Subgroups — C₂×#22559580 #30838166

Order 17152.

702 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×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863682	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
4	#22559580	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863141	C₂	1	2	yes	maximal
6	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	—	2	1	no	—
8	C₂²×Dic₂₆₈	—	2	1	no	—
9	C₂×#14524043	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	#16863776	—	2	2	no	—
12	#16863776	C₂²	1	4	yes	—
13	#16863682	—	2	4	no	—
14	#16863682	C₂²	1	8	yes	—
15	C₈×Dic₁₃₄	—	2	2	no	—
16	C₈×Dic₁₃₄	C₂²	1	4	yes	—
17	C₂×#14523975	—	2	2	no	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×Dic₆₇⋊C₈	—	2	1	no	—
20	C₂×#14523968	—	2	2	no	—
21	C₂×C₈×Dic₆₇	—	2	1	no	—
22	#16863141	C₂²	1	8	yes	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#335×C₆₇	C₂²	1	4	yes	—
26	C₂²×Dic₁₃₄	—	2	2	no	—
27	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
28	C₂×Dic₂₆₈	—	2	4	no	—
29	C₂×Dic₂₆₈	—	2	4	no	—
30	C₂×Dic₂₆₈	—	4	4	no	—
31	C₂×Dic₂₆₈	—	4	4	no	—
32	C₂²×C₆₇⋊C₈	—	2	1	no	—
33	#14524043	—	2	2	no	—
34	#14524043	—	2	2	no	—
35	#14524043	—	2	2	no	—
36	#14524043	C₂³	1	4	yes	—
37	C₄×Dic₁₃₄	—	2	1	no	—
38	C₄×Dic₁₃₄	—	2	1	no	—
39	C₄×Dic₁₃₄	D₄	1	2	yes	—
40	C₄×Dic₁₃₄	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₆₇	—	2	2	no	—
42	C₂×C₄⋊Dic₆₇	—	2	2	no	—
43	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
44	C₂×Dic₆₇⋊C₄	—	2	1	no	—
45	C₂×C₄×Dic₆₇	—	2	1	no	—
46	#14523975	—	2	8	no	—
47	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
49	Dic₆₇⋊C₈	—	2	4	no	—
50	#14523968	—	2	8	no	—
51	C₈×Dic₆₇	—	2	4	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₇	—	2	2	no	—
57	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
58	C₄×C₈×C₆₇	D₄	1	2	yes	—
59	C₄×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂×Dic₁₃₄	—	2	1	no	—
61	C₂×Dic₁₃₄	—	2	2	no	—
62	C₂×Dic₁₃₄	—	2	4	no	—
63	C₂×Dic₁₃₄	—	4	4	no	—
64	C₂×Dic₁₃₄	—	4	4	no	—
65	C₂×Dic₁₃₄	—	4	2	no	—
66	C₂×Dic₁₃₄	Q₁₆	1	4	yes	complemented
67	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
68	C₂²×Dic₆₇	—	2	1	no	—
69	C₂²×Dic₆₇	—	2	1	no	—
70	C₂²×Dic₆₇	—	4	2	no	—
71	Dic₂₆₈	—	4	8	no	—
72	Dic₂₆₈	—	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	Dic₆₇⋊C₄	—	2	2	no	—
82	Dic₆₇⋊C₄	—	2	2	no	—
83	C₄×Dic₆₇	—	2	2	no	—
84	C₄×Dic₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	4	4	no	—
91	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₆₇	#41	1	1	yes	characteristic
97	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₂³×C₆₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₆₇	#135	1	2	yes	—
106	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	2	4	no	—
112	C₂×Dic₆₇	—	2	2	no	—
113	C₂×Dic₆₇	—	4	2	no	—
114	C₂×Dic₆₇	—	4	4	no	—
115	C₂×Dic₆₇	—	8	4	no	—
116	Dic₁₃₄	—	2	2	no	—
117	Dic₁₃₄	—	2	4	no	—
