Subgroups — #31966606

Order 18176.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22596973	C₂	1	2	yes	maximal
2	#22587120	C₂	1	8	yes	maximal, complemented
3	#22587099	C₂	1	2	yes	maximal, complemented
4	#22587096	C₂	1	2	yes	maximal, complemented
5	C₂×#431×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986056	—	2	2	no	—
7	#17986056	C₂²	1	4	yes	—
8	#17986052	—	2	2	no	—
9	#17986052	C₂²	1	4	yes	—
10	#17986023	C₂²	1	4	yes	—
11	#17985963	—	2	2	no	—
12	C₄×#11765849	—	2	2	no	—
13	#17985752	C₂²	1	8	yes	complemented
14	#17985750	C₂²	1	8	yes	complemented
15	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	#431×C₇₁	C₄	1	4	yes	—
17	#431×C₇₁	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈×C₇₁	C₄	1	2	yes	—
19	C₂×C₂²⋊C₈×C₇₁	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×#11765849	—	2	4	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
24	C₄⋊(C₇₁⋊C₈)	C₂³	1	8	yes	—
25	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
27	#14556039	—	2	4	no	—
28	C₄×C₇₁⋊C₈	—	2	4	no	—
29	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₇₁	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇₁	D₄	1	2	yes	—
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
40	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
41	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
43	C₂²⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₁	C₂³	1	2	yes	—
47	C₄×C₈×C₇₁	C₂³	1	2	yes	—
48	#11765849	—	4	8	no	—
49	C₂×C₇₁⋊C₈	—	2	8	no	—
50	C₂⁴×C₇₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	—	2	4	no	—
52	C₂²×C₄×C₇₁	—	4	2	no	—
53	C₂²×C₄×C₇₁	—	4	2	no	—
54	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
66	C₂×C₈×C₇₁	—	4	2	no	—
67	C₂×C₈×C₇₁	—	4	2	no	—
68	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
70	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
71	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
72	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₇₁	C₂×D₄	1	2	yes	—
76	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
78	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
80	C₂³×C₇₁	—	2	2	no	—
81	C₂³×C₇₁	—	4	1	no	—
82	C₂³×C₇₁	—	4	1	no	—
83	C₂³×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₁	—	2	2	no	—
86	C₂×C₄×C₇₁	—	2	2	no	—
87	C₂×C₄×C₇₁	—	2	4	no	—
88	C₂×C₄×C₇₁	—	2	4	no	—
89	C₂×C₄×C₇₁	—	2	4	no	—
90	C₂×C₄×C₇₁	—	2	4	no	—
91	C₂×C₄×C₇₁	—	2	2	no	—
92	C₂×C₄×C₇₁	—	4	1	no	—
93	C₂×C₄×C₇₁	—	4	2	no	—
94	C₂×C₄×C₇₁	—	4	1	no	—
95	C₂×C₄×C₇₁	—	4	2	no	—
96	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₁	#135	1	4	yes	—
103	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₇₁	—	2	4	no	—
107	C₈×C₇₁	—	2	4	no	—
108	C₇₁⋊C₈	—	4	8	no	—
109	C₂²×C₇₁	—	2	1	no	—
110	C₂²×C₇₁	—	2	1	no	—
111	C₂²×C₇₁	—	2	2	no	—
112	C₂²×C₇₁	—	2	2	no	—
113	C₂²×C₇₁	—	2	2	no	—
114	C₂²×C₇₁	—	4	4	no	—
115	C₂²×C₇₁	#427	1	2	yes	—
116	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₇₁	—	2	1	no	—
118	C₄×C₇₁	—	2	2	no	—
119	C₄×C₇₁	—	2	2	no	—
120	C₄×C₇₁	—	2	2	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	#427	1	2	yes	—
123	C₄×C₇₁	#430	1	2	yes	—
124	C₄×C₇₁	#568	1	2	yes	—
125	C₄×C₇₁	C₂×#135	1	2	yes	—
126	#17815	—	71	1	no	maximal, Sylow
127	C₂×C₇₁	—	2	2	no	—
128	C₂×C₇₁	—	4	2	no	—
129	C₂×C₇₁	#2905	1	1	yes	characteristic
130	C₂×C₇₁	C₂×#427	1	1	yes	characteristic
131	C₂×C₇₁	C₂×#430	1	1	yes	characteristic
132	#2920	—	71	2	no	—
133	C₂×#431	D₇₁	1	1	yes	characteristic
134	#1572	—	71	8	no	—
135	#1567	—	71	2	no	—
136	#1566	—	71	2	no	—
