Subgroups — C₂×#22562363 #31010443

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic
2	#22562363	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863223	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863221	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863196	C₂	1	1	yes	maximal, characteristic
7	C₂×#16863196	C₂	1	1	yes	maximal, characteristic
8	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₆₈	—	2	1	no	—
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄×D₅₃₆	—	2	2	no	—
14	C₄×D₅₃₆	C₂²	1	4	yes	—
15	C₂×#14523977	C₂²	1	1	yes	characteristic
16	C₂×#14523965	—	2	1	no	—
17	#16863223	C₂²	1	4	yes	complemented
18	#16863221	C₂²	1	4	yes	complemented
19	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	complemented
20	#16863196	C₂²	1	4	yes	—
21	#16863196	C₂²	1	4	yes	—
22	C₂×#125×C₆₇	—	2	1	no	—
23	#499×C₆₇	—	2	2	no	—
24	#499×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₇	—	2	1	no	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×D₂₆₈	—	2	1	no	—
30	C₂²×C₄×D₆₇	—	4	1	no	—
31	C₂×D₅₃₆	—	2	1	no	—
32	C₂×D₅₃₆	—	2	2	no	—
33	C₂×D₅₃₆	Q₈	1	4	yes	—
34	C₂×D₅₃₆	C₂³	1	2	yes	—
35	C₄×D₂₆₈	—	2	4	no	—
36	C₄×D₂₆₈	—	4	2	no	—
37	C₂×C₄⋊Dic₆₇	—	2	1	no	—
38	C₂×(C₂×D₆₇)⋊C₄	—	4	1	no	—
39	C₂×C₆₇⋊C₁₆	—	2	2	no	—
40	C₂×C₆₇⋊C₁₆	—	2	2	no	—
41	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
43	#14523977	—	2	1	no	—
44	#14523977	C₂³	1	2	yes	—
45	#14523965	—	2	4	no	—
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	#125×C₆₇	—	2	4	no	—
49	#125×C₆₇	—	4	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	#107×C₆₇	—	2	1	no	—
56	#107×C₆₇	—	2	1	no	—
57	#107×C₆₇	C₂³	1	2	yes	—
58	#107×C₆₇	C₂³	1	2	yes	—
59	#106×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	D₄	1	2	yes	—
61	C₄×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂³×D₆₇	—	4	1	no	—
63	C₂×D₂₆₈	—	2	4	no	—
64	C₂×D₂₆₈	—	2	2	no	—
65	C₂×D₂₆₈	—	4	1	no	—
66	C₂×D₂₆₈	—	4	2	no	—
67	C₂²×Dic₆₇	—	4	1	no	—
68	C₂×C₄×D₆₇	—	4	4	no	—
69	C₂×C₄×D₆₇	—	8	4	no	—
70	D₅₃₆	—	2	4	no	—
71	D₅₃₆	—	2	2	no	—
72	D₅₃₆	C₂×Q₈	1	4	yes	—
73	C₄⋊Dic₆₇	—	2	2	no	—
74	C₄⋊Dic₆₇	—	4	1	no	—
75	(C₂×D₆₇)⋊C₄	—	4	4	no	—
76	C₆₇⋊C₁₆	—	2	2	no	—
77	C₆₇⋊C₁₆	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	4	1	no	—
79	C₂²×C₄×C₆₇	—	4	1	no	—
80	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₆₇	#41	1	2	yes	—
88	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	—	4	1	no	—
92	C₄⋊C₄×C₆₇	—	4	1	no	—
93	C₄⋊C₄×C₆₇	—	4	4	no	—
94	C₄⋊C₄×C₆₇	—	4	4	no	—
95	C₄²×C₆₇	D₈	1	2	yes	—
96	C₄²×C₆₇	C₂×D₄	1	2	yes	—
97	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₆₇	—	4	2	no	—
99	C₂²×D₆₇	—	4	4	no	—
100	C₂²×D₆₇	—	8	2	no	—
101	C₂²×D₆₇	—	8	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	8	2	no	—
107	C₂×C₄×C₆₇	—	8	2	no	—
108	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#137	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₆₇	—	4	2	no	—
