Subgroups — C₂×#22559733 #30838367

Order 17152.

694 conjugacy classes of subgroups.

The subgroup lattice has 298 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, complemented
2	C₂×#16863687	C₂	1	2	yes	maximal
3	C₂×#16863475	C₂	1	1	yes	maximal, characteristic
4	#22559733	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863144	C₂	1	2	yes	maximal, complemented
6	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11761611	—	2	1	no	—
8	C₂×#14524043	C₂²	1	2	yes	—
9	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
10	#16863801	—	2	2	no	—
11	#16863801	C₂²	1	4	yes	complemented
12	#16863687	—	2	4	no	—
13	#16863687	C₂²	1	8	yes	—
14	#16863475	—	2	2	no	—
15	#16863475	C₂²	1	4	yes	—
16	C₂×#14523977	—	2	1	no	—
17	C₂×#14523977	—	2	1	no	—
18	C₂×#14523975	—	2	2	no	—
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#14523965	—	2	2	no	—
21	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
22	#16863144	C₂²	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#334×C₆₇	C₂²	1	4	yes	complemented
26	C₂²×Dic₁₃₄	—	2	2	no	—
27	C₂²×D₂₆₈	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₆₇	—	2	1	no	—
29	C₂×#11761611	—	2	4	no	—
30	C₂×#11761611	—	4	4	no	—
31	C₂²×C₆₇⋊C₈	—	2	1	no	—
32	#14524043	—	2	2	no	—
33	#14524043	—	2	2	no	—
34	#14524043	—	2	2	no	—
35	#14524043	C₂³	1	4	yes	—
36	C₄×D₂₆₈	—	2	1	no	—
37	C₄×D₂₆₈	—	2	1	no	—
38	C₄×D₂₆₈	D₄	1	2	yes	—
39	C₄×D₂₆₈	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₆₇	—	2	2	no	—
41	C₂×C₄⋊Dic₆₇	—	2	2	no	—
42	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
43	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
44	#14523977	—	2	4	no	—
45	#14523977	—	2	4	no	—
46	#14523975	—	2	8	no	—
47	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
49	#14523965	—	2	8	no	—
50	(C₂×D₆₇)⋊C₈	—	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	C₂³×D₆₇	—	2	1	no	—
60	C₂×Dic₁₃₄	—	2	4	no	—
61	C₂×Dic₁₃₄	—	4	2	no	—
62	C₂×Dic₁₃₄	—	8	4	no	—
63	C₂×D₂₆₈	—	2	4	no	—
64	C₂×D₂₆₈	—	2	1	no	—
65	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
66	C₂²×Dic₆₇	—	2	1	no	—
67	C₂²×Dic₆₇	—	4	2	no	—
68	C₂×C₄×D₆₇	—	2	2	no	—
69	C₂×C₄×D₆₇	—	2	2	no	—
70	C₂×C₄×D₆₇	—	4	2	no	—
71	C₂×C₄×D₆₇	—	4	2	no	—
72	#11761611	—	4	8	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	4	2	no	—
75	C₄⋊Dic₆₇	—	2	1	no	—
76	C₄⋊Dic₆₇	—	2	4	no	—
77	C₄⋊Dic₆₇	—	2	4	no	—
78	C₄⋊Dic₆₇	—	4	2	no	—
79	C₄⋊Dic₆₇	—	4	2	no	—
80	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
81	(C₂×D₆₇)⋊C₄	—	2	2	no	—
82	(C₂×D₆₇)⋊C₄	—	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	SD₁₆	1	4	yes	—
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	C₂²×D₆₇	—	2	2	no	—
98	C₂²×D₆₇	—	4	2	no	—
99	C₂²×D₆₇	—	4	2	no	—
100	C₂²×D₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	2	2	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×SD₁₆	1	2	yes	—
106	C₂×C₄×C₆₇	#137	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	—	4	2	no	—
113	C₂×Dic₆₇	—	4	4	no	—
114	C₂×Dic₆₇	—	8	4	no	—
115	D₂₆₈	—	2	4	no	—
116	D₂₆₈	—	4	2	no	—
