Subgroups — C₂×#22586303 #31907186

Order 18176.

406 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	C₂×C₄×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22586303	C₂	1	4	yes	maximal, direct product factor
4	C₂×#335×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₈₄	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₄₂	—	2	1	no	—
7	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
8	C₄×Dic₂₈₄	—	2	1	no	—
9	C₄×Dic₂₈₄	—	2	1	no	—
10	C₄×Dic₂₈₄	C₄	1	2	yes	—
11	C₄×Dic₂₈₄	C₂²	1	2	yes	—
12	C₂×#14556062	C₄	1	1	yes	characteristic
13	C₂×#14556053	—	2	1	no	—
14	C₄×C₇₁⋊C₁₆	—	2	1	no	—
15	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#335×C₇₁	—	2	1	no	—
19	#335×C₇₁	C₂²	1	2	yes	—
20	C₂²×Dic₁₄₂	—	2	1	no	—
21	C₂×Dic₂₈₄	—	2	1	no	—
22	C₂×Dic₂₈₄	—	2	2	no	—
23	C₂×Dic₂₈₄	C₈	1	4	yes	complemented
24	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₄₂	—	2	2	no	—
26	C₄×Dic₁₄₂	—	2	2	no	—
27	C₄×Dic₁₄₂	—	4	1	no	—
28	C₄×Dic₁₄₂	—	4	1	no	—
29	C₂×C₄⋊Dic₇₁	—	2	1	no	—
30	C₂×Dic₇₁⋊C₄	—	4	1	no	—
31	C₂×C₄×Dic₇₁	—	4	1	no	—
32	C₂×C₇₁⋊C₁₆	—	2	4	no	—
33	C₂×C₇₁⋊C₁₆	—	2	2	no	—
34	#14556062	—	2	1	no	—
35	#14556062	C₂×C₄	1	2	yes	—
36	#14556053	—	2	2	no	—
37	#14556053	—	2	2	no	—
38	C₂²×C₈×C₇₁	—	4	1	no	—
39	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₄⋊C₈×C₇₁	—	2	2	no	—
43	C₄⋊C₈×C₇₁	—	4	1	no	—
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₄₂	—	2	4	no	—
49	C₂×Dic₁₄₂	—	2	2	no	—
50	C₂×Dic₁₄₂	—	4	1	no	—
51	C₂×Dic₁₄₂	—	4	2	no	—
52	C₂²×Dic₇₁	—	4	1	no	—
53	C₂²×Dic₇₁	—	4	1	no	—
54	Dic₂₈₄	—	2	4	no	—
55	Dic₂₈₄	—	2	2	no	—
56	Dic₂₈₄	C₂×C₈	1	4	yes	complemented
57	C₄⋊Dic₇₁	—	2	2	no	—
58	C₄⋊Dic₇₁	—	4	1	no	—
59	Dic₇₁⋊C₄	—	4	2	no	—
60	Dic₇₁⋊C₄	—	4	2	no	—
61	C₄×Dic₇₁	—	4	2	no	—
62	C₄×Dic₇₁	—	4	2	no	—
63	C₇₁⋊C₁₆	—	2	4	no	—
64	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
67	C₂×C₈×C₇₁	—	2	2	no	—
68	C₂×C₈×C₇₁	—	4	2	no	—
69	C₂×C₈×C₇₁	—	8	2	no	—
70	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
74	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇₁	D₈	1	1	yes	characteristic
78	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
79	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
80	C₂³×C₇₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₁	—	2	2	no	—
82	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#104	1	2	yes	—
87	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
88	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
89	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×Dic₇₁	—	4	4	no	—
93	C₂×Dic₇₁	—	4	2	no	—
94	C₂×Dic₇₁	—	4	2	no	—
95	C₂×Dic₇₁	—	8	2	no	—
96	Dic₁₄₂	—	2	4	no	—
97	Dic₁₄₂	—	4	4	no	—
98	Dic₁₄₂	—	4	2	no	—
99	C₈×C₇₁	—	2	1	no	—
100	C₈×C₇₁	—	2	1	no	—
101	C₈×C₇₁	—	8	2	no	—
102	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
103	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
104	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
105	C₂²×C₇₁	D₄⋊C₈	1	2	yes	—
106	C₂²×C₇₁	#358	1	2	yes	—
107	C₂²×C₇₁	#359	1	2	yes	—
108	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
109	C₄×C₇₁	—	2	1	no	—
110	C₄×C₇₁	—	2	1	no	—
111	C₄×C₇₁	D₄⋊C₈	1	2	yes	—
112	C₄×C₇₁	#358	1	2	yes	—
113	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
115	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
116	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
117	Dic₇₁	—	4	4	no	—
118	Dic₇₁	—	8	2	no	—
119	C₂×#1332	—	71	1	no	maximal, Sylow
120	C₂×C₇₁	#1332	1	4	yes	complemented
121	C₂×C₇₁	C₂×D₄⋊C₈	1	1	yes	characteristic
122	C₂×C₇₁	C₂×#358	1	1	yes	characteristic
123	C₂×C₇₁	C₂×#359	1	1	yes	characteristic
124	C₂×C₄×Q₁₆	—	71	1	no	—
125	C₂×C₄×C₁₆	—	71	1	no	—
126	C₂×#335	D₇₁	1	1	yes	characteristic
127	#1332	—	71	4	no	—
128	C₇₁	C₂×#1332	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	—	71	1	no	—
130	C₂×C₄×Q₈	—	142	1	no	—
131	C₂²×C₁₆	—	142	1	no	—
132	C₄×Q₁₆	—	71	2	no	—
133	C₄×Q₁₆	—	71	2	no	—
134	C₄×Q₁₆	—	142	1	no	—
135	C₄×Q₁₆	—	142	1	no	—
136	C₂×#107	—	71	1	no	—
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×#103	—	142	1	no	—
139	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
