Subgroups — #31913648

Order 18176.

358 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986417	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985828	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590122	C₂	1	1	yes	maximal, characteristic, complemented
4	#22588395	C₂	1	1	yes	maximal, characteristic
5	#22588391	C₂	1	1	yes	maximal, characteristic
6	#22586455	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	C₂²	1	1	yes	characteristic
9	#17986417	—	2	1	no	—
10	#17986417	—	2	1	no	—
11	#17986417	C₄	1	2	yes	complemented
12	#17986417	C₄	1	2	yes	complemented
13	C₂×#14556067	—	2	1	no	—
14	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₇₁	—	2	1	no	—
16	C₂×#14556064	C₂²	1	1	yes	characteristic
17	#17986037	C₂²	1	1	yes	characteristic
18	#17986017	—	2	1	no	—
19	#17986015	—	2	1	no	—
20	#17985828	—	2	1	no	—
21	#17985828	—	2	1	no	—
22	#17985826	—	2	1	no	—
23	#17985824	—	2	1	no	—
24	#17985770	—	2	1	no	—
25	#17985769	—	2	1	no	—
26	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
28	#427×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×#11765865	D₄	1	1	yes	characteristic
30	#14556134	—	2	1	no	—
31	#14556134	—	2	1	no	—
32	#14556134	C₂×C₄	1	2	yes	—
33	#14556134	Q₈	1	2	yes	—
34	C₂×#11765849	—	2	1	no	—
35	C₂×#11765849	D₄	1	1	yes	characteristic
36	C₂×#11765848	—	2	1	no	—
37	C₂×C₈×D₇₁	—	2	1	no	—
38	#14556087	—	2	1	no	—
39	#14556067	—	2	2	no	—
40	#14556067	—	2	2	no	—
41	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
44	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
45	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
46	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
47	C₁₆×D₇₁	—	2	2	no	—
48	C₁₆×D₇₁	—	2	2	no	—
49	#14556064	—	2	1	no	—
50	#14556064	—	2	1	no	—
51	#14556062	—	2	1	no	—
52	#14556060	—	2	1	no	—
53	#14556056	—	4	2	no	—
54	#14556054	—	4	2	no	—
55	C₂×M₁₆×C₇₁	—	2	1	no	—
56	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
57	#117×C₇₁	—	2	1	no	—
58	M₃₂×C₇₁	—	2	1	no	—
59	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₇₁	—	2	1	no	—
61	#108×C₇₁	—	2	1	no	—
62	#108×C₇₁	—	2	1	no	—
63	#107×C₇₁	—	2	1	no	—
64	#106×C₇₁	—	2	1	no	—
65	#11765865	—	2	1	no	—
66	#11765865	—	2	1	no	—
67	#11765865	C₄⋊C₄	1	2	yes	—
68	#11765865	D₈	1	2	yes	—
69	C₂×Dic₁₄₂	Q₁₆	1	1	yes	characteristic
70	C₂×D₂₈₄	Q₁₆	1	1	yes	characteristic
71	C₂×C₂²⋊D₇₁	—	2	1	no	—
72	C₂×C₄×D₇₁	—	2	1	no	—
73	#11765849	—	2	1	no	—
74	#11765849	—	4	1	no	—
75	#11765849	—	4	1	no	—
76	#11765849	C₄⋊C₄	1	2	yes	—
77	#11765848	—	2	2	no	—
78	#11765848	—	2	2	no	—
79	C₂×C₇₁⋊C₈	—	2	1	no	—
80	C₂×C₇₁⋊C₈	—	4	1	no	—
81	C₈×D₇₁	—	2	2	no	—
82	C₈×D₇₁	—	2	2	no	—
83	C₄⋊Dic₇₁	—	4	1	no	—
84	C₂²⋊Dic₇₁	—	4	1	no	—
85	C₄×Dic₇₁	—	4	1	no	—
86	C₇₁⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
87	C₇₁⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
88	C₇₁⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
89	C₇₁⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
91	M₁₆×C₇₁	—	4	1	no	—
92	M₁₆×C₇₁	—	4	1	no	—
93	C₂×C₈×C₇₁	—	4	1	no	—
94	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
99	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	—	4	1	no	—