117	C₄×D₆₇	—	4	2	no	—
118	C₄×D₆₇	—	4	2	no	—
119	C₄×D₆₇	—	4	4	no	—
120	Dic₁₃₄	—	8	4	no	—
121	Dic₁₃₄	—	8	4	no	—
122	C₈×C₆₇	—	2	2	no	—
123	C₈×C₆₇	—	4	4	no	—
124	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
125	C₆₇⋊C₈	—	4	2	no	—
126	C₂²×C₆₇	#461	1	2	yes	—
127	C₂²×C₆₇	#470	1	2	yes	—
128	C₂²×C₆₇	#472	1	2	yes	—
129	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
130	C₂×D₆₇	—	4	2	no	—
131	C₂×D₆₇	—	4	2	no	—
132	C₂×D₆₇	—	4	2	no	—
133	C₂×D₆₇	—	8	4	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	#461	1	2	yes	—
137	C₄×C₆₇	#470	1	2	yes	—
138	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
139	C₄×C₆₇	C₂×#137	1	2	yes	—
140	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
141	Dic₆₇	—	4	2	no	—
142	Dic₆₇	—	8	4	no	—
143	C₂×#1695	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	#1695	1	4	yes	complemented
145	C₂×C₆₇	C₂×#461	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#470	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
148	D₆₇	—	8	2	no	—
149	C₂×#500	—	67	1	no	—
150	C₂×#476	—	67	2	no	—
151	C₂×#436	—	67	1	no	—
152	#1695	—	67	8	no	—
153	C₂×#334	D₆₇	1	1	yes	characteristic, complemented
154	C₂×#333	—	67	2	no	—
155	C₆₇	C₂×#1695	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	134	1	no	—
157	C₂×C₄⋊Q₈	—	67	2	no	—
158	C₂×C₄×D₄	—	67	1	no	—
159	#500	—	67	4	no	—
160	#500	—	134	2	no	—
161	#476	—	67	8	no	—
162	#476	—	134	4	no	—
163	#436	—	67	4	no	—
164	#436	—	134	2	no	—
165	C₂×#107	—	134	1	no	—
166	C₂×#107	—	134	1	no	—
167	C₂×#106	—	134	2	no	—
168	C₂×C₄⋊C₈	—	67	1	no	—
169	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
170	C₂×#102	—	134	2	no	—
171	C₂×C₂²⋊C₈	—	134	1	no	—
172	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
173	#334	C₂×D₆₇	1	4	yes	complemented
174	#333	—	67	8	no	—
175	C₂²×Q₈	—	134	2	no	—
176	C₂²×D₄	—	67	1	no	—
177	C₂³×C₄	—	134	1	no	—
178	C₂×M₁₆	—	268	4	no	—
179	C₂×M₁₆	—	134	4	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	134	1	no	—
182	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
183	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
184	C₄⋊Q₈	—	67	4	no	—
185	C₄⋊Q₈	—	134	2	no	—
186	C₄⋊Q₈	—	134	2	no	—
187	C₄⋊Q₈	—	134	2	no	—
188	C₄×D₄	—	67	2	no	—
189	C₄×D₄	—	67	2	no	—
190	C₄×D₄	—	134	1	no	—
191	C₄×D₄	—	134	1	no	—
192	C₂×C₄⋊C₄	—	67	1	no	—
193	C₂×C₄⋊C₄	—	134	2	no	—
194	C₂×C₄⋊C₄	—	134	2	no	—
195	C₂×C₂²⋊C₄	—	134	1	no	—
196	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
197	#107	—	134	4	no	—
198	#107	—	134	4	no	—
199	#106	—	134	8	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	67	2	no	—
202	C₄⋊C₈	—	134	1	no	—
203	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
204	#102	—	134	8	no	—
205	C₂²⋊C₈	—	134	4	no	—
206	C₄×C₈	D₂₆₈	1	2	yes	—
207	C₄×C₈	C₂²×D₆₇	1	2	yes	—
208	C₂⁴	—	134	1	no	—
209	C₂×Q₈	—	268	2	no	—
210	C₂×Q₈	—	536	4	no	—
211	C₂×Q₈	—	134	4	no	—
212	C₂×D₄	—	67	2	no	—
213	C₂×D₄	—	134	1	no	—
214	C₂×D₄	—	134	4	no	—
215	C₂²×C₄	—	268	2	no	—
216	C₂²×C₄	—	268	2	no	—
217	C₂²×C₄	—	268	2	no	—
218	C₂²×C₄	—	134	1	no	—
