	2	2	no	—
27	C₂×#11765848	—	2	1	no	—
28	C₂×#11765848	—	2	1	no	—
29	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₄₂	Q₈	1	1	yes	characteristic
31	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₈₄	Q₈	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₁	—	2	1	no	—
34	C₄×C₂²⋊D₇₁	—	2	1	no	—
35	#14556087	D₄	1	1	yes	characteristic
36	#14556085	—	2	1	no	—
37	C₄²×D₇₁	—	2	1	no	—
38	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
39	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
40	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	Dic₇₁⋊C₈	—	2	1	no	—
43	Dic₇₁⋊C₈	—	2	1	no	—
44	#14556056	—	4	4	no	—
45	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
46	(C₂×D₇₁)⋊C₈	—	2	1	no	—
47	(C₂×D₇₁)⋊C₈	—	2	1	no	—
48	#14556038	—	2	1	no	—
49	#14556038	—	2	1	no	—
50	C₄×C₇₁⋊C₈	—	2	1	no	—
51	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
54	C₂×C₄²×C₇₁	—	2	2	no	—
55	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
56	#106×C₇₁	—	2	4	no	—
57	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
61	#95×C₇₁	—	2	2	no	—
62	#11765865	—	2	2	no	—
63	#11765865	—	2	2	no	—
64	C₂×Dic₁₄₂	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₂₈₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	—	2	1	no	—
70	#11765849	—	4	2	no	—
71	#11765849	—	4	2	no	—
72	#11765848	—	4	2	no	—
73	#11765848	—	4	2	no	—
74	C₂×C₇₁⋊C₈	—	2	1	no	—
75	C₂×C₇₁⋊C₈	—	2	1	no	—
76	C₂×C₇₁⋊C₈	—	4	2	no	—
77	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
78	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
79	Dic₇₁⋊C₄	—	2	1	no	—
80	Dic₇₁⋊C₄	—	2	1	no	—
81	C₂²⋊Dic₇₁	—	2	1	no	—
82	(C₂×D₇₁)⋊C₄	—	2	1	no	—
83	(C₂×D₇₁)⋊C₄	—	2	1	no	—
84	C₄×Dic₇₁	—	2	1	no	—
85	C₄×Dic₇₁	—	2	1	no	—
86	C₄×Dic₇₁	—	2	1	no	—
87	C₂²×C₄×C₇₁	—	4	2	no	—
88	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₇₁	—	2	2	no	—
92	C₂×C₈×C₇₁	—	4	1	no	—
93	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₇₁	—	2	4	no	—
98	C₄⋊C₄×C₇₁	—	4	2	no	—
99	C₄²×C₇₁	—	4	2	no	—
100	C₄²×C₇₁	—	4	2	no	—
101	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₇₁	#41	1	1	yes	characteristic
105	C₂³×C₇₁	#121	1	1	yes	characteristic
106	C₂²×D₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	4	1	no	—
110	C₂×C₄×C₇₁	—	4	4	no	—
111	C₂×C₄×C₇₁	—	8	2	no	—
112	C₂×C₄×C₇₁	—	8	2	no	—
113	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
116	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₇₁	#137	1	2	yes	—
121	C₂²⋊D₇₁	—	4	2	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	D₂₈₄	—	2	2	no	—
126	C₄×D₇₁	—	4	2	no	—
127	C₄×D₇₁	—	4	1	no	—
128	C₄×D₇₁	—	4	1	no	—
129	C₄×D₇₁	—	4	1	no	—
130	C₄×D₇₁	—	4	1	no	—
131	Dic₁₄₂	—	2	2	no	—
132	C₈×C₇₁	—	2	1	no	—
133	C₈×C₇₁	—	2	1	no	—
134	C₈×C₇₁	—	2	1	no	—
135	C₈×C₇₁	#104	1	2	yes	complemented
136	C₇₁⋊C₈	—	4	1	no	—
137	C₇₁⋊C₈	—	4	1	no	—
138	C₇₁⋊C₈	—	8	2	no	—
139	C₂²×C₇₁	—	2	1	no	—
140	C₂²×C₇₁	—	2	1	no	—
141	C₂²×C₇₁	#380	1	1	yes	characteristic
