Subgroups — C₂×#22559604 #30838188

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 288 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
2	C₂×#16863607	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
4	#22559604	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863143	C₂	1	2	yes	maximal
6	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	—	2	2	no	—
8	C₂×#14524043	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	#16863801	—	2	2	no	—
11	#16863801	C₂²	1	4	yes	—
12	#16863607	—	2	4	no	—
13	#16863607	C₂²	1	8	yes	—
14	C₈×Dic₁₃₄	—	2	2	no	—
15	C₈×Dic₁₃₄	C₂²	1	4	yes	—
16	C₂×#14523977	—	2	1	no	—
17	C₂×#14523977	—	2	1	no	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×Dic₆₇⋊C₈	—	2	1	no	—
20	C₂×#14523968	—	2	2	no	—
21	C₂×C₈×Dic₆₇	—	2	1	no	—
22	#16863143	C₂²	1	8	yes	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#335×C₆₇	C₂²	1	4	yes	—
26	C₂²×Dic₁₃₄	—	2	2	no	—
27	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
28	C₂×Dic₂₆₈	—	2	8	no	—
29	C₂×Dic₂₆₈	—	4	8	no	—
30	C₂²×C₆₇⋊C₈	—	2	1	no	—
31	#14524043	—	2	2	no	—
32	#14524043	—	2	2	no	—
33	#14524043	—	2	2	no	—
34	#14524043	C₂³	1	4	yes	—
35	C₄×Dic₁₃₄	—	2	1	no	—
36	C₄×Dic₁₃₄	—	2	1	no	—
37	C₄×Dic₁₃₄	D₄	1	2	yes	—
38	C₄×Dic₁₃₄	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₆₇	—	2	2	no	—
40	C₂×C₄⋊Dic₆₇	—	2	2	no	—
41	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
42	C₂×Dic₆₇⋊C₄	—	2	1	no	—
43	C₂×C₄×Dic₆₇	—	2	1	no	—
44	#14523977	—	2	4	no	—
45	#14523977	—	2	4	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
47	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
48	Dic₆₇⋊C₈	—	2	4	no	—
49	#14523968	—	2	8	no	—
50	C₈×Dic₆₇	—	2	4	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₆₇	—	2	2	no	—
56	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
57	C₄×C₈×C₆₇	D₄	1	2	yes	—
58	C₄×C₈×C₆₇	C₂³	1	2	yes	—
59	C₂×Dic₁₃₄	—	2	1	no	—
60	C₂×Dic₁₃₄	—	2	2	no	—
61	C₂×Dic₁₃₄	—	2	4	no	—
62	C₂×Dic₁₃₄	—	4	4	no	—
63	C₂×Dic₁₃₄	—	4	4	no	—
64	C₂×Dic₁₃₄	—	4	2	no	—
65	C₂×Dic₁₃₄	D₈	1	4	yes	—
66	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
67	C₂²×Dic₆₇	—	2	1	no	—
68	C₂²×Dic₆₇	—	2	1	no	—
69	C₂²×Dic₆₇	—	4	2	no	—
70	Dic₂₆₈	—	4	16	no	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	4	2	no	—
73	C₄⋊Dic₆₇	—	2	1	no	—
74	C₄⋊Dic₆₇	—	2	4	no	—
75	C₄⋊Dic₆₇	—	2	4	no	—
76	C₄⋊Dic₆₇	—	4	2	no	—
77	C₄⋊Dic₆₇	—	4	2	no	—
78	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
79	Dic₆₇⋊C₄	—	2	2	no	—
80	Dic₆₇⋊C₄	—	2	2	no	—
81	C₄×Dic₆₇	—	2	2	no	—
82	C₄×Dic₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	—	4	4	no	—
89	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
90	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
92	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₆₇	#41	1	1	yes	characteristic
95	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₂³×C₆₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₆₇	#135	1	2	yes	—
