Subgroups — C₂×#22562357 #31010441

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic
3	#22562357	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863224	C₂	1	2	yes	maximal, complemented
5	C₂×#16863198	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
8	C₂×#14524043	—	2	1	no	—
9	C₂×C₄×D₂₆₈	—	2	1	no	—
10	#16863801	—	2	2	no	—
11	#16863801	C₂²	1	4	yes	complemented
12	C₄×D₅₃₆	—	2	2	no	—
13	C₄×D₅₃₆	C₂²	1	4	yes	—
14	C₂×#14523977	—	2	1	no	—
15	C₂×#14523977	C₂²	1	1	yes	characteristic
16	C₂×#14523977	C₂²	1	2	yes	—
17	C₂×#14523965	—	2	1	no	—
18	#16863224	C₂²	1	8	yes	complemented
19	#16863198	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	complemented
23	C₂²×Dic₁₃₄	—	4	1	no	—
24	C₂²×D₂₆₈	—	2	1	no	—
25	C₂²×C₄×D₆₇	—	4	1	no	—
26	C₂×D₅₃₆	—	2	1	no	—
27	C₂×D₅₃₆	—	2	2	no	—
28	C₂×D₅₃₆	Q₈	1	4	yes	complemented
29	C₂×D₅₃₆	C₂³	1	2	yes	—
30	#14524043	—	2	4	no	—
31	#14524043	—	4	2	no	—
32	C₄×D₂₆₈	—	2	4	no	—
33	C₄×D₂₆₈	—	4	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	1	no	—
35	C₂×C₄⋊Dic₆₇	—	2	2	no	—
36	C₂×C₄⋊Dic₆₇	—	4	1	no	—
37	C₂×(C₂×D₆₇)⋊C₄	—	4	1	no	—
38	#14523977	—	2	2	no	—
39	#14523977	—	2	4	no	—
40	#14523977	—	2	1	no	—
41	#14523977	C₂³	1	4	yes	—
42	#14523977	C₂³	1	2	yes	—
43	#14523965	—	2	4	no	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₁₆×C₆₇	—	2	4	no	—
48	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	C₄×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂³×D₆₇	—	4	1	no	—
52	C₂×Dic₁₃₄	—	4	4	no	—
53	C₂×Dic₁₃₄	—	8	4	no	—
54	C₂×D₂₆₈	—	2	4	no	—
55	C₂×D₂₆₈	—	2	2	no	—
56	C₂×D₂₆₈	—	4	1	no	—
57	C₂×D₂₆₈	—	4	2	no	—
58	C₂²×Dic₆₇	—	4	1	no	—
59	C₂²×Dic₆₇	—	4	2	no	—
60	C₂×C₄×D₆₇	—	4	4	no	—
61	C₂×C₄×D₆₇	—	8	4	no	—
62	D₅₃₆	—	2	4	no	—
63	D₅₃₆	—	2	2	no	—
64	D₅₃₆	C₂×Q₈	1	4	yes	complemented
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₄⋊Dic₆₇	—	2	4	no	—
67	C₄⋊Dic₆₇	—	4	1	no	—
68	C₄⋊Dic₆₇	—	4	2	no	—
69	C₄⋊Dic₆₇	—	4	4	no	—
70	(C₂×D₆₇)⋊C₄	—	4	4	no	—
71	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₆₇	#41	1	2	yes	—
79	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₄²×C₆₇	D₈	1	2	yes	—
81	C₄²×C₆₇	C₂×D₄	1	2	yes	—
82	C₁₆×C₆₇	—	2	4	no	—
83	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₆₇	—	4	2	no	—
85	C₂²×D₆₇	—	4	4	no	—
86	C₂²×D₆₇	—	8	2	no	—
87	C₂²×D₆₇	—	8	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
90	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₆₇	#137	1	2	yes	—
95	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×Dic₆₇	—	4	2	no	—
97	C₂×Dic₆₇	—	4	4	no	—
98	C₂×Dic₆₇	—	8	2	no	—
99	C₂×Dic₆₇	—	8	4	no	—
100	D₂₆₈	—	2	4	no	—
101	D₂₆₈	—	4	2	no	—
102	D₂₆₈	—	4	4	no	—
103	C₄×D₆₇	—	8	8	no	—
104	Dic₁₃₄	—	8	8	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
107	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂²×C₆₇	#474	1	2	yes	—
110	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
111	C₂²×C₆₇	#509	1	2	yes	—
112	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₆₇	—	4	4	no	—
114	C₂×D₆₇	—	8	2	no	—
115	C₂×D₆₇	—	8	2	no	—
116	C₂×D₆₇	—	8	8	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	#474	1	2	yes	—
119	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
120	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
122	Dic₆₇	—	8	2	no	—
123	Dic₆₇	—	8	4	no	—
124	C₂×#2226	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	#2226	1	4	yes	complemented
126	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
127	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
128	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
129	D₆₇	—	8	4	no	—
130	C₂×#500	—	67	1	no	—
131	C₂×C₄×D₈	—	67	1	no	—
132	#2226	—	67	8	no	—
133	C₂×#367	—	67	2	no	—
134	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic, complemented
135	C₂×#357	—	67	2	no	—
136	C₆₇	C₂×#2226	1	1	yes	characteristic, complemented, Sylow
137	C₂²×D₈	—	67	1	no	—
138	C₂×C₄⋊Q₈	—	134	1	no	—
139	C₂×C₄×D₄	—	134	1	no	—
140	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
141	#500	—	67	4	no	—
142	#500	—	134	2	no	—
143	C₄×D₈	—	67	4	no	—
144	C₄×D₈	—	134	2	no	—
