Subgroups — C₄×#16863773 #30864261

Order 17152.

1422 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863773	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₅₃₆	C₂	1	4	yes	maximal, complemented
3	C₄×#14524040	C₂	1	2	yes	maximal, complemented
4	#22561075	C₂	1	4	yes	maximal, complemented
5	#22561015	C₂	1	2	yes	maximal, complemented
6	#22560369	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅₃₆	C₂²	1	4	yes	—
9	C₂×#14524040	C₂²	1	2	yes	—
10	C₂×C₄×D₂₆₈	—	2	4	no	—
11	C₂×C₄×D₂₆₈	C₂²	1	4	yes	—
12	#16863773	C₄	1	8	yes	direct product factor
13	C₄×D₅₃₆	—	2	16	no	—
14	C₂×#14523977	C₂²	1	4	yes	—
15	C₂×#14523965	C₂²	1	8	yes	—
16	#16863298	—	2	4	no	—
17	#16863298	C₂²	1	4	yes	—
18	#16863266	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×D₂₆₈	—	2	4	no	—
25	C₂²×D₂₆₈	C₂³	1	4	yes	—
26	C₂²×C₄×D₆₇	—	2	4	no	—
27	C₂×D₅₃₆	—	2	8	no	—
28	C₂×D₅₃₆	—	2	8	no	—
29	C₂×D₅₃₆	C₂×C₄	1	16	yes	complemented
30	#14524040	—	2	4	no	—
31	#14524040	C₂×C₄	1	8	yes	complemented
32	C₄×D₂₆₈	—	2	16	no	—
33	C₄×D₂₆₈	—	4	16	no	—
34	C₂×C₄⋊Dic₆₇	—	2	4	no	—
35	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
36	C₂×(C₂×D₆₇)⋊C₄	—	2	4	no	—
37	C₂×(C₂×D₆₇)⋊C₄	—	2	8	no	—
38	#14523977	—	2	8	no	—
39	#14523965	—	2	16	no	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	D₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₆₇	—	2	4	no	—
48	C₄×C₈×C₆₇	—	2	2	no	—
49	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
50	C₄×C₈×C₆₇	D₄	1	8	yes	complemented
51	C₂³×D₆₇	—	2	4	no	—
52	C₂×D₂₆₈	—	2	8	no	—
53	C₂×D₂₆₈	—	2	8	no	—
54	C₂×D₂₆₈	—	2	16	no	—
55	C₂×D₂₆₈	—	2	4	no	—
56	C₂×D₂₆₈	—	4	8	no	—
57	C₂×D₂₆₈	—	4	8	no	—
58	C₂×D₂₆₈	C₂²×C₄	1	8	yes	—
59	C₂²×Dic₆₇	—	2	4	no	—
60	C₂×C₄×D₆₇	—	4	8	no	—
61	C₂×C₄×D₆₇	—	4	16	no	—
62	D₅₃₆	—	2	32	no	—
63	C₄⋊Dic₆₇	—	2	8	no	—
64	C₄⋊Dic₆₇	—	4	8	no	—
65	(C₂×D₆₇)⋊C₄	—	4	8	no	—
66	(C₂×D₆₇)⋊C₄	—	4	16	no	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₆₇	#41	1	4	yes	—
78	C₄²×C₆₇	—	2	4	no	—
79	C₄²×C₆₇	—	2	2	no	—
80	C₄²×C₆₇	D₈	1	8	yes	complemented
81	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₆₇	C₂×D₄	1	4	yes	—
83	C₄²×C₆₇	#41	1	2	yes	—
84	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×D₆₇	—	2	8	no	—
86	C₂²×D₆₇	—	4	4	no	—
87	C₂²×D₆₇	—	4	4	no	—
88	C₂²×D₆₇	—	4	8	no	—
89	C₂²×D₆₇	—	4	8	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₆₇	#127	1	2	yes	—
95	C₂×C₄×C₆₇	#127	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
97	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