101	C₂²⋊C₄×C₇₁	—	4	1	no	—
102	C₄²×C₇₁	—	4	1	no	—
103	C₁₆×C₇₁	—	2	2	no	—
104	C₂³×C₇₁	#135	1	1	yes	characteristic
105	C₂²×D₇₁	—	2	1	no	—
106	C₂×C₄×C₇₁	—	8	1	no	—
107	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
113	C₂²⋊D₇₁	—	2	2	no	—
114	C₂²⋊D₇₁	—	2	2	no	—
115	C₂×Dic₇₁	—	2	1	no	—
116	C₂×Dic₇₁	—	8	1	no	—
117	D₂₈₄	—	2	1	no	—
118	D₂₈₄	#107	1	2	yes	—
119	C₄×D₇₁	—	2	2	no	—
120	C₄×D₇₁	—	2	2	no	—
121	Dic₁₄₂	—	2	1	no	—
122	Dic₁₄₂	#107	1	2	yes	—
123	C₈×C₇₁	—	8	1	no	—
124	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₇₁	#117	1	1	yes	characteristic
126	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
127	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
128	C₇₁⋊C₈	—	2	2	no	—
129	C₇₁⋊C₈	—	8	1	no	—
130	C₂²×C₇₁	—	4	1	no	—
131	C₂²×C₇₁	#427	1	1	yes	characteristic
132	C₂²×C₇₁	#466	1	1	yes	characteristic
133	C₂²×C₇₁	#467	1	1	yes	characteristic
134	C₂×D₇₁	—	2	2	no	—
135	C₂×D₇₁	—	4	2	no	—
136	C₄×C₇₁	—	16	1	no	—
137	C₄×C₇₁	#380	1	1	yes	characteristic
138	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
139	C₄×C₇₁	#466	1	1	yes	characteristic
140	C₄×C₇₁	#467	1	1	yes	characteristic
141	Dic₇₁	—	2	2	no	—
142	Dic₇₁	—	16	1	no	—
143	#12841	—	71	1	no	maximal, Sylow
144	C₂×C₇₁	—	2	1	no	—
145	C₂×C₇₁	—	2	1	no	—
146	C₂×C₇₁	—	2	1	no	—
147	C₂×C₇₁	#1808	1	1	yes	characteristic
148	D₇₁	—	4	2	no	—
149	C₂×#504	—	71	1	no	—
150	C₂×#368	—	71	1	no	—
151	#2158	—	71	1	no	—
152	#1814	—	71	1	no	—
153	#1814	—	71	1	no	—
154	#1389	—	71	1	no	—
155	#1389	D₇₁	1	1	yes	characteristic, complemented
156	C₇₁	#12841	1	1	yes	characteristic, complemented, Sylow
157	C₂×#131	—	71	1	no	—
158	#504	—	71	2	no	—
159	#504	—	71	2	no	—
160	#504	—	142	1	no	—
161	#504	—	142	1	no	—
162	C₂×M₃₂	—	142	1	no	—
163	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
164	C₂²×C₁₆	—	71	1	no	—
165	C₂²×C₁₆	—	142	1	no	—
166	C₂×#108	—	71	1	no	—
167	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
168	#427	—	71	1	no	—
169	#427	C₂×D₇₁	1	1	yes	characteristic
170	#421	—	142	1	no	—
171	#421	—	142	1	no	—
172	#368	—	142	1	no	—
173	#368	—	142	1	no	—
174	#367	—	142	1	no	—
175	#366	—	142	1	no	—
176	#344	—	142	1	no	—
177	#344	—	142	1	no	—
178	C₂×#41	—	71	1	no	—
179	#131	—	71	2	no	—
180	#131	—	71	2	no	—
181	#131	—	142	1	no	—
182	#131	—	142	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	71	1	no	—
185	C₂×M₁₆	—	142	1	no	—
186	C₂×M₁₆	—	142	1	no	—
187	C₂²×C₈	—	142	1	no	—
188	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	#117	—	142	1	no	—
191	M₃₂	—	2	1	no	—
192	M₃₂	—	142	2	no	—
193	M₃₂	—	142	2	no	—
194	M₃₂	C₄×D₇₁	1	2	yes	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	71	1	no	—
197	C₂×C₁₆	—	71	1	no	—
198	C₂×C₁₆	—	71	1	no	—
199	C₂×C₁₆	—	71	1	no	—
200	C₂×C₁₆	—	71	1	no	—
201	C₂×C₁₆	—	71	1	no	—
202	C₂×C₁₆	—	142	2	no	—
203	C₂×C₁₆	—	142	2	no	—
204	#108	—	2	1	no	—
205	#108	—	2	1	no	—
206	#108	—	142	1	no	—
207	#108	—	142	1	no	—
208	#107	—	2	1	no	—
209	#107	—	142	1	no	—
210	#106	—	2	1	no	—
211	#106	—	142	1	no	—
212	#104	—	284	2	no	—
213	#104	—	284	2	no	—
214	#41	—	71	2	no	—
215	#41	—	71	2	no	—
216	#41	—	142	1	no	—
217	#41	—	142	1	no	—
218	C₂×Q₈	—	71	1	no	—