115	C₂×Dic₆₇	—	8	2	no	—
116	D₂₆₈	—	2	4	no	—
117	D₂₆₈	—	4	2	no	—
118	D₂₆₈	—	4	4	no	—
119	C₄×D₆₇	—	8	8	no	—
120	C₈×C₆₇	—	2	2	no	—
121	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
122	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂²×C₆₇	#474	1	2	yes	—
125	C₂²×C₆₇	#508	1	2	yes	—
126	C₂²×C₆₇	#509	1	2	yes	—
127	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₆₇	—	4	4	no	—
129	C₂×D₆₇	—	8	2	no	—
130	C₂×D₆₇	—	8	2	no	—
131	C₂×D₆₇	—	8	8	no	—
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	8	2	no	—
135	C₄×C₆₇	—	8	2	no	—
136	C₄×C₆₇	#474	1	2	yes	—
137	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
138	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
139	Dic₆₇	—	8	2	no	—
140	C₂×#2228	—	67	1	no	maximal, Sylow
141	C₂×C₆₇	#2228	1	4	yes	complemented
142	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
143	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
145	D₆₇	—	8	4	no	—
146	C₂×#499	D₆₇	1	1	yes	characteristic, complemented
147	C₂×C₄×D₈	—	67	1	no	—
148	#2228	—	67	8	no	—
149	C₂×#367	—	67	1	no	—
150	C₂×#366	—	67	1	no	—
151	C₂×C₄⋊C₁₆	—	67	1	no	—
152	C₂×#357	—	67	1	no	—
153	C₂×#357	—	67	1	no	—
154	C₆₇	C₂×#2228	1	1	yes	characteristic, complemented, Sylow
155	C₂²×D₈	—	67	1	no	—
156	C₂×#125	—	2	1	no	—
157	C₂×C₄×D₄	—	134	1	no	—
158	C₂²×C₁₆	—	67	1	no	—
159	C₂²×C₁₆	—	67	1	no	—
160	#499	—	2	2	no	—
161	#499	C₂×D₆₇	1	4	yes	complemented
162	C₄×D₈	—	67	4	no	—
163	C₄×D₈	—	134	2	no	—
164	C₂×#107	—	67	1	no	—
165	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
166	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
167	C₂×#106	—	2	1	no	—
168	C₂×#102	—	134	1	no	—
169	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
170	#367	—	67	4	no	—
171	#366	—	67	4	no	—
172	C₄⋊C₁₆	—	67	4	no	—
173	#357	—	67	4	no	—
174	#357	—	67	4	no	—
175	C₂²×D₄	—	134	1	no	—
176	C₂³×C₄	—	268	1	no	—
177	C₂×D₈	—	67	4	no	—
178	C₂×D₈	—	67	2	no	—
179	C₂×D₈	—	134	1	no	—
180	C₂×D₈	—	134	2	no	—
181	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
183	#125	—	2	4	no	—
184	#125	—	4	2	no	—
185	C₄×D₄	—	268	2	no	—
186	C₄×D₄	—	134	4	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	4	1	no	—
190	C₂×C₄⋊C₄	—	4	1	no	—
191	C₂×C₄⋊C₄	—	134	1	no	—
192	C₂×C₂²⋊C₄	—	268	1	no	—
193	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂×C₁₆	—	67	2	no	—
195	C₂×C₁₆	—	67	2	no	—
196	C₂×C₁₆	—	134	2	no	—
197	C₂×C₁₆	—	134	2	no	—
198	#107	—	2	1	no	—
199	#107	—	2	1	no	—
200	#107	—	67	2	no	—
201	#107	—	134	1	no	—
202	#107	C₂²×D₆₇	1	2	yes	—
203	#107	C₂²×D₆₇	1	2	yes	—
204	#106	—	2	4	no	—
205	#102	—	134	4	no	—
206	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
207	C₄×C₈	C₂²×D₆₇	1	2	yes	—
208	C₂⁴	—	268	1	no	—
209	C₂×D₄	—	268	1	no	—
210	C₂×D₄	—	268	2	no	—
211	C₂×D₄	—	134	4	no	—