118	Dic₁₃₄	—	4	4	no	—
119	Dic₁₃₄	—	4	4	no	—
120	Dic₁₃₄	—	8	4	no	—
121	Dic₁₃₄	C₂×Q₁₆	1	4	yes	complemented
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	#460	1	2	yes	—
127	C₂²×C₆₇	#467	1	2	yes	—
128	C₂²×C₆₇	#473	1	2	yes	—
129	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	#460	1	2	yes	—
133	C₄×C₆₇	#467	1	2	yes	—
134	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
135	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
136	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
137	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
138	Dic₆₇	—	2	4	no	—
139	Dic₆₇	—	4	2	no	—
140	Dic₆₇	—	8	4	no	—
141	C₂×#1670	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	#1670	1	4	yes	complemented
143	C₂×C₆₇	C₂×#460	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#467	1	1	yes	characteristic
145	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
146	C₂×#494	—	67	1	no	—
147	C₂×#475	—	67	2	no	—
148	C₂×C₈×Q₈	—	67	1	no	—
149	#1670	—	67	8	no	—
150	C₂×#335	D₆₇	1	1	yes	characteristic
151	C₂×#332	—	67	2	no	—
152	C₆₇	C₂×#1670	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₁₆	—	134	1	no	—
154	C₂²×Q₁₆	—	134	1	no	—
155	C₂×C₄⋊Q₈	—	67	2	no	—
156	C₂×C₄×Q₈	—	67	1	no	—
157	#494	—	67	4	no	—
158	#494	—	134	2	no	—
159	#475	—	67	8	no	—
160	#475	—	134	4	no	—
161	C₈×Q₈	—	67	4	no	—
162	C₈×Q₈	—	134	2	no	—
163	C₂×#106	—	134	2	no	—
164	C₂×C₄⋊C₈	—	67	1	no	—
165	C₂×C₄⋊C₈	—	134	1	no	—
166	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
167	C₂×#103	—	134	2	no	—
168	C₂×C₄×C₈	—	134	1	no	—
169	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
170	#335	C₂×D₆₇	1	4	yes	—
171	#332	—	67	8	no	—
172	C₂²×Q₈	—	67	1	no	—
173	C₂²×Q₈	—	134	2	no	—
174	C₂×Q₁₆	—	268	4	no	—
175	C₂×Q₁₆	—	268	4	no	—
176	C₂×Q₁₆	—	134	4	no	—
177	C₂×Q₁₆	—	134	4	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	134	1	no	—
180	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
181	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
182	C₄⋊Q₈	—	67	4	no	—
183	C₄⋊Q₈	—	134	2	no	—
184	C₄⋊Q₈	—	134	2	no	—
185	C₄⋊Q₈	—	134	2	no	—
186	C₄×Q₈	—	67	2	no	—
187	C₄×Q₈	—	67	2	no	—
188	C₄×Q₈	—	134	1	no	—
189	C₄×Q₈	—	134	1	no	—
190	C₂×C₄⋊C₄	—	67	1	no	—
191	C₂×C₄⋊C₄	—	134	2	no	—
192	C₂×C₄⋊C₄	—	134	2	no	—
193	C₂×C₄⋊C₄	—	134	1	no	—
194	C₂×C₄²	—	134	1	no	—
195	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
196	#106	—	134	8	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	67	2	no	—
199	C₄⋊C₈	—	134	4	no	—
200	C₄⋊C₈	—	134	1	no	—
201	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
202	#103	—	134	8	no	—
203	C₄×C₈	—	134	4	no	—
204	C₄×C₈	D₂₆₈	1	2	yes	—
205	C₄×C₈	C₂²×D₆₇	1	2	yes	—
206	C₂×Q₈	—	268	2	no	—
207	C₂×Q₈	—	268	4	no	—
208	C₂×Q₈	—	268	4	no	—
209	C₂×Q₈	—	67	4	no	—
210	C₂×Q₈	—	67	2	no	—
211	C₂×Q₈	—	134	4	no	—
212	C₂×Q₈	—	134	1	no	—
213	C₂×Q₈	—	134	2	no	—
