Subgroups — C₂×#22562377 #31010558

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863514	C₂	1	1	yes	maximal, characteristic
2	#22562377	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863218	C₂	1	2	yes	maximal, complemented
4	C₂×#16863217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863205	C₂	1	2	yes	maximal
6	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14524065	C₂²	1	1	yes	characteristic
8	C₂×#14524049	—	2	1	no	—
9	C₂×#14523983	C₂²	1	2	yes	—
10	#16863514	—	2	2	no	—
11	#16863514	C₂²	1	4	yes	—
12	C₂×#14523979	C₂²	1	1	yes	characteristic
13	C₂×#14523971	—	2	1	no	—
14	#16863218	C₂²	1	8	yes	complemented
15	#16863217	C₂²	1	4	yes	complemented
16	#16863205	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
18	#498×C₆₇	—	2	2	no	—
19	#498×C₆₇	C₂²	1	4	yes	complemented
20	C₂×#106×C₆₇	—	2	1	no	—
21	C₂×#106×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761628	—	2	1	no	—
24	#14524065	—	2	1	no	—
25	#14524065	—	2	1	no	—
26	#14524065	Q₈	1	2	yes	complemented
27	#14524065	C₂³	1	2	yes	—
28	C₂×Dic₂₆₈	Q₈	1	1	yes	characteristic, complemented
29	C₂×#11761617	—	2	1	no	—
30	C₂×D₅₃₆	Q₈	1	1	yes	characteristic, complemented
31	#14524049	—	2	4	no	—
32	#14524049	—	4	2	no	—
33	C₂×#11761612	—	2	1	no	—
34	C₂×#11761611	—	4	1	no	—
35	C₂×C₈×D₆₇	—	4	1	no	—
36	#14523983	—	2	2	no	—
37	#14523983	C₂³	1	4	yes	—
38	C₂×C₆₇⋊C₁₆	—	2	2	no	—
39	#14523979	—	2	1	no	—
40	#14523979	C₂³	1	2	yes	—
41	#14523971	—	2	4	no	—
42	C₂²×Q₈×C₆₇	—	4	1	no	—
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₆₇	—	2	4	no	—
46	C₄⋊Q₈×C₆₇	—	4	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	#106×C₆₇	—	2	2	no	—
51	#106×C₆₇	—	2	4	no	—
52	#106×C₆₇	C₂³	1	4	yes	—
53	C₄×C₈×C₆₇	D₄	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	#11761628	—	2	2	no	—
56	#11761628	—	2	2	no	—
57	#11761628	—	4	1	no	—
58	#11761628	—	4	1	no	—
59	C₂×Dic₁₃₄	—	2	1	no	—
60	C₂×D₂₆₈	—	2	1	no	—
61	C₂×C₂²⋊D₆₇	—	4	1	no	—
62	C₂×C₄×D₆₇	—	4	1	no	—
63	Dic₂₆₈	—	2	1	no	—
64	Dic₂₆₈	C₂×Q₈	1	2	yes	complemented
65	#11761617	—	2	2	no	—
66	#11761617	—	2	2	no	—
67	D₅₃₆	—	2	1	no	—
68	D₅₃₆	C₂×Q₈	1	2	yes	complemented
69	#11761612	—	2	2	no	—
70	#11761612	—	4	1	no	—
71	#11761611	—	4	4	no	—
72	C₂×C₆₇⋊C₈	—	4	1	no	—
73	C₈×D₆₇	—	4	4	no	—
74	C₆₇⋊C₁₆	—	2	4	no	—
75	C₂×Q₈×C₆₇	—	4	4	no	—
76	C₂×Q₈×C₆₇	—	8	4	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
81	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
82	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₄⋊C₄×C₆₇	—	4	2	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	C₂×D₄	1	2	yes	—
95	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂²×D₆₇	—	4	1	no	—
97	Q₈×C₆₇	—	8	8	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	8	4	no	—
102	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂²⋊D₆₇	—	4	2	no	—
109	C₂²⋊D₆₇	—	4	2	no	—
110	C₂×Dic₆₇	—	4	1	no	—
111	D₂₆₈	—	2	2	no	—
112	D₂₆₈	—	4	1	no	—
113	C₄×D₆₇	—	4	2	no	—
