Subgroups — C₂×#22562345 #31010433

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863800	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863478	C₂	1	1	yes	maximal, characteristic
3	#22562345	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863221	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863198	C₂	1	1	yes	maximal, characteristic
7	C₂×#16863196	C₂	1	1	yes	maximal, characteristic
8	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
10	C₂×#14524042	—	2	1	no	—
11	C₂×#14524005	—	2	1	no	—
12	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16863800	—	2	2	no	—
14	#16863800	C₂²	1	4	yes	complemented
15	#16863478	—	2	2	no	—
16	#16863478	C₂²	1	4	yes	—
17	C₂×#14523977	C₂²	1	1	yes	characteristic
18	C₂×#14523976	—	2	1	no	—
19	C₂×#14523963	—	2	1	no	—
20	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
21	#16863221	C₂²	1	4	yes	complemented
22	Dic₆₇⋊C₁₆	C₂²	1	4	yes	complemented
23	#16863198	C₂²	1	4	yes	—
24	#16863196	C₂²	1	4	yes	—
25	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
28	#366×C₆₇	C₂²	1	4	yes	complemented
29	C₂²×Dic₁₃₄	—	4	1	no	—
30	C₂²×D₂₆₈	—	2	1	no	—
31	C₂²×C₄×D₆₇	—	4	1	no	—
32	C₂×D₅₃₆	—	2	1	no	—
33	C₂×D₅₃₆	—	2	2	no	—
34	C₂×D₅₃₆	Q₈	1	4	yes	complemented
35	C₂×D₅₃₆	C₂³	1	2	yes	—
36	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
37	#14524042	—	2	4	no	—
38	#14524042	—	4	2	no	—
39	#14524005	—	2	4	no	—
40	#14524005	—	4	2	no	—
41	C₂×C₄⋊Dic₆₇	—	2	1	no	—
42	C₂×Dic₆₇⋊C₄	—	4	1	no	—
43	C₂×(C₂×D₆₇)⋊C₄	—	4	1	no	—
44	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
45	C₂×C₆₇⋊C₁₆	—	2	2	no	—
46	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
47	#14523977	—	2	1	no	—
48	#14523977	C₂³	1	2	yes	—
49	#14523976	—	2	4	no	—
50	#14523963	—	2	4	no	—
51	C₈×Dic₆₇	D₄	1	2	yes	—
52	C₈×Dic₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₁₆×C₆₇	—	2	2	no	—
57	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
58	#107×C₆₇	—	2	1	no	—
59	#107×C₆₇	—	2	1	no	—
60	#107×C₆₇	C₂³	1	2	yes	—
61	#107×C₆₇	C₂³	1	2	yes	—
62	C₂³×D₆₇	—	4	1	no	—
63	C₂×Dic₁₃₄	—	4	4	no	—
64	C₂×Dic₁₃₄	—	8	4	no	—
65	C₂×D₂₆₈	—	2	4	no	—
66	C₂×D₂₆₈	—	2	2	no	—
67	C₂×D₂₆₈	—	4	1	no	—
68	C₂×D₂₆₈	—	4	2	no	—
69	C₂²×Dic₆₇	—	4	1	no	—
70	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
71	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
72	C₂×C₄×D₆₇	—	4	4	no	—
73	C₂×C₄×D₆₇	—	8	4	no	—
74	D₅₃₆	—	2	4	no	—
75	D₅₃₆	—	2	2	no	—
76	D₅₃₆	C₂×Q₈	1	4	yes	complemented
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
79	C₄⋊Dic₆₇	—	2	2	no	—
80	C₄⋊Dic₆₇	—	4	1	no	—
81	Dic₆₇⋊C₄	—	4	4	no	—
82	(C₂×D₆₇)⋊C₄	—	4	4	no	—
83	C₄×Dic₆₇	D₈	1	2	yes	—
84	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
85	C₆₇⋊C₁₆	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	4	1	no	—
87	C₂²×C₄×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₆₇	#41	1	2	yes	—
92	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₄⋊C₄×C₆₇	—	4	1	no	—
96	C₄⋊C₄×C₆₇	—	4	1	no	—
97	C₁₆×C₆₇	—	2	2	no	—
98	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
99	C₂²×D₆₇	—	4	2	no	—
100	C₂²×D₆₇	—	4	4	no	—
101	C₂²×D₆₇	—	8	2	no	—