137	C₇₁	#17815	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
139	#432	—	71	4	no	—
140	#432	—	142	2	no	—
141	#431	—	71	4	no	—
142	#431	—	142	2	no	—
143	#431	Dic₇₁	1	4	yes	—
144	#431	C₂×D₇₁	1	4	yes	—
145	#423	—	71	4	no	—
146	C₂×C₄⋊C₈	Dic₇₁	1	2	yes	—
147	#407	—	142	2	no	—
148	C₂×C₂²⋊C₈	Dic₇₁	1	2	yes	—
149	C₄×M₁₆	—	142	2	no	—
150	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
152	#335	—	71	8	no	—
153	#334	—	71	8	no	—
154	C₂³×C₄	Dic₁₄₂	1	1	yes	characteristic
155	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
157	C₂×M₁₆	—	142	4	no	—
158	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
159	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
160	C₂×C₄²	Dic₁₄₂	1	1	yes	characteristic
161	C₂×C₄²	Dic₁₄₂	1	2	yes	—
162	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
163	C₂×C₄²	D₂₈₄	1	2	yes	—
164	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
166	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
168	C₄⋊C₈	—	71	8	no	—
169	C₄⋊C₈	—	142	4	no	—
170	C₄⋊C₈	C₂×Dic₇₁	1	8	yes	—
171	C₂²⋊C₈	—	142	4	no	—
172	C₂²⋊C₈	—	142	4	no	—
173	C₂²⋊C₈	C₂×Dic₇₁	1	8	yes	—
174	#97	—	142	4	no	—
175	C₄×C₈	—	142	4	no	—
176	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
177	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
178	C₄×C₈	C₂²×D₇₁	1	2	yes	—
179	C₄×C₈	C₂²×D₇₁	1	2	yes	—
180	C₂⁴	C₄⋊Dic₇₁	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
186	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
191	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
193	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
194	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
195	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
196	M₁₆	—	284	8	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	142	8	no	—
200	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
201	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
202	C₄²	C₄⋊Dic₇₁	1	2	yes	—
203	C₄²	C₄⋊Dic₇₁	1	2	yes	—
204	C₄²	C₄⋊Dic₇₁	1	2	yes	—
205	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
206	C₄²	C₂²×Dic₇₁	1	2	yes	—
207	C₄²	C₂×D₂₈₄	1	2	yes	—
208	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
209	C₄²	C₂×Dic₁₄₂	1	2	yes	—
210	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
211	C₄²	C₂³×D₇₁	1	1	yes	characteristic
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄⋊Dic₇₁	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
234	C₂×C₄	#14556150	1	4	yes	—
235	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₈	—	284	8	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	#17986037	1	2	yes	—
248	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	#17986037	1	2	yes	—
255	C₄	#17986048	1	2	yes	—
256	C₄	#17986766	1	2	yes	—
257	C₄	C₂×#14556150	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	#22596904	1	1	yes	characteristic
261	C₂	C₂×#17986037	1	1	yes	characteristic
262	C₂	C₂×#17986048	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₂×#431×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇₁	C₄⋊C₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	C₂×#431	D₇₁
hypocenter	C₇₁	#17815
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #22587120	C₂	semidirect
3. #22587099	C₂	semidirect
4. #22587096	C₂	semidirect
13. #17985752	C₂²	semidirect
14. #17985750	C₂²	semidirect
137. C₇₁	#17815	semidirect

Maximal subgroups

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

subgroup	index
#22596973	2
#22587120	2
#22587099	2
#22587096	2
C₂×#431×C₇₁	2
#17815	71