142	C₂²×C₇₁	#468	1	2	yes	—
143	C₂×D₇₁	—	4	1	no	—
144	C₂×D₇₁	—	4	1	no	—
145	C₂×D₇₁	—	4	1	no	—
146	C₄×C₇₁	—	2	1	no	—
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	8	4	no	—
150	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
151	C₄×C₇₁	#421	1	1	yes	characteristic
152	C₄×C₇₁	#428	1	2	yes	—
153	C₄×C₇₁	#469	1	1	yes	characteristic
154	C₄×C₇₁	#470	1	1	yes	characteristic
155	Dic₇₁	—	4	1	no	—
156	Dic₇₁	—	4	1	no	—
157	Dic₇₁	—	4	1	no	—
158	#8669	—	71	1	no	maximal, Sylow
159	C₂×C₇₁	—	2	2	no	—
160	C₂×C₇₁	#1806	1	1	yes	characteristic
161	C₂×C₇₁	#1809	1	1	yes	characteristic
162	C₂×C₇₁	#1814	1	1	yes	characteristic
163	D₇₁	—	8	1	no	—
164	#2965	—	71	1	no	—
165	#1806	—	71	2	no	—
166	C₄×#106	D₇₁	1	1	yes	characteristic, complemented
167	#1567	—	71	1	no	—
168	#1303	—	71	2	no	—
169	C₇₁	#8669	1	1	yes	characteristic, complemented, Sylow
170	C₄×#41	—	71	1	no	—
171	#446	—	71	1	no	—
172	#446	—	71	1	no	—
173	#436	—	71	1	no	—
174	#436	—	71	1	no	—
175	#435	—	142	1	no	—
176	#435	—	142	1	no	—
177	#431	—	71	1	no	—
178	#431	—	142	1	no	—
179	C₂×#106	—	2	1	no	—
180	#423	—	71	2	no	—
181	C₂×#104	—	142	2	no	—
182	C₄×M₁₆	—	142	1	no	—
183	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
184	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
185	#340	—	2	1	no	—
186	#339	—	142	2	no	—
187	#335	—	142	1	no	—
188	#335	—	142	1	no	—
189	C₂×#41	—	71	1	no	—
190	C₂×M₁₆	—	142	2	no	—
191	C₂×M₁₆	—	142	1	no	—
192	C₂×M₁₆	—	142	1	no	—
193	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
195	C₄×Q₈	—	71	1	no	—
196	C₄×Q₈	—	71	1	no	—
197	C₄×D₄	—	71	1	no	—
198	C₄×D₄	—	71	1	no	—
199	C₄×D₄	—	142	1	no	—
200	C₄×D₄	—	142	1	no	—
201	#117	—	71	1	no	—
202	#117	—	142	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	142	1	no	—
206	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
207	#106	—	2	4	no	—
208	C₄⋊C₈	—	71	1	no	—
209	C₄⋊C₈	—	71	1	no	—
210	C₄⋊C₈	—	142	1	no	—
211	C₄⋊C₈	—	142	1	no	—
212	C₄⋊C₈	—	142	2	no	—
213	C₄⋊C₈	—	142	2	no	—
214	#104	—	284	4	no	—
215	C₂²⋊C₈	—	142	1	no	—
216	C₂²⋊C₈	—	142	1	no	—
217	C₂²⋊C₈	—	142	1	no	—
218	#97	—	142	1	no	—
219	#97	—	142	1	no	—
220	C₄×C₈	—	142	1	no	—
221	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
222	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
223	C₄×C₈	D₂₈₄	1	1	yes	characteristic
224	C₄×C₈	D₂₈₄	1	1	yes	characteristic
225	#95	—	2	2	no	—
226	#41	—	142	2	no	—
227	#41	—	142	2	no	—
228	C₂×Q₈	—	71	1	no	—
229	C₂×D₄	—	71	1	no	—
230	C₂×D₄	—	142	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	142	1	no	—
233	C₂²×C₄	—	142	1	no	—
234	C₂²×C₄	—	142	1	no	—
235	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
237	C₂²×C₄	#11765864	1	1	yes	characteristic
238	M₁₆	—	284	2	no	—
239	M₁₆	—	284	2	no	—
240	M₁₆	—	284	2	no	—
241	M₁₆	—	284	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	284	2	no	—
245	C₂×C₈	—	142	1	no	—
246	C₂×C₈	—	142	1	no	—
247	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
