Subgroups — #31968175

Order 18176.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597275	C₂	1	2	yes	maximal
2	#22587191	C₂	1	8	yes	maximal, complemented
3	#22587099	C₂	1	1	yes	maximal, characteristic
4	#22586795	C₂	1	2	yes	maximal
5	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
6	C₇₁×#3170	C₂	1	1	yes	maximal, characteristic
7	#17986130	—	2	2	no	—
8	#17986130	C₂²	1	4	yes	—
9	Q₈×C₇₁⋊C₈	—	2	2	no	—
10	Q₈×C₇₁⋊C₈	C₂²	1	4	yes	—
11	#17986026	—	2	2	no	—
12	#17986023	C₂²	1	2	yes	—
13	#17985955	—	2	2	no	—
14	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	#17985752	C₂²	1	4	yes	—
17	Q₈⋊(C₇₁⋊C₈)	C₂²	1	8	yes	—
18	#17985721	C₂²	1	4	yes	—
19	#533×C₇₁	C₂²	1	2	yes	—
20	#500×C₇₁	C₄	1	1	yes	characteristic
21	#494×C₇₁	C₄	1	1	yes	characteristic
22	#487×C₇₁	C₄	1	2	yes	—
23	#467×C₇₁	C₄	1	4	yes	—
24	C₄×Q₁₆×C₇₁	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×#11765849	—	2	2	no	—
27	C₂²×C₇₁⋊C₈	—	2	1	no	—
28	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
32	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇₁⋊C₈)	C₂³	1	4	yes	—
34	#14556039	—	2	4	no	—
35	C₄×C₇₁⋊C₈	—	2	4	no	—
36	C₄×C₇₁⋊C₈	—	2	1	no	—
37	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
38	C₄×C₇₁⋊C₈	C₂³	1	2	yes	—
39	C₄×C₇₁⋊C₈	C₂³	1	2	yes	—
40	C₂×Q₁₆×C₇₁	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
42	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
43	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
44	#125×C₇₁	—	2	2	no	—
45	C₂²⋊Q₈×C₇₁	—	2	2	no	—
46	C₂²⋊Q₈×C₇₁	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₇₁	—	2	2	no	—
48	C₄×Q₈×C₇₁	C₂³	1	4	yes	—
49	#117×C₇₁	—	2	2	no	—
50	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
51	#107×C₇₁	C₂×C₄	1	4	yes	—
52	#103×C₇₁	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₇₁	C₂³	1	2	yes	—
56	#11765849	—	4	4	no	—
57	C₂×C₇₁⋊C₈	—	2	2	no	—
58	C₂×C₇₁⋊C₈	—	2	4	no	—
59	C₂×C₇₁⋊C₈	—	2	2	no	—
60	C₂×C₇₁⋊C₈	—	4	2	no	—
61	C₂×C₇₁⋊C₈	Q₁₆	1	4	yes	complemented
62	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
63	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
64	C₂×Q₈×C₇₁	—	2	2	no	—
65	C₂×Q₈×C₇₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
69	Q₁₆×C₇₁	—	4	4	no	—
70	C₂×C₈×C₇₁	—	4	2	no	—
71	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
73	C₄⋊C₄×C₇₁	—	2	2	no	—
74	C₄⋊C₄×C₇₁	—	2	2	no	—
75	C₄⋊C₄×C₇₁	—	2	2	no	—
76	C₄⋊C₄×C₇₁	—	2	4	no	—
77	C₄⋊C₄×C₇₁	—	2	4	no	—
78	C₄⋊C₄×C₇₁	C₂²×C₄	1	4	yes	—
79	C₂²⋊C₄×C₇₁	—	2	4	no	—
80	C₄²×C₇₁	—	2	4	no	—
81	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇₁	#41	1	1	yes	characteristic
84	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
85	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₇₁	—	2	8	no	—
87	Q₈×C₇₁	—	4	4	no	—
88	C₂×C₄×C₇₁	—	2	2	no	—
89	C₂×C₄×C₇₁	—	2	2	no	—
90	C₂×C₄×C₇₁	—	2	4	no	—
91	C₂×C₄×C₇₁	—	2	4	no	—
92	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₇₁	#135	1	2	yes	—
97	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
100	C₈×C₇₁	—	2	2	no	—
101	C₈×C₇₁	—	2	2	no	—
102	C₇₁⋊C₈	—	2	2	no	—
103	C₇₁⋊C₈	—	4	2	no	—
104	C₇₁⋊C₈	—	4	4	no	—
105	C₇₁⋊C₈	C₂×Q₁₆	1	4	yes	—
106	C₂²×C₇₁	—	2	1	no	—
107	C₂²×C₇₁	—	2	1	no	—
108	C₂²×C₇₁	C₄×Q₁₆	1	2	yes	—
109	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₇₁	—	2	1	no	—
111	C₄×C₇₁	—	2	1	no	—
112	C₄×C₇₁	—	4	4	no	—
113	C₄×C₇₁	—	4	4	no	—
114	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
115	C₄×C₇₁	#444	1	2	yes	—
116	C₄×C₇₁	#568	1	2	yes	—
117	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
118	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
119	#17977	—	71	1	no	maximal, Sylow
120	C₂×C₇₁	—	2	2	no	—
121	C₂×C₇₁	#2932	1	1	yes	characteristic
122	C₂×C₇₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
123	C₂×C₇₁	C₂×#444	1	1	yes	characteristic
124	#3170	D₇₁	1	1	yes	characteristic
125	#2960	—	71	2	no	—