98	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
99	C₂×C₄×C₆₇	#135	1	4	yes	—
100	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
104	C₂×Dic₆₇	—	4	4	no	—
105	C₂×Dic₆₇	—	4	8	no	—
106	D₂₆₈	—	2	16	no	—
107	D₂₆₈	—	4	8	no	—
108	D₂₆₈	—	4	32	no	—
109	C₄×D₆₇	—	8	16	no	—
110	C₈×C₆₇	—	2	4	no	—
111	C₈×C₆₇	C₄×D₄	1	8	yes	complemented
112	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
114	C₂²×C₆₇	#532	1	1	yes	characteristic
115	C₂²×C₆₇	C₂²×D₈	1	2	yes	—
116	C₂²×C₆₇	C₂×#135	1	2	yes	—
117	C₂×D₆₇	—	4	8	no	—
118	C₂×D₆₇	—	4	16	no	—
119	C₂×D₆₇	—	8	4	no	—
120	C₂×D₆₇	—	8	4	no	—
121	C₂×D₆₇	—	8	8	no	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	2	4	no	—
124	C₄×C₆₇	C₄×D₈	1	8	yes	complemented
125	C₄×C₆₇	#493	1	4	yes	complemented
126	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
127	C₄×C₆₇	#532	1	2	yes	—
128	Dic₆₇	—	8	4	no	—
129	C₄×#493	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
131	C₂×C₆₇	C₂×C₄×D₈	1	2	yes	—
132	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
133	C₂×C₆₇	#3147	1	1	yes	characteristic
134	C₂×C₆₇	#3169	1	2	yes	—
135	D₆₇	—	8	8	no	—
136	C₂×#493	—	67	1	no	—
137	C₂×C₄×D₈	—	67	4	no	—
138	C₄×#127	—	67	2	no	—
139	#1970	—	67	4	no	—
140	#1957	—	67	2	no	—
141	#1845	—	67	1	no	—
142	C₄²×C₈	D₆₇	1	1	yes	characteristic, complemented
143	C₆₇	C₄×#493	1	1	yes	characteristic, complemented, Sylow
144	C₂²×D₈	—	67	4	no	—
145	C₂×#127	—	67	2	no	—
146	C₂×C₄×D₄	—	67	4	no	—
147	C₂×C₄×D₄	—	134	4	no	—
148	#493	—	67	8	no	—
149	C₄×D₈	—	134	16	no	—
150	C₂×#107	—	67	4	no	—
151	C₂×#102	—	67	8	no	—
152	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
154	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
156	#390	—	67	4	no	—
157	#390	—	134	4	no	—
158	#384	—	67	2	no	—
159	C₄³	C₂×D₆₇	1	1	yes	characteristic
160	C₂²×D₄	—	67	4	no	—
161	C₂²×D₄	—	134	4	no	—
162	C₂³×C₄	—	134	4	no	—
163	C₂×D₈	—	67	16	no	—
164	C₂×D₈	—	134	8	no	—
165	C₂×D₈	—	134	8	no	—
166	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
167	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
168	#127	—	67	8	no	—
169	#127	—	134	4	no	—
170	C₄×D₄	—	268	16	no	—
171	C₄×D₄	—	134	16	no	—
172	C₂×C₄⋊C₄	—	67	4	no	—
173	C₂×C₄⋊C₄	—	134	4	no	—
174	C₂×C₂²⋊C₄	—	134	4	no	—
175	C₂×C₂²⋊C₄	—	134	8	no	—
176	C₂×C₄²	D₂₆₈	1	2	yes	—
177	C₂×C₄²	D₂₆₈	1	2	yes	—
178	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
181	#107	—	134	8	no	—
182	#102	—	134	16	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₄×D₆₇	1	4	yes	complemented
186	C₄×C₈	D₂₆₈	1	8	yes