102	C₂²×D₆₇	—	8	2	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	—	8	2	no	—
106	C₂×C₄×C₆₇	—	8	2	no	—
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#137	1	2	yes	—
110	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₆₇	—	2	2	no	—
112	C₂×Dic₆₇	—	4	2	no	—
113	C₂×Dic₆₇	—	8	2	no	—
114	C₂×Dic₆₇	D₁₆	1	4	yes	—
115	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
116	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
117	D₂₆₈	—	2	4	no	—
118	D₂₆₈	—	4	2	no	—
119	D₂₆₈	—	4	4	no	—
120	C₄×D₆₇	—	8	8	no	—
121	Dic₁₃₄	—	8	8	no	—
122	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
123	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₆₇⋊C₈	—	2	2	no	—
126	C₂²×C₆₇	#474	1	2	yes	—
127	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
128	C₂²×C₆₇	#510	1	2	yes	—
129	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
130	C₂×D₆₇	—	4	4	no	—
131	C₂×D₆₇	—	8	2	no	—
132	C₂×D₆₇	—	8	2	no	—
133	C₂×D₆₇	—	8	8	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	—	2	2	no	—
140	Dic₆₇	—	8	2	no	—
141	Dic₆₇	C₂×D₁₆	1	4	yes	—
142	C₂×#2224	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	#2224	1	4	yes	complemented
144	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
145	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
147	D₆₇	—	8	4	no	—
148	C₂×#500	—	67	1	no	—
149	C₂×C₄×D₈	—	67	1	no	—
150	#2224	—	67	8	no	—
151	C₂×#366	—	67	1	no	—
152	C₂×#366	D₆₇	1	1	yes	characteristic, complemented
153	C₂×C₄⋊C₁₆	—	67	1	no	—
154	C₂×#357	—	67	1	no	—
155	C₂×#357	—	67	1	no	—
156	C₆₇	C₂×#2224	1	1	yes	characteristic, complemented, Sylow
157	C₂²×D₈	—	67	1	no	—
158	C₂×C₄⋊Q₈	—	134	1	no	—
159	C₂×C₄×D₄	—	134	1	no	—
160	C₂²×C₁₆	—	67	1	no	—
161	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
162	#500	—	67	4	no	—
163	#500	—	134	2	no	—
164	C₄×D₈	—	67	4	no	—
165	C₄×D₈	—	134	2	no	—
166	C₂×#107	—	67	1	no	—
167	C₂×#107	—	134	1	no	—
168	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
169	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
170	C₂×#102	—	134	1	no	—
171	C₂×C₄×C₈	—	67	1	no	—
172	#366	—	67	4	no	—
173	#366	C₂×D₆₇	1	4	yes	complemented
174	C₄⋊C₁₆	—	67	4	no	—
175	#357	—	67	4	no	—
176	#357	—	67	4	no	—
177	C₂²×Q₈	—	268	1	no	—
178	C₂²×D₄	—	134	1	no	—
179	C₂³×C₄	—	268	1	no	—
180	C₂×D₈	—	67	4	no	—
181	C₂×D₈	—	67	2	no	—
182	C₂×D₈	—	134	1	no	—
183	C₂×D₈	—	134	2	no	—
184	C₂²×C₈	—	67	1	no	—
185	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
186	C₄⋊Q₈	—	268	2	no	—
187	C₄⋊Q₈	—	134	4	no	—
188	C₄×D₄	—	268	2	no	—
189	C₄×D₄	—	134	4	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	268	1	no	—
193	C₂×C₄⋊C₄	—	134	1	no	—
194	C₂×C₂²⋊C₄	—	268	1	no	—
195	C₂×C₄²	—	67	1	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	67	2	no	—
198	C₂×C₁₆	—	134	2	no	—
199	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
200	#107	—	2	1	no	—
201	#107	—	2	1	no	—
202	#107	—	67	2	no	—
203	#107	—	134	4	no	—
204	#107	—	134	1	no	—
205	#107	C₂²×D₆₇	1	2	yes	—
206	#107	C₂²×D₆₇	1	2	yes	—
207	#102	—	134	4	no	—
208	C₄×C₈	—	67	2	no	—
209	C₄×C₈	—	67	2	no	—
210	C₂⁴	—	268	1	no	—