219	C₂²×C₄	—	134	2	no	—
220	C₂²×C₄	—	134	2	no	—
221	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
223	C₂²×C₄	#11761629	1	1	yes	characteristic
224	M₁₆	—	268	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	—	268	2	no	—
229	C₂×C₈	—	134	2	no	—
230	C₂×C₈	#11761617	1	4	yes	—
231	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
232	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
233	C₄⋊C₄	—	268	2	no	—
234	C₄⋊C₄	—	268	2	no	—
235	C₄⋊C₄	—	67	2	no	—
236	C₄⋊C₄	—	134	4	no	—
237	C₄⋊C₄	—	134	4	no	—
238	C₄⋊C₄	—	134	1	no	—
239	C₂²⋊C₄	—	134	2	no	—
240	C₂²⋊C₄	—	134	2	no	—
241	C₄²	D₄×D₆₇	1	1	yes	characteristic
242	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
243	C₄²	#11761629	1	1	yes	characteristic
244	C₄²	C₂³×D₆₇	1	1	yes	characteristic
245	C₂³	—	268	2	no	—
246	C₂³	—	268	2	no	—
247	C₂³	—	268	2	no	—
248	C₂³	—	134	2	no	—
249	C₂³	#14524018	1	1	yes	characteristic
250	Q₈	—	536	4	no	—
251	Q₈	—	536	4	no	—
252	D₄	—	268	2	no	—
253	D₄	—	134	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	268	2	no	—
256	C₂×C₄	—	268	4	no	—
257	C₂×C₄	—	268	2	no	—
258	C₂×C₄	—	268	2	no	—
259	C₂×C₄	—	268	4	no	—
260	C₂×C₄	—	536	4	no	—
261	C₂×C₄	—	134	2	no	—
262	C₂×C₄	#14524018	1	1	yes	characteristic
263	C₂×C₄	#14524018	1	1	yes	characteristic
264	C₂×C₄	#14524018	1	1	yes	characteristic
265	C₂×C₄	C₂×#11761617	1	2	yes	—
266	C₂×C₄	#14524072	1	4	yes	—
267	C₂×C₄	#14524076	1	2	yes	—
268	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
270	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	4	4	no	—
273	C₈	—	268	2	no	—
274	C₈	C₂×#11761617	1	4	yes	—
275	C₂²	—	268	2	no	—
276	C₂²	—	268	2	no	—
277	C₂²	—	268	2	no	—
278	C₂²	—	536	4	no	—
279	C₂²	#16863602	1	2	yes	—
280	C₂²	#16863655	1	2	yes	—
281	C₂²	#16863667	1	2	yes	—
282	C₂²	C₂×#14524018	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	268	2	no	—
286	C₄	—	536	4	no	—
287	C₄	#16863602	1	2	yes	—
288	C₄	#16863655	1	2	yes	—
289	C₄	C₂²×#11761617	1	1	yes	characteristic
290	C₄	C₂×#14524072	1	2	yes	—
291	C₄	C₂×#14524076	1	1	yes	characteristic
292	C₂	—	536	2	no	—
293	C₂	#22559733	1	4	yes	direct product factor
294	C₂	C₂×#16863602	1	1	yes	characteristic
295	C₂	C₂×#16863655	1	1	yes	characteristic
296	C₂	C₂×#16863667	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂²×D₂₆₈	D₄
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	C₂×#1695
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
1. C₂×#16863801	C₂	semidirect
4. #22559733	C₂	direct (N × H)
5. C₂×#16863144	C₂	semidirect
6. C₂×#334×C₆₇	C₂	semidirect
11. #16863801	C₂²	semidirect
22. #16863144	C₂²	semidirect
25. #334×C₆₇	C₂²	semidirect
144. C₂×C₆₇	#1695	semidirect
153. C₂×#334	D₆₇	semidirect
155. C₆₇	C₂×#1695	semidirect
173. #334	C₂×D₆₇	semidirect
293. C₂	#22559733	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×#16863687	2
C₂×#16863475	2
#22559733	2
C₂×#16863144	2
C₂×#334×C₆₇	2
C₂×#1695	67

Subgroups — C2×#22559733 #30838367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559733 #30838367