145	C₂×#107	—	67	2	no	—
146	C₂×#107	—	67	1	no	—
147	C₂×#107	—	134	1	no	—
148	C₂×#102	—	134	1	no	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	#367	—	67	8	no	—
151	C₄⋊C₁₆	C₂×D₆₇	1	4	yes	complemented
152	#357	—	67	8	no	—
153	C₂²×Q₈	—	268	1	no	—
154	C₂²×D₄	—	134	1	no	—
155	C₂³×C₄	—	268	1	no	—
156	C₂×D₈	—	67	4	no	—
157	C₂×D₈	—	67	2	no	—
158	C₂×D₈	—	134	1	no	—
159	C₂×D₈	—	134	2	no	—
160	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	268	2	no	—
163	C₄⋊Q₈	—	134	4	no	—
164	C₄×D₄	—	268	2	no	—
165	C₄×D₄	—	134	4	no	—
166	C₂×C₄⋊C₄	—	268	1	no	—
167	C₂×C₄⋊C₄	—	134	1	no	—
168	C₂×C₄⋊C₄	—	134	2	no	—
169	C₂×C₂²⋊C₄	—	268	1	no	—
170	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
173	#107	—	67	4	no	—
174	#107	—	67	2	no	—
175	#107	—	134	1	no	—
176	#107	—	134	2	no	—
177	#107	—	134	4	no	—
178	#102	—	134	4	no	—
179	C₄×C₈	D₂₆₈	1	2	yes	—
180	C₄×C₈	C₂²×D₆₇	1	2	yes	—
181	C₂⁴	—	268	1	no	—
182	C₂×Q₈	—	268	4	no	—
183	C₂×Q₈	—	536	4	no	—
184	C₂×D₄	—	268	1	no	—
185	C₂×D₄	—	268	2	no	—
186	C₂×D₄	—	134	4	no	—
187	C₂×D₄	—	134	2	no	—
188	C₂²×C₄	—	268	1	no	—
189	C₂²×C₄	—	268	2	no	—
190	C₂²×C₄	—	268	4	no	—
191	C₂²×C₄	—	536	4	no	—
192	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
193	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
195	D₈	—	67	4	no	—
196	D₈	—	134	4	no	—
197	D₈	—	134	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
200	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₆₇	1	2	yes	—
202	C₂×C₈	#11761627	1	2	yes	—
203	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
204	C₄⋊C₄	—	268	4	no	—
205	C₄⋊C₄	—	268	1	no	—
206	C₄⋊C₄	—	268	2	no	—
207	C₄⋊C₄	—	134	2	no	—
208	C₄⋊C₄	—	134	4	no	—
209	C₂²⋊C₄	—	268	4	no	—
210	C₄²	D₅₃₆	1	2	yes	—
211	C₄²	C₂×D₂₆₈	1	2	yes	—
212	C₁₆	—	2	4	no	—
213	C₂³	—	268	2	no	—
214	C₂³	—	268	4	no	—
215	C₂³	—	536	2	no	—
216	C₂³	—	536	2	no	—
217	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
218	Q₈	—	536	8	no	—
219	D₄	—	268	4	no	—
220	D₄	—	268	2	no	—
221	D₄	—	134	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	268	2	no	—
224	C₂×C₄	—	268	4	no	—
225	C₂×C₄	—	536	2	no	—
226	C₂×C₄	—	536	4	no	—
227	C₂×C₄	—	536	8	no	—
228	C₂×C₄	#14523988	1	4	yes	—
229	C₂×C₄	#14524023	1	2	yes	—
230	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
231	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
232	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
233	C₂×C₄	#14524076	1	2	yes	—
234	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	#14524023	1	2	yes	—
237	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
238	C₈	C₂×#11761627	1	1	yes	characteristic
239	C₂²	—	268	4	no	—
240	C₂²	—	536	2	no	—
241	C₂²	—	536	2	no	—
242	C₂²	—	536	8	no	—
243	C₂²	#16863677	1	2	yes	—
244	C₂²	C₂×#14523988	1	2	yes	—
245	C₂²	#16863834	1	2	yes	—
246	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	536	2	no	—
249	C₄	—	536	4	no	—
250	C₄	#16863677	1	2	yes	—
251	C₄	C₂×#14523988	1	4	yes	—
252	C₄	C₂×#14524023	1	1	yes	characteristic
253	C₄	C₂×#14524076	1	1	yes	characteristic
254	C₂	—	536	4	no	—
255	C₂	#22562357	1	4	yes	direct product factor
256	C₂	C₂×#16863677	1	1	yes	characteristic
257	C₂	C₂²×#14523988	1	1	yes	characteristic
258	C₂	C₂×#16863834	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₅₃₆	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2226
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16863801	C₂	semidirect
3. #22562357	C₂	direct (N × H)
4. C₂×#16863224	C₂	semidirect
6. C₂×C₄⋊C₁₆×C₆₇	C₂	semidirect
11. #16863801	C₂²	semidirect
18. #16863224	C₂²	semidirect
22. C₄⋊C₁₆×C₆₇	C₂²	semidirect
28. C₂×D₅₃₆	Q₈	semidirect
64. D₅₃₆	C₂×Q₈	semidirect
125. C₂×C₆₇	#2226	semidirect
134. C₂×C₄⋊C₁₆	D₆₇	semidirect
136. C₆₇	C₂×#2226	semidirect
151. C₄⋊C₁₆	C₂×D₆₇	semidirect
255. C₂	#22562357	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×#16863801	2
C₂×C₄×D₅₃₆	2
#22562357	2
C₂×#16863224	2
C₂×#16863198	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#2226	67

Subgroups — C2×#22562357 #31010441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562357 #31010441