248	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
249	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
250	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	71	1	no	—
254	C₄⋊C₄	—	71	1	no	—
255	C₄⋊C₄	—	142	1	no	—
256	C₄⋊C₄	—	142	1	no	—
257	C₂²⋊C₄	—	142	1	no	—
258	C₂²⋊C₄	—	142	1	no	—
259	C₂²⋊C₄	—	142	1	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	142	1	no	—
263	C₄²	—	142	1	no	—
264	C₄²	—	142	1	no	—
265	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
266	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
267	C₄²	Q₈×D₇₁	1	1	yes	characteristic
268	C₄²	#11765866	1	1	yes	characteristic
269	C₂³	—	142	1	no	—
270	C₂³	#14556099	1	1	yes	characteristic
271	Q₈	—	142	2	no	—
272	D₄	—	284	2	no	—
273	D₄	—	142	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	284	2	no	—
281	C₂×C₄	—	284	1	no	—
282	C₂×C₄	—	284	1	no	—
283	C₂×C₄	—	284	1	no	—
284	C₂×C₄	—	284	1	no	—
285	C₂×C₄	—	142	1	no	—
286	C₂×C₄	—	142	1	no	—
287	C₂×C₄	—	142	1	no	—
288	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
289	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
290	C₂×C₄	#14556086	1	1	yes	characteristic
291	C₂×C₄	#14556103	1	1	yes	characteristic
292	C₂×C₄	#14556104	1	1	yes	characteristic
293	C₂×C₄	#14556108	1	1	yes	characteristic
294	C₂×C₄	#14556155	1	2	yes	—
295	C₂×C₄	#14556157	1	2	yes	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	284	1	no	—
300	C₈	—	284	1	no	—
301	C₈	—	568	2	no	—
302	C₈	#14556056	1	2	yes	complemented
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	284	1	no	—
306	C₂²	—	284	1	no	—
307	C₂²	—	284	1	no	—
308	C₂²	#17985856	1	1	yes	characteristic
309	C₂²	#17986242	1	2	yes	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	4	no	—
314	C₄	—	284	1	no	—
315	C₄	—	284	1	no	—
316	C₄	—	284	1	no	—
317	C₄	C₂×#14556056	1	1	yes	characteristic
318	C₄	#17986017	1	1	yes	characteristic
319	C₄	#17986038	1	2	yes	—
320	C₄	#17986253	1	1	yes	characteristic
321	C₄	#17986258	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	568	1	no	—
324	C₂	#22588361	1	1	yes	characteristic
325	C₂	#22588370	1	1	yes	characteristic
326	C₂	#22588391	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556103
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#106×C₇₁	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂×#11765865	D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#106	D₇₁
hypocenter	C₇₁	#8669
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
3. #22587099	C₂	semidirect
4. #22586243	C₂	semidirect
5. C₄×#106×C₇₁	C₂	semidirect
7. #17986129	C₄	semidirect
8. #17986129	C₄	semidirect
9. #17986078	C₄	semidirect
10. #17986078	C₄	semidirect
135. C₈×C₇₁	#104	semidirect
166. C₄×#106	D₇₁	semidirect
169. C₇₁	#8669	semidirect
302. C₈	#14556056	semidirect

Maximal subgroups

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

subgroup	index
#22597306	2
#22588361	2
#22587099	2
#22586243	2
C₄×#106×C₇₁	2
#8669	71