104	C₂×C₄×C₆₇	#137	1	2	yes	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	2	4	no	—
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	4	2	no	—
112	C₂×Dic₆₇	—	4	4	no	—
113	C₂×Dic₆₇	—	8	4	no	—
114	Dic₁₃₄	—	2	2	no	—
115	Dic₁₃₄	—	2	4	no	—
116	Dic₁₃₄	—	4	4	no	—
117	Dic₁₃₄	—	4	4	no	—
118	Dic₁₃₄	—	8	4	no	—
119	Dic₁₃₄	C₂×D₈	1	4	yes	—
120	C₈×C₆₇	—	2	2	no	—
121	C₈×C₆₇	—	4	4	no	—
122	C₈×C₆₇	C₂×Q₁₆	1	4	yes	complemented
123	C₆₇⋊C₈	—	4	2	no	—
124	C₂²×C₆₇	#462	1	2	yes	—
125	C₂²×C₆₇	#466	1	2	yes	—
126	C₂²×C₆₇	#473	1	2	yes	—
127	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	#462	1	2	yes	—
131	C₄×C₆₇	#466	1	2	yes	—
132	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
134	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
136	Dic₆₇	—	2	4	no	—
137	Dic₆₇	—	4	2	no	—
138	Dic₆₇	—	8	4	no	—
139	C₂×#1674	—	67	1	no	maximal, Sylow
140	C₂×C₆₇	#1674	1	4	yes	complemented
141	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
142	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
143	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
144	C₂×#500	—	67	1	no	—
145	C₂×#462	—	67	2	no	—
146	C₂×C₈×Q₈	—	67	1	no	—
147	#1674	—	67	8	no	—
148	C₂×#335	D₆₇	1	1	yes	characteristic
149	C₂×#332	—	67	2	no	—
150	C₆₇	C₂×#1674	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	134	2	no	—
152	C₂×C₄⋊Q₈	—	67	2	no	—
153	C₂×C₄×Q₈	—	67	1	no	—
154	#500	—	67	4	no	—
155	#500	—	134	2	no	—
156	#462	—	67	8	no	—
157	#462	—	134	4	no	—
158	C₈×Q₈	—	67	4	no	—
159	C₈×Q₈	—	134	2	no	—
160	C₂×#107	—	134	1	no	—
161	C₂×#107	—	134	1	no	—
162	C₂×C₄⋊C₈	—	67	1	no	—
163	C₂×C₄⋊C₈	—	134	1	no	—
164	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
165	C₂×#103	—	134	2	no	—
166	C₂×C₄×C₈	—	134	1	no	—
167	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
168	#335	C₂×D₆₇	1	4	yes	—
169	#332	—	67	8	no	—
170	C₂²×Q₈	—	67	1	no	—
171	C₂²×Q₈	—	134	2	no	—
172	C₂×Q₁₆	—	268	8	no	—
173	C₂×Q₁₆	—	134	8	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	134	1	no	—
176	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
177	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
178	C₄⋊Q₈	—	67	4	no	—
179	C₄⋊Q₈	—	134	2	no	—
180	C₄⋊Q₈	—	134	2	no	—
181	C₄⋊Q₈	—	134	2	no	—
182	C₄×Q₈	—	67	2	no	—
183	C₄×Q₈	—	67	2	no	—
184	C₄×Q₈	—	134	1	no	—
185	C₄×Q₈	—	134	1	no	—
186	C₂×C₄⋊C₄	—	67	1	no	—
187	C₂×C₄⋊C₄	—	134	2	no	—
188	C₂×C₄⋊C₄	—	134	2	no	—
189	C₂×C₄⋊C₄	—	134	1	no	—
190	C₂×C₄²	—	134	1	no	—
191	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
192	#107	—	134	4	no	—
193	#107	—	134	4	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	67	2	no	—
196	C₄⋊C₈	—	134	4	no	—
197	C₄⋊C₈	—	134	1	no	—
198	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
199	#103	—	134	8	no	—
200	C₄×C₈	—	134	4	no	—
201	C₄×C₈	D₂₆₈	1	2	yes	—
202	C₄×C₈	C₂²×D₆₇	1	2	yes	—
203	C₂×Q₈	—	268	2	no	—
204	C₂×Q₈	—	268	4	no	—
205	C₂×Q₈	—	268	4	no	—
206	C₂×Q₈	—	67	4	no	—
207	C₂×Q₈	—	67	2	no	—
208	C₂×Q₈	—	134	4	no	—