114	C₄×D₆₇	—	4	2	no	—
115	Dic₁₃₄	—	2	2	no	—
116	Dic₁₃₄	—	4	1	no	—
117	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
120	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
121	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
122	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₆₇⋊C₈	—	4	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	#476	1	1	yes	characteristic
126	C₂²×C₆₇	#476	1	1	yes	characteristic
127	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×D₆₇	—	4	2	no	—
129	C₂×D₆₇	—	8	2	no	—
130	C₄×C₆₇	—	2	4	no	—
131	C₄×C₆₇	—	8	4	no	—
132	C₄×C₆₇	#476	1	1	yes	characteristic
133	C₄×C₆₇	#476	1	1	yes	characteristic
134	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
135	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
136	Dic₆₇	—	4	2	no	—
137	C₂×#2230	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	—	2	1	no	—
140	C₂×C₆₇	#2230	1	2	yes	complemented
141	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
142	D₆₇	—	8	2	no	—
143	C₂×#498	D₆₇	1	1	yes	characteristic, complemented
144	C₂×#443	—	67	1	no	—
145	#2230	—	67	8	no	—
146	C₂×#365	—	67	2	no	—
147	C₂×#364	—	67	1	no	—
148	C₂×#360	—	67	2	no	—
149	C₆₇	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
150	C₂×#135	—	67	1	no	—
151	C₂×#131	—	134	1	no	—
152	C₂×C₄⋊Q₈	—	2	1	no	—
153	C₂×M₃₂	—	67	2	no	—
154	#498	—	2	2	no	—
155	#498	C₂×D₆₇	1	4	yes	complemented
156	#443	—	67	4	no	—
157	#443	—	134	2	no	—
158	C₂×#108	—	67	1	no	—
159	C₂×#106	—	2	1	no	—
160	C₂×#106	C₂×D₆₇	1	2	yes	—
161	C₂×#104	—	134	1	no	—
162	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
163	#365	—	67	8	no	—
164	#364	—	67	4	no	—
165	#360	—	67	8	no	—
166	C₂×#41	—	134	1	no	—
167	C₂²×Q₈	—	4	1	no	—
168	#135	—	67	2	no	—
169	#135	—	67	2	no	—
170	#135	—	134	1	no	—
171	#135	—	134	1	no	—
172	C₂×Q₁₆	—	67	1	no	—
173	C₂×SD₁₆	—	134	1	no	—
174	C₂×D₈	—	67	1	no	—
175	#131	—	268	2	no	—
176	#131	—	134	4	no	—
177	C₂×M₁₆	—	268	1	no	—
178	C₂×M₁₆	—	134	1	no	—
179	C₂²×C₈	—	268	1	no	—
180	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
182	C₄⋊Q₈	—	2	4	no	—
183	C₄⋊Q₈	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
187	M₃₂	—	67	4	no	—
188	M₃₂	—	134	2	no	—
189	C₂×C₁₆	—	134	2	no	—
190	#108	—	67	2	no	—
191	#108	—	134	1	no	—
192	#106	—	2	4	no	—
193	#106	—	2	2	no	—
194	#106	C₂²×D₆₇	1	4	yes	—
195	#104	—	134	4	no	—
196	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
197	C₄×C₈	C₂²×D₆₇	1	2	yes	—
198	#41	—	268	1	no	—
199	#41	—	268	1	no	—
200	#41	—	134	2	no	—
201	#41	—	134	2	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	8	4	no	—
204	C₂×Q₈	—	134	1	no	—
205	C₂×D₄	—	268	1	no	—
206	C₂×D₄	—	134	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	268	1	no	—
210	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	Q₁₆	—	67	2	no	—
212	Q₁₆	—	134	1	no	—
213	SD₁₆	—	134	2	no	—
214	SD₁₆	—	134	2	no	—
215	D₈	—	67	2	no	—
216	D₈	—	134	1	no	—
217	M₁₆	—	268	1	no	—
218	M₁₆	—	268	4	no	—