211	C₂×Q₈	—	268	4	no	—
212	C₂×Q₈	—	536	4	no	—
213	C₂×D₄	—	268	1	no	—
214	C₂×D₄	—	268	2	no	—
215	C₂×D₄	—	134	4	no	—
216	C₂×D₄	—	134	2	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	268	4	no	—
220	C₂²×C₄	—	268	1	no	—
221	C₂²×C₄	—	536	4	no	—
222	C₂²×C₄	—	67	1	no	—
223	C₂²×C₄	—	67	1	no	—
224	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
225	D₈	—	67	4	no	—
226	D₈	—	134	4	no	—
227	D₈	—	134	2	no	—
228	C₂×C₈	—	67	2	no	—
229	C₂×C₈	—	134	2	no	—
230	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
231	C₂×C₈	Q₈×D₆₇	1	2	yes	—
232	C₂×C₈	#11761628	1	2	yes	—
233	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	268	4	no	—
239	C₄⋊C₄	—	268	1	no	—
240	C₄⋊C₄	—	134	2	no	—
241	C₂²⋊C₄	—	268	4	no	—
242	C₄²	—	67	2	no	—
243	C₄²	—	67	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	134	2	no	—
246	C₂³	—	268	2	no	—
247	C₂³	—	268	4	no	—
248	C₂³	—	536	2	no	—
249	C₂³	—	536	2	no	—
250	C₂³	D₈×D₆₇	1	1	yes	characteristic
251	Q₈	—	536	8	no	—
252	D₄	—	268	4	no	—
253	D₄	—	268	2	no	—
254	D₄	—	134	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	268	2	no	—
260	C₂×C₄	—	536	8	no	—
261	C₂×C₄	—	536	2	no	—
262	C₂×C₄	—	67	4	no	—
263	C₂×C₄	—	67	2	no	—
264	C₂×C₄	—	67	2	no	—
265	C₂×C₄	—	134	2	no	—
266	C₂×C₄	#14524021	1	2	yes	—
267	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
268	C₂×C₄	#14524075	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
270	C₈	—	134	2	no	—
271	C₈	#14524021	1	2	yes	—
272	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
273	C₈	C₂×#11761628	1	1	yes	characteristic
274	C₂²	—	268	4	no	—
275	C₂²	—	536	2	no	—
276	C₂²	—	536	2	no	—
277	C₂²	—	536	8	no	—
278	C₂²	#16863675	1	2	yes	—
279	C₂²	D₁₆×D₆₇	1	2	yes	—
280	C₂²	#16863838	1	2	yes	—
281	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	536	2	no	—
285	C₄	—	67	4	no	—
286	C₄	—	134	2	no	—
287	C₄	#16863675	1	2	yes	—
288	C₄	C₂×#14524021	1	1	yes	characteristic
289	C₄	C₂×#14524075	1	1	yes	characteristic
290	C₂	—	536	4	no	—
291	C₂	#22562345	1	4	yes	direct product factor
292	C₂	C₂×#16863675	1	1	yes	characteristic
293	C₂	C₂×D₁₆×D₆₇	1	1	yes	characteristic
294	C₂	C₂×#16863838	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#366×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₅₃₆	C₂²
hypercenter	C₂×#366	D₆₇
hypocenter	C₆₇	C₂×#2224
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
1. C₂×#16863800	C₂	semidirect
3. #22562345	C₂	direct (N × H)
4. C₂×#16863221	C₂	semidirect
5. C₂×Dic₆₇⋊C₁₆	C₂	semidirect
8. C₂×#366×C₆₇	C₂	semidirect
14. #16863800	C₂²	semidirect
21. #16863221	C₂²	semidirect
22. Dic₆₇⋊C₁₆	C₂²	semidirect
28. #366×C₆₇	C₂²	semidirect
34. C₂×D₅₃₆	Q₈	semidirect
76. D₅₃₆	C₂×Q₈	semidirect
143. C₂×C₆₇	#2224	semidirect
152. C₂×#366	D₆₇	semidirect
156. C₆₇	C₂×#2224	semidirect
173. #366	C₂×D₆₇	semidirect
291. C₂	#22562345	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×#16863800	2
C₂×#16863478	2
#22562345	2
C₂×#16863221	2
C₂×Dic₆₇⋊C₁₆	2
C₂×#16863198	2
C₂×#16863196	2
C₂×#366×C₆₇	2
C₂×#2224	67

Subgroups — C2×#22562345 #31010433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562345 #31010433