209	C₂×Q₈	—	134	1	no	—
210	C₂×Q₈	—	134	2	no	—
211	C₂²×C₄	—	268	2	no	—
212	C₂²×C₄	—	134	1	no	—
213	C₂²×C₄	—	134	1	no	—
214	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
216	C₂²×C₄	#11761629	1	1	yes	characteristic
217	Q₁₆	—	268	16	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	268	2	no	—
222	C₂×C₈	—	134	2	no	—
223	C₂×C₈	Dic₂₆₈	1	4	yes	complemented
224	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
225	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
226	C₄⋊C₄	—	268	2	no	—
227	C₄⋊C₄	—	268	2	no	—
228	C₄⋊C₄	—	67	2	no	—
229	C₄⋊C₄	—	134	4	no	—
230	C₄⋊C₄	—	134	4	no	—
231	C₄⋊C₄	—	134	1	no	—
232	C₄⋊C₄	—	134	2	no	—
233	C₄⋊C₄	—	134	2	no	—
234	C₄²	—	134	2	no	—
235	C₄²	—	134	2	no	—
236	C₄²	D₄×D₆₇	1	1	yes	characteristic
237	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
238	C₄²	#11761629	1	1	yes	characteristic
239	C₄²	C₂³×D₆₇	1	1	yes	characteristic
240	C₂³	#14524018	1	1	yes	characteristic
241	Q₈	—	268	4	no	—
242	Q₈	—	268	4	no	—
243	Q₈	—	536	4	no	—
244	Q₈	—	67	4	no	—
245	Q₈	—	134	4	no	—
246	Q₈	—	134	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	268	2	no	—
249	C₂×C₄	—	268	4	no	—
250	C₂×C₄	—	536	4	no	—
251	C₂×C₄	—	134	2	no	—
252	C₂×C₄	—	134	4	no	—
253	C₂×C₄	—	134	2	no	—
254	C₂×C₄	#14524018	1	1	yes	characteristic
255	C₂×C₄	#14524018	1	1	yes	characteristic
256	C₂×C₄	#14524018	1	1	yes	characteristic
257	C₂×C₄	D₈×D₆₇	1	2	yes	—
258	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
259	C₂×C₄	#14524066	1	2	yes	—
260	C₂×C₄	#14524076	1	2	yes	—
261	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
263	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	4	no	—
266	C₈	—	268	2	no	—
267	C₈	C₂×Dic₂₆₈	1	4	yes	complemented
268	C₂²	#16863607	1	2	yes	—
269	C₂²	#16863633	1	2	yes	—
270	C₂²	#16863672	1	2	yes	—
271	C₂²	C₂×#14524018	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	268	2	no	—
275	C₄	—	536	4	no	—
276	C₄	—	134	4	no	—
277	C₄	#16863607	1	2	yes	—
278	C₄	#16863633	1	2	yes	—
279	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
280	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
281	C₄	C₂×#14524066	1	1	yes	characteristic
282	C₄	C₂×#14524076	1	1	yes	characteristic
283	C₂	#22559604	1	4	yes	direct product factor
284	C₂	C₂×#16863607	1	1	yes	characteristic
285	C₂	C₂×#16863633	1	1	yes	characteristic
286	C₂	C₂×#16863672	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	C₂×#1674
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #22559604	C₂	direct (N × H)
89. C₂×C₈×C₆₇	Q₁₆	semidirect
122. C₈×C₆₇	C₂×Q₁₆	semidirect
140. C₂×C₆₇	#1674	semidirect
150. C₆₇	C₂×#1674	semidirect
223. C₂×C₈	Dic₂₆₈	semidirect
267. C₈	C₂×Dic₂₆₈	semidirect
283. C₂	#22559604	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×#16863607	2
C₂×C₈×Dic₁₃₄	2
#22559604	2
C₂×#16863143	2
C₂×#335×C₆₇	2
C₂×#1674	67

Subgroups — C2×#22559604 #30838188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559604 #30838188