Subgroups — C₂×#22559727 #30838459

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×#16863801	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863606	C₂	1	2	yes	maximal
3	#22559727	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863141	C₂	1	2	yes	maximal
6	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761618	—	2	2	no	—
8	C₂×#14524043	C₂²	1	2	yes	—
9	#16863801	—	2	2	no	—
10	#16863801	C₂²	1	4	yes	—
11	#16863606	—	2	4	no	—
12	#16863606	C₂²	1	8	yes	—
13	C₂×#14523977	—	2	1	no	—
14	C₂×#14523977	—	2	1	no	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14523966	—	2	2	no	—
17	#16863147	C₂²	1	4	yes	—
18	#16863141	C₂²	1	8	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#446×C₆₇	—	2	2	no	—
21	#446×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₆₇	—	2	1	no	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₃₄	—	2	2	no	—
27	C₂×#11761618	—	2	8	no	—
28	C₂×#11761618	—	4	8	no	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	#14524043	—	2	2	no	—
31	#14524043	—	2	2	no	—
32	#14524043	—	2	2	no	—
33	#14524043	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄⋊Dic₆₇	—	2	2	no	—
36	#14523977	—	2	4	no	—
37	#14523977	—	2	4	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
40	#14523966	—	2	8	no	—
41	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	—	2	1	no	—
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₆₇	—	2	1	no	—
46	C₄×Q₈×C₆₇	—	2	1	no	—
47	C₄×Q₈×C₆₇	D₄	1	2	yes	—
48	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	—	2	1	no	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	—	2	4	no	—
54	C₄⋊C₈×C₆₇	—	2	1	no	—
55	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
56	#97×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	—	4	4	no	—
61	C₂×Dic₁₃₄	—	4	4	no	—
62	C₂×Dic₁₃₄	—	4	2	no	—
63	C₂²×Dic₆₇	—	4	2	no	—
64	#11761618	—	4	16	no	—
65	C₂×C₆₇⋊C₈	—	2	4	no	—
66	C₂×C₆₇⋊C₈	—	4	4	no	—
67	C₄⋊Dic₆₇	—	2	4	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	4	2	no	—
70	C₄⋊Dic₆₇	—	4	2	no	—
71	C₂×Q₈×C₆₇	—	2	1	no	—
72	C₂×Q₈×C₆₇	—	2	4	no	—
73	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
83	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₄²×C₆₇	—	2	2	no	—
90	C₄²×C₆₇	—	2	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	—	2	4	no	—
97	Q₈×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
107	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
108	C₂×C₄×C₆₇	#137	1	2	yes	—
109	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₆₇	—	4	4	no	—
113	C₂×Dic₆₇	—	8	4	no	—
114	Dic₁₃₄	—	4	4	no	—
115	Dic₁₃₄	—	4	4	no	—
116	Dic₁₃₄	—	8	4	no	—
117	C₈×C₆₇	—	2	2	no	—
118	C₈×C₆₇	—	4	2	no	—
119	C₈×C₆₇	C₂×Q₁₆	1	4	yes	complemented
120	C₆₇⋊C₈	—	4	4	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	—	4	2	no	—
128	C₄×C₆₇	—	4	2	no	—
129	C₄×C₆₇	#462	1	2	yes	—
130	C₄×C₆₇	#469	1	2	yes	—
131	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
132	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
133	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
134	Dic₆₇	—	8	4	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	—	67	1	no	—
141	C₂×#462	—	67	2	no	—
142	C₂×#446	D₆₇	1	1	yes	characteristic
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	—	67	2	no	—
149	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
150	#500	—	67	4	no	—
151	#500	—	134	2	no	—
152	#462	—	67	8	no	—
153	#462	—	134	4	no	—
154	#446	—	2	2	no	—
155	#446	C₂×D₆₇	1	4	yes	—
156	C₂×#107	—	134	1	no	—
157	C₂×#107	—	134	1	no	—
158	C₂×C₄⋊C₈	—	2	1	no	—
159	C₂×C₄⋊C₈	—	67	2	no	—