219	C₂×D₄	—	71	1	no	—
220	C₂×D₄	—	142	1	no	—
221	C₂²×C₄	—	142	1	no	—
222	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	284	1	no	—
226	M₁₆	—	284	1	no	—
227	M₁₆	—	71	2	no	—
228	M₁₆	—	142	1	no	—
229	M₁₆	—	142	2	no	—
230	M₁₆	—	142	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	284	1	no	—
233	C₂×C₈	—	142	1	no	—
234	C₂×C₈	—	142	2	no	—
235	C₂×C₈	—	142	2	no	—
236	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
237	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
238	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
239	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
240	C₂×C₈	#11765865	1	1	yes	characteristic
241	C₂×C₈	#11765865	1	1	yes	characteristic
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	284	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₂²⋊C₄	—	284	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	284	1	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	71	1	no	—
250	C₁₆	—	71	1	no	—
251	C₁₆	—	71	1	no	—
252	C₁₆	—	71	1	no	—
253	C₂³	—	142	1	no	—
254	C₂³	#14556149	1	1	yes	characteristic
255	Q₈	—	71	2	no	—
256	Q₈	—	142	1	no	—
257	D₄	—	71	2	no	—
258	D₄	—	142	2	no	—
259	D₄	—	142	2	no	—
260	D₄	—	142	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	568	1	no	—
263	C₂×C₄	—	142	1	no	—
264	C₂×C₄	—	142	2	no	—
265	C₂×C₄	—	142	2	no	—
266	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
267	C₂×C₄	#14556101	1	1	yes	characteristic
268	C₂×C₄	#14556101	1	1	yes	characteristic
269	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
270	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
271	C₂×C₄	#14556149	1	1	yes	characteristic
272	C₈	—	8	1	no	—
273	C₈	—	568	1	no	—
274	C₈	—	142	2	no	—
275	C₈	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
276	C₈	#14556085	1	1	yes	characteristic
277	C₈	#14556106	1	1	yes	characteristic
278	C₈	#14556106	1	1	yes	characteristic
279	C₂²	—	4	1	no	—
280	C₂²	—	284	2	no	—
281	C₂²	—	142	2	no	—
282	C₂²	#17986034	1	1	yes	characteristic
283	C₂²	#17986234	1	1	yes	characteristic
284	C₂²	#17986239	1	1	yes	characteristic
285	C₄	—	16	1	no	—
286	C₄	—	1136	1	no	—
287	C₄	—	142	2	no	—
288	C₄	#17985858	1	1	yes	characteristic
289	C₄	#107×D₇₁	1	1	yes	characteristic
290	C₄	#17986234	1	1	yes	characteristic
291	C₄	#17986239	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	284	2	no	—
296	C₂	#22588368	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17986234
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1389	C₂
socle	C₂×C₇₁	#1808
prime radical	C₂×#11765865	D₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1389	D₇₁
hypocenter	C₇₁	#12841
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1389

Complements

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

N	H ≅ G/N	type
2. C₂×#17985828	C₂	semidirect
3. #22590122	C₂	semidirect
6. #22586455	C₂	semidirect
7. C₇₁×#1389	C₂	semidirect
11. #17986417	C₄	semidirect
12. #17986417	C₄	semidirect
155. #1389	D₇₁	semidirect
156. C₇₁	#12841	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986417	2
C₂×#17985828	2
#22590122	2
#22588395	2
#22588391	2
#22586455	2
C₇₁×#1389	2
#12841	71