Subgroups — C₂×#22559724 #30838456

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863603	C₂	1	2	yes	maximal
2	C₂×#16863526	C₂	1	1	yes	maximal, characteristic
3	#22559724	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863142	C₂	1	2	yes	maximal
6	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761618	—	2	2	no	—
8	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	#16863603	—	2	4	no	—
10	#16863603	C₂²	1	8	yes	—
11	#16863526	—	2	2	no	—
12	#16863526	C₂²	1	4	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₆₇⋊C₈	—	2	1	no	—
16	C₂×#14523967	—	2	2	no	—
17	C₂×#14523953	—	2	1	no	—
18	#16863147	C₂²	1	4	yes	—
19	#16863142	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	#500×C₆₇	—	2	2	no	—
22	#500×C₆₇	C₂²	1	4	yes	—
23	C₂×#107×C₆₇	—	2	1	no	—
24	C₂×#107×C₆₇	—	2	1	no	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
27	C₂×#11761618	—	2	8	no	—
28	C₂×#11761618	—	4	8	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	2	no	—
31	C₄×Dic₁₃₄	—	2	1	no	—
32	C₄×Dic₁₃₄	—	2	1	no	—
33	C₄×Dic₁₃₄	D₄	1	2	yes	—
34	C₄×Dic₁₃₄	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
36	C₂×Dic₆₇⋊C₄	—	2	1	no	—
37	C₂×C₄×Dic₆₇	—	2	1	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
41	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
42	Dic₆₇⋊C₈	—	2	4	no	—
43	#14523967	—	2	8	no	—
44	#14523953	—	2	4	no	—
45	C₂²×Q₈×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#107×C₆₇	—	2	4	no	—
56	#107×C₆₇	—	2	4	no	—
57	C₄×C₈×C₆₇	D₄	1	2	yes	—
58	C₄×C₈×C₆₇	C₂³	1	2	yes	—
59	C₂×Dic₁₃₄	—	2	4	no	—
60	C₂×Dic₁₃₄	—	2	1	no	—
61	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
62	C₂²×Dic₆₇	—	2	1	no	—
63	C₂²×Dic₆₇	—	2	1	no	—
64	#11761618	—	4	16	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	4	no	—
67	C₂×C₆₇⋊C₈	—	4	4	no	—
68	C₂×C₆₇⋊C₈	—	4	2	no	—
69	C₄⋊Dic₆₇	—	2	1	no	—
70	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
71	Dic₆₇⋊C₄	—	2	2	no	—
72	Dic₆₇⋊C₄	—	2	2	no	—
73	C₄×Dic₆₇	—	2	2	no	—
74	C₄×Dic₆₇	—	2	2	no	—
75	C₂×Q₈×C₆₇	—	2	4	no	—
76	C₂×Q₈×C₆₇	—	4	2	no	—
77	C₂×Q₈×C₆₇	—	4	4	no	—
78	C₂×Q₈×C₆₇	—	4	4	no	—
79	C₂²×C₄×C₆₇	—	4	2	no	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
85	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	4	2	no	—
90	C₄⋊C₄×C₆₇	—	4	2	no	—
91	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₆₇	#41	1	1	yes	characteristic
94	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₂³×C₆₇	#121	1	1	yes	characteristic
96	Q₈×C₆₇	—	4	4	no	—
97	Q₈×C₆₇	—	4	4	no	—
98	Q₈×C₆₇	—	8	4	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	8	4	no	—
102	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₆₇	#137	1	2	yes	—
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₆₇	—	2	2	no	—
112	C₂×Dic₆₇	—	2	2	no	—
113	C₂×Dic₆₇	—	4	2	no	—
114	C₂×Dic₆₇	—	4	2	no	—
115	Dic₁₃₄	—	2	4	no	—
116	Dic₁₃₄	—	4	2	no	—
117	C₈×C₆₇	—	2	2	no	—
118	C₈×C₆₇	C₂×Q₁₆	1	4	yes	complemented
119	C₆₇⋊C₈	—	4	4	no	—
120	C₆₇⋊C₈	—	4	2	no	—
121	C₂²×C₆₇	#462	1	2	yes	—
122	C₂²×C₆₇	#469	1	2	yes	—
123	C₂²×C₆₇	#473	1	2	yes	—
124	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
125	C₄×C₆₇	—	2	1	no	—
126	C₄×C₆₇	—	2	1	no	—
127	C₄×C₆₇	—	8	4	no	—
128	C₄×C₆₇	#462	1	2	yes	—
129	C₄×C₆₇	#469	1	2	yes	—
130	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
131	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
132	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
133	Dic₆₇	—	4	2	no	—
134	Dic₆₇	—	4	2	no	—
135	C₂×#1694	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	#1694	1	4	yes	complemented
137	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
138	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
139	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
140	C₂×#500	D₆₇	1	1	yes	characteristic
141	C₂×#462	—	67	2	no	—
142	C₂×#446	—	67	1	no	—
143	#1694	—	67	8	no	—
144	C₂×#334	—	67	1	no	—
145	C₂×#332	—	67	2	no	—
146	C₆₇	C₂×#1694	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₁₆	—	134	2	no	—
148	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
149	C₂×C₄×Q₈	—	67	1	no	—