160	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
161	C₂×#103	—	134	2	no	—
162	C₂×#97	—	2	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	—	134	2	no	—
167	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
168	C₂×Q₁₆	—	268	8	no	—
169	C₂×Q₁₆	—	134	8	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	134	2	no	—
172	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₄⋊Q₈	—	67	4	no	—
175	C₄⋊Q₈	—	134	2	no	—
176	C₄⋊Q₈	—	134	2	no	—
177	C₄⋊Q₈	—	134	2	no	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×Q₈	—	2	1	no	—
180	C₄×Q₈	D₂₆₈	1	2	yes	—
181	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	134	2	no	—
184	C₂×C₄⋊C₄	—	134	2	no	—
185	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
188	#107	—	134	4	no	—
189	#107	—	134	4	no	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	67	4	no	—
193	C₄⋊C₈	—	134	2	no	—
194	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
195	#103	—	134	8	no	—
196	#97	—	2	4	no	—
197	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
198	C₄×C₈	C₂²×D₆₇	1	2	yes	—
199	C₂×Q₈	—	2	1	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	268	2	no	—
202	C₂×Q₈	—	268	4	no	—
203	C₂×Q₈	—	268	4	no	—
204	C₂×Q₈	—	134	4	no	—
205	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	268	2	no	—
209	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
211	C₂²×C₄	#11761628	1	1	yes	characteristic
212	Q₁₆	—	268	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	268	4	no	—
217	C₂×C₈	—	134	4	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	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	268	2	no	—
225	C₄⋊C₄	—	268	2	no	—
226	C₄⋊C₄	—	134	4	no	—
227	C₄⋊C₄	—	134	4	no	—
228	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
232	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
233	C₄²	#11761628	1	1	yes	characteristic
234	C₄²	C₂³×D₆₇	1	1	yes	characteristic
235	C₂³	#14524017	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	268	4	no	—
239	Q₈	—	268	4	no	—
240	Q₈	—	536	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	268	4	no	—
247	C₂×C₄	—	536	4	no	—
248	C₂×C₄	#14524017	1	1	yes	characteristic
249	C₂×C₄	#14524017	1	1	yes	characteristic
250	C₂×C₄	#14524017	1	1	yes	characteristic
251	C₂×C₄	C₂×#11761618	1	2	yes	—
252	C₂×C₄	#14524071	1	4	yes	—
253	C₂×C₄	#14524073	1	2	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
256	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	268	4	no	—
260	C₈	C₂×#11761618	1	4	yes	complemented
261	C₂²	#16863606	1	2	yes	—
262	C₂²	#16863649	1	2	yes	—
263	C₂²	#16863671	1	2	yes	—
264	C₂²	C₂×#14524017	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	536	4	no	—
270	C₄	#16863606	1	2	yes	—
271	C₄	#16863649	1	2	yes	—
272	C₄	C₂²×#11761618	1	1	yes	characteristic
273	C₄	C₂×#14524071	1	2	yes	—
274	C₄	C₂×#14524073	1	1	yes	characteristic
275	C₂	#22559727	1	4	yes	direct product factor
276	C₂	C₂×#16863606	1	1	yes	characteristic
277	C₂	C₂×#16863649	1	1	yes	characteristic
278	C₂	C₂×#16863671	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#446×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	C₂×#446	D₆₇
hypocenter	C₆₇	C₂×#1694
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #22559727	C₂	direct (N × H)
82. C₂×C₈×C₆₇	Q₁₆	semidirect
119. 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	#22559727	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×#16863606	2
#22559727	2
C₂×#16863147	2
C₂×#16863141	2
C₂×#446×C₆₇	2
C₂×#1694	67

Subgroups — C2×#22559727 #30838459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559727 #30838459