212	C₂×D₄	—	134	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	268	1	no	—
216	C₂²×C₄	—	268	4	no	—
217	C₂²×C₄	—	536	4	no	—
218	C₂²×C₄	#11761613	1	1	yes	characteristic
219	C₂²×C₄	#11761613	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
221	D₈	—	67	4	no	—
222	D₈	—	134	4	no	—
223	D₈	—	134	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
227	C₂×C₈	Q₈×D₆₇	1	2	yes	—
228	C₂×C₈	#11761629	1	2	yes	—
229	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	268	1	no	—
237	C₄⋊C₄	—	134	2	no	—
238	C₂²⋊C₄	—	268	4	no	—
239	C₄²	#11761613	1	2	yes	—
240	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
241	C₁₆	—	134	2	no	—
242	C₁₆	—	134	2	no	—
243	C₂³	—	268	2	no	—
244	C₂³	—	268	4	no	—
245	C₂³	—	536	2	no	—
246	C₂³	—	536	2	no	—
247	C₂³	C₂×#11761613	1	1	yes	characteristic
248	D₄	—	268	4	no	—
249	D₄	—	268	2	no	—
250	D₄	—	134	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	268	2	no	—
258	C₂×C₄	—	536	8	no	—
259	C₂×C₄	—	536	2	no	—
260	C₂×C₄	#14524019	1	2	yes	—
261	C₂×C₄	C₂×#11761613	1	2	yes	—
262	C₂×C₄	C₂×#11761613	1	2	yes	—
263	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
264	C₂×C₄	#14524074	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	#14524019	1	2	yes	—
268	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
269	C₈	C₂×#11761629	1	1	yes	characteristic
270	C₂²	—	268	4	no	—
271	C₂²	—	536	2	no	—
272	C₂²	—	536	2	no	—
273	C₂²	—	536	8	no	—
274	C₂²	#16863673	1	2	yes	—
275	C₂²	#16863826	1	2	yes	—
276	C₂²	#16863831	1	2	yes	—
277	C₂²	C₂²×#11761613	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	536	2	no	—
283	C₄	#16863673	1	2	yes	—
284	C₄	C₂×#14524019	1	1	yes	characteristic
285	C₄	C₂×#14524074	1	1	yes	characteristic
286	C₂	—	536	4	no	—
287	C₂	#22562363	1	4	yes	direct product factor
288	C₂	C₂×#16863673	1	1	yes	characteristic
289	C₂	C₂×#16863826	1	1	yes	characteristic
290	C₂	C₂×#16863831	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₅₃₆	C₂²
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	C₂×#2228
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #22562363	C₂	direct (N × H)
3. C₂×#16863223	C₂	semidirect
4. C₂×#16863221	C₂	semidirect
5. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
8. C₂×#499×C₆₇	C₂	semidirect
17. #16863223	C₂²	semidirect
18. #16863221	C₂²	semidirect
19. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
24. #499×C₆₇	C₂²	semidirect
141. C₂×C₆₇	#2228	semidirect
146. C₂×#499	D₆₇	semidirect
154. C₆₇	C₂×#2228	semidirect
161. #499	C₂×D₆₇	semidirect
287. C₂	#22562363	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×D₅₃₆	2
#22562363	2
C₂×#16863223	2
C₂×#16863221	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863196	2
C₂×#499×C₆₇	2
C₂×#2228	67

Subgroups — C2×#22562363 #31010443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562363 #31010443