140	C₄×C₁₆	—	71	2	no	—
141	C₄×C₁₆	—	142	1	no	—
142	#335	—	2	1	no	—
143	#335	C₂×D₇₁	1	2	yes	—
144	C₂²×Q₈	—	142	1	no	—
145	C₂×Q₁₆	—	71	4	no	—
146	C₂×Q₁₆	—	71	2	no	—
147	C₂×Q₁₆	—	142	1	no	—
148	C₂×Q₁₆	—	142	2	no	—
149	C₂²×C₈	—	4	1	no	—
150	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
151	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
152	C₄×Q₈	—	284	1	no	—
153	C₄×Q₈	—	284	1	no	—
154	C₄×Q₈	—	142	2	no	—
155	C₄×Q₈	—	142	2	no	—
156	C₂×C₄⋊C₄	—	284	1	no	—
157	C₂×C₄⋊C₄	—	142	1	no	—
158	C₂×C₄²	—	284	1	no	—
159	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
160	C₂×C₁₆	—	142	4	no	—
161	C₂×C₁₆	—	142	2	no	—
162	#107	—	71	2	no	—
163	#107	—	142	1	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	4	1	no	—
166	#103	—	142	2	no	—
167	#103	—	142	2	no	—
168	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
169	C₄×C₈	D₂₈₄	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
172	C₂×Q₈	—	284	1	no	—
173	C₂×Q₈	—	284	2	no	—
174	C₂×Q₈	—	142	4	no	—
175	C₂×Q₈	—	142	2	no	—
176	C₂²×C₄	—	284	1	no	—
177	C₂²×C₄	—	284	1	no	—
178	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	#11765850	1	1	yes	characteristic
180	C₂²×C₄	#11765852	1	1	yes	characteristic
181	Q₁₆	—	71	4	no	—
182	Q₁₆	—	142	4	no	—
183	Q₁₆	—	142	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	8	2	no	—
187	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
188	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
189	C₂×C₈	C₈×D₇₁	1	2	yes	—
190	C₂×C₈	#11765848	1	2	yes	—
191	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
192	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
193	C₄⋊C₄	—	284	2	no	—
194	C₄⋊C₄	—	284	1	no	—
195	C₄⋊C₄	—	284	2	no	—
196	C₄⋊C₄	—	142	2	no	—
197	C₄²	—	284	2	no	—
198	C₄²	—	284	2	no	—
199	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
200	C₄²	#11765850	1	1	yes	characteristic
201	C₄²	#11765852	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
203	C₁₆	—	142	4	no	—
204	C₂³	#14556048	1	1	yes	characteristic
205	Q₈	—	284	4	no	—
206	Q₈	—	284	2	no	—
207	Q₈	—	142	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	284	4	no	—
210	C₂×C₄	—	284	2	no	—
211	C₂×C₄	—	284	2	no	—
212	C₂×C₄	—	568	2	no	—
213	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
214	C₂×C₄	#14556048	1	1	yes	characteristic
215	C₂×C₄	#14556048	1	1	yes	characteristic
216	C₂×C₄	#14556048	1	1	yes	characteristic
217	C₂×C₄	#14556054	1	2	yes	—
218	C₂×C₄	#14556072	1	2	yes	—
219	C₂×C₄	#14556074	1	2	yes	—
220	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
221	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
222	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	8	2	no	—
226	C₈	(C₂×D₇₁)⋊C₈	1	2	yes	—
227	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
228	C₈	C₂×#11765848	1	1	yes	characteristic
229	C₂²	#17985726	1	2	yes	—
230	C₂²	#17985802	1	2	yes	—
231	C₂²	#17985805	1	2	yes	—
232	C₂²	C₂×#14556048	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	284	4	no	—
236	C₄	—	568	2	no	—
237	C₄	#17985726	1	2	yes	—
238	C₄	#17985802	1	2	yes	—
239	C₄	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
240	C₄	C₂×#14556054	1	1	yes	characteristic
241	C₄	C₂×#14556072	1	1	yes	characteristic
242	C₄	C₂×#14556074	1	1	yes	characteristic
243	C₂	#22586303	1	4	yes	direct product factor
244	C₂	C₂×#17985726	1	1	yes	characteristic
245	C₂	C₂×#17985802	1	1	yes	characteristic
246	C₂	C₂×#17985805	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11765850
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×#335×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×#335	D₇₁
hypocenter	C₇₁	C₂×#1332
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #22586303	C₂	direct (N × H)
23. C₂×Dic₂₈₄	C₈	semidirect
56. Dic₂₈₄	C₂×C₈	semidirect
120. C₂×C₇₁	#1332	semidirect
128. C₇₁	C₂×#1332	semidirect
243. C₂	#22586303	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₈₄	2
C₂×C₄×C₇₁⋊C₁₆	2
#22586303	2
C₂×#335×C₇₁	2
C₂×#1332	71

Subgroups — C2×#22586303 #31907186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586303 #31907186