126	#1584	—	71	8	no	—
127	#1567	—	71	1	no	—
128	#1493	—	71	2	no	—
129	C₂×#322	—	71	1	no	—
130	C₇₁	#17977	1	1	yes	characteristic, complemented, Sylow
131	#533	C₂×D₇₁	1	2	yes	—
132	#500	Dic₇₁	1	1	yes	characteristic
133	#494	Dic₇₁	1	1	yes	characteristic
134	#487	Dic₇₁	1	2	yes	—
135	#467	Dic₇₁	1	4	yes	—
136	#446	—	71	4	no	—
137	#446	—	142	2	no	—
138	C₈×Q₈	—	71	4	no	—
139	C₈×Q₈	—	142	2	no	—
140	C₄×Q₁₆	C₂×D₇₁	1	4	yes	—
141	#424	—	142	2	no	—
142	#423	—	71	2	no	—
143	#405	—	142	2	no	—
144	C₂×C₄×C₈	—	71	1	no	—
145	C₂×C₄×C₈	—	71	1	no	—
146	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
147	#335	—	71	4	no	—
148	Q₈⋊C₈	—	71	8	no	—
149	#322	—	71	4	no	—
150	C₂×Q₁₆	C₂×Dic₇₁	1	4	yes	—
151	C₂×M₁₆	—	142	2	no	—
152	C₂²×C₈	—	71	1	no	—
153	C₂²×C₈	—	71	1	no	—
154	C₂²×C₈	—	142	1	no	—
155	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
156	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
157	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
158	#125	—	2	2	no	—
159	C₂²⋊Q₈	—	2	2	no	—
160	C₂²⋊Q₈	C₂×Dic₇₁	1	4	yes	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	C₂²×D₇₁	1	4	yes	—
163	#117	—	2	2	no	—
164	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
165	#107	C₂×Dic₇₁	1	4	yes	—
166	C₄⋊C₈	—	71	4	no	—
167	C₄⋊C₈	—	142	4	no	—
168	C₄⋊C₈	—	142	4	no	—
169	C₄⋊C₈	—	142	2	no	—
170	#103	C₂×Dic₇₁	1	8	yes	—
171	#97	—	142	4	no	—
172	C₄×C₈	—	71	2	no	—
173	C₄×C₈	—	71	2	no	—
174	C₄×C₈	—	71	2	no	—
175	C₄×C₈	—	142	4	no	—
176	C₄×C₈	—	142	1	no	—
177	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
178	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₇₁	1	2	yes	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂²×Dic₇₁	1	4	yes	—
182	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
184	C₂²×C₄	#11765864	1	1	yes	characteristic
185	Q₁₆	—	4	4	no	—
186	M₁₆	—	284	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	284	2	no	—
189	C₂×C₈	—	71	2	no	—
190	C₂×C₈	—	71	4	no	—
191	C₂×C₈	—	71	2	no	—
192	C₂×C₈	—	142	4	no	—
193	C₂×C₈	—	142	2	no	—
194	C₂×C₈	—	142	2	no	—
195	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
196	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	C₂²×Dic₇₁	1	4	yes	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
206	C₄²	D₄×D₇₁	1	1	yes	characteristic
207	C₄²	#11765864	1	1	yes	characteristic
208	C₄²	C₂³×D₇₁	1	1	yes	characteristic
209	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
210	Q₈	—	2	8	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
217	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
218	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
219	C₂×C₄	Q₁₆×D₇₁	1	2	yes	—
220	C₂×C₄	#14556151	1	2	yes	—
221	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
223	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	284	2	no	—
227	C₈	—	284	4	no	—
228	C₈	—	71	4	no	—
229	C₈	—	142	2	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	Q₁₆×Dic₇₁	1	2	yes	—
233	C₂²	C₂×D₄×Dic₇₁	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	4	no	—
237	C₄	—	4	4	no	—
238	C₄	Q₁₆×Dic₇₁	1	2	yes	—
239	C₄	#17986121	1	2	yes	—
240	C₄	#17986766	1	2	yes	—
241	C₄	C₂×Q₁₆×D₇₁	1	1	yes	characteristic
242	C₄	C₂×#14556151	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#22597058	1	1	yes	characteristic
245	C₂	C₂×Q₁₆×Dic₇₁	1	1	yes	characteristic
246	C₂	C₂×#17986121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₇₁×#3170	C₂
socle	C₂²×C₇₁	C₂×C₄×D₄
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	#3170	D₇₁
hypocenter	C₇₁	#17977
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #22587191	C₂	semidirect
61. C₂×C₇₁⋊C₈	Q₁₆	semidirect
130. C₇₁	#17977	semidirect

Maximal subgroups

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

subgroup	index
#22597275	2
#22587191	2
#22587099	2
#22586795	2
C₂×#17985721	2
C₇₁×#3170	2
#17977	71