219	M₁₆	—	134	2	no	—
220	C₂×C₈	—	268	1	no	—
221	C₂×C₈	—	268	4	no	—
222	C₂×C₈	#11761615	1	1	yes	characteristic
223	C₂×C₈	#11761615	1	1	yes	characteristic
224	C₂×C₈	#11761615	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
227	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
228	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
229	C₂×C₈	#11761629	1	1	yes	characteristic
230	C₂×C₈	#11761629	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
237	C₁₆	—	134	4	no	—
238	C₂³	—	268	1	no	—
239	C₂³	#14524069	1	1	yes	characteristic
240	Q₈	—	8	8	no	—
241	Q₈	—	268	1	no	—
242	Q₈	—	134	2	no	—
243	D₄	—	268	2	no	—
244	D₄	—	268	2	no	—
245	D₄	—	268	1	no	—
246	D₄	—	134	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	268	1	no	—
252	C₂×C₄	—	268	2	no	—
253	C₂×C₄	—	268	2	no	—
254	C₂×C₄	#14524019	1	1	yes	characteristic
255	C₂×C₄	#14524019	1	1	yes	characteristic
256	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
257	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
258	C₂×C₄	#14524069	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
260	C₈	—	268	2	no	—
261	C₈	#14524019	1	1	yes	characteristic
262	C₈	#14524019	1	1	yes	characteristic
263	C₈	C₂×#11761615	1	2	yes	—
264	C₈	C₂×#11761615	1	2	yes	—
265	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
266	C₈	C₂×#11761629	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	268	2	no	—
269	C₂²	—	536	2	no	—
270	C₂²	#16863683	1	1	yes	characteristic
271	C₂²	#16863683	1	1	yes	characteristic
272	C₂²	C₂×#14524069	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	8	4	no	—
275	C₄	—	268	2	no	—
276	C₄	#16863683	1	1	yes	characteristic
277	C₄	#16863683	1	1	yes	characteristic
278	C₄	C₂×#14524019	1	1	yes	characteristic
279	C₄	C₂²×#11761615	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	536	2	no	—
283	C₂	#22562377	1	2	yes	direct product factor
284	C₂	C₂×#16863683	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863683
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#498×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂×#14524065	C₂²
hypercenter	C₂×#498	D₆₇
hypocenter	C₆₇	C₂×#2230
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
2. #22562377	C₂	direct (N × H)
3. C₂×#16863218	C₂	semidirect
4. C₂×#16863217	C₂	semidirect
6. C₂×#498×C₆₇	C₂	semidirect
14. #16863218	C₂²	semidirect
15. #16863217	C₂²	semidirect
19. #498×C₆₇	C₂²	semidirect
26. #14524065	Q₈	semidirect
28. C₂×Dic₂₆₈	Q₈	semidirect
30. C₂×D₅₃₆	Q₈	semidirect
64. Dic₂₆₈	C₂×Q₈	semidirect
68. D₅₃₆	C₂×Q₈	semidirect
140. C₂×C₆₇	#2230	semidirect
143. C₂×#498	D₆₇	semidirect
149. C₆₇	C₂×#2230	semidirect
155. #498	C₂×D₆₇	semidirect
283. C₂	#22562377	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×#16863514	2
#22562377	2
C₂×#16863218	2
C₂×#16863217	2
C₂×#16863205	2
C₂×#498×C₆₇	2
C₂×#2230	67

Subgroups — C2×#22562377 #31010558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562377 #31010558