214	C₂²×C₄	—	268	2	no	—
215	C₂²×C₄	—	134	1	no	—
216	C₂²×C₄	—	134	1	no	—
217	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
219	C₂²×C₄	#11761629	1	1	yes	characteristic
220	Q₁₆	—	268	8	no	—
221	Q₁₆	—	268	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	4	no	—
225	C₂×C₈	—	268	2	no	—
226	C₂×C₈	—	134	2	no	—
227	C₂×C₈	#11761617	1	4	yes	—
228	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
229	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
230	C₄⋊C₄	—	268	2	no	—
231	C₄⋊C₄	—	268	2	no	—
232	C₄⋊C₄	—	67	2	no	—
233	C₄⋊C₄	—	134	4	no	—
234	C₄⋊C₄	—	134	4	no	—
235	C₄⋊C₄	—	134	1	no	—
236	C₄⋊C₄	—	134	2	no	—
237	C₄⋊C₄	—	134	2	no	—
238	C₄²	—	134	2	no	—
239	C₄²	—	134	2	no	—
240	C₄²	D₄×D₆₇	1	1	yes	characteristic
241	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
242	C₄²	#11761629	1	1	yes	characteristic
243	C₄²	C₂³×D₆₇	1	1	yes	characteristic
244	C₂³	#14524018	1	1	yes	characteristic
245	Q₈	—	268	4	no	—
246	Q₈	—	268	4	no	—
247	Q₈	—	536	4	no	—
248	Q₈	—	67	4	no	—
249	Q₈	—	134	4	no	—
250	Q₈	—	134	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	268	2	no	—
253	C₂×C₄	—	268	4	no	—
254	C₂×C₄	—	536	4	no	—
255	C₂×C₄	—	134	2	no	—
256	C₂×C₄	—	134	2	no	—
257	C₂×C₄	—	134	4	no	—
258	C₂×C₄	#14524018	1	1	yes	characteristic
259	C₂×C₄	#14524018	1	1	yes	characteristic
260	C₂×C₄	#14524018	1	1	yes	characteristic
261	C₂×C₄	C₂×#11761617	1	2	yes	—
262	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
263	C₂×C₄	#14524062	1	2	yes	—
264	C₂×C₄	#14524071	1	2	yes	—
265	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
267	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	268	2	no	—
271	C₈	C₂×#11761617	1	4	yes	—
272	C₂²	#16863597	1	2	yes	—
273	C₂²	#16863638	1	2	yes	—
274	C₂²	#16863672	1	2	yes	—
275	C₂²	C₂×#14524018	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	268	2	no	—
279	C₄	—	536	4	no	—
280	C₄	—	134	4	no	—
281	C₄	#16863597	1	2	yes	—
282	C₄	#16863638	1	2	yes	—
283	C₄	C₂²×#11761617	1	1	yes	characteristic
284	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
285	C₄	C₂×#14524062	1	1	yes	characteristic
286	C₄	C₂×#14524071	1	1	yes	characteristic
287	C₂	#22559580	1	4	yes	direct product factor
288	C₂	C₂×#16863597	1	1	yes	characteristic
289	C₂	C₂×#16863638	1	1	yes	characteristic
290	C₂	C₂×#16863672	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	C₂×#1670
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #22559580	C₂	direct (N × H)
66. C₂×Dic₁₃₄	Q₁₆	semidirect
121. Dic₁₃₄	C₂×Q₁₆	semidirect
142. C₂×C₆₇	#1670	semidirect
152. C₆₇	C₂×#1670	semidirect
287. C₂	#22559580	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×#16863776	2
C₂×#16863682	2
C₂×C₈×Dic₁₃₄	2
#22559580	2
C₂×#16863141	2
C₂×#335×C₆₇	2
C₂×#1670	67

Subgroups — C2×#22559580 #30838166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559580 #30838166