150	#500	—	2	2	no	—
151	#500	C₂×D₆₇	1	4	yes	—
152	#462	—	67	8	no	—
153	#462	—	134	4	no	—
154	#446	—	67	4	no	—
155	#446	—	134	2	no	—
156	C₂×#107	—	2	1	no	—
157	C₂×#107	—	2	1	no	—
158	C₂×C₄⋊C₈	—	67	1	no	—
159	C₂×C₄⋊C₈	—	67	2	no	—
160	C₂×C₄⋊C₈	—	134	1	no	—
161	C₂×#103	—	134	2	no	—
162	C₂×#97	—	134	1	no	—
163	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
164	#334	—	67	4	no	—
165	#332	—	67	8	no	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	—	67	1	no	—
168	C₂×Q₁₆	—	268	8	no	—
169	C₂×Q₁₆	—	134	8	no	—
170	C₂²×C₈	—	134	2	no	—
171	C₂²×C₈	—	134	1	no	—
172	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	C₂²×D₆₇	1	4	yes	—
178	C₄×Q₈	—	67	2	no	—
179	C₄×Q₈	—	67	2	no	—
180	C₄×Q₈	—	134	1	no	—
181	C₄×Q₈	—	134	1	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	67	1	no	—
185	C₂×C₄⋊C₄	—	134	1	no	—
186	C₂×C₄²	—	134	1	no	—
187	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
188	#107	—	2	4	no	—
189	#107	—	2	4	no	—
190	C₄⋊C₈	—	67	2	no	—
191	C₄⋊C₈	—	67	4	no	—
192	C₄⋊C₈	—	134	4	no	—
193	C₄⋊C₈	—	134	2	no	—
194	C₄⋊C₈	—	134	1	no	—
195	#103	—	134	8	no	—
196	#97	—	134	4	no	—
197	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
198	C₄×C₈	C₂²×D₆₇	1	2	yes	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	67	2	no	—
204	C₂×Q₈	—	134	4	no	—
205	C₂×Q₈	—	134	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	134	1	no	—
208	C₂²×C₄	—	134	1	no	—
209	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	C₂²×C₄	#11761627	1	1	yes	characteristic
212	Q₁₆	—	268	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	268	4	no	—
215	C₂×C₈	—	268	2	no	—
216	C₂×C₈	—	134	4	no	—
217	C₂×C₈	—	134	2	no	—
218	C₂×C₈	#11761618	1	4	yes	complemented
219	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
220	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	67	2	no	—
226	C₄⋊C₄	—	134	1	no	—
227	C₄⋊C₄	—	134	2	no	—
228	C₄⋊C₄	—	134	2	no	—
229	C₄²	—	134	2	no	—
230	C₄²	—	134	2	no	—
231	C₄²	D₄×D₆₇	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
233	C₄²	#11761627	1	1	yes	characteristic
234	C₄²	C₂³×D₆₇	1	1	yes	characteristic
235	C₂³	#14524014	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	4	4	no	—
238	Q₈	—	8	4	no	—
239	Q₈	—	268	2	no	—
240	Q₈	—	134	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	268	2	no	—
245	C₂×C₄	—	268	2	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	—	134	2	no	—
248	C₂×C₄	#14524014	1	1	yes	characteristic
249	C₂×C₄	#14524014	1	1	yes	characteristic
250	C₂×C₄	#14524014	1	1	yes	characteristic
251	C₂×C₄	C₂×#11761618	1	2	yes	—
252	C₂×C₄	#14524067	1	4	yes	—
253	C₂×C₄	#14524074	1	2	yes	—
254	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
256	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	268	4	no	—
259	C₈	—	268	2	no	—
260	C₈	C₂×#11761618	1	4	yes	complemented
261	C₂²	#16863603	1	2	yes	—
262	C₂²	#16863646	1	2	yes	—
263	C₂²	#16863668	1	2	yes	—
264	C₂²	C₂×#14524014	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	8	4	no	—
268	C₄	—	268	2	no	—
269	C₄	—	268	2	no	—
270	C₄	#16863603	1	2	yes	—
271	C₄	#16863646	1	2	yes	—
272	C₄	C₂²×#11761618	1	1	yes	characteristic
273	C₄	C₂×#14524067	1	2	yes	—
274	C₄	C₂×#14524074	1	1	yes	characteristic
275	C₂	#22559724	1	4	yes	direct product factor
276	C₂	C₂×#16863603	1	1	yes	characteristic
277	C₂	C₂×#16863646	1	1	yes	characteristic
278	C₂	C₂×#16863668	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524014
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	C₂×#1694
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #22559724	C₂	direct (N × H)
84. C₂×C₈×C₆₇	Q₁₆	semidirect
118. C₈×C₆₇	C₂×Q₁₆	semidirect
136. C₂×C₆₇	#1694	semidirect
146. C₆₇	C₂×#1694	semidirect
218. C₂×C₈	#11761618	semidirect
260. C₈	C₂×#11761618	semidirect
275. C₂	#22559724	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×#16863603	2
C₂×#16863526	2
#22559724	2
C₂×#16863147	2
C₂×#16863142	2
C₂×#500×C₆₇	2
C₂×#1694	67

Subgroups — C2×#22559724 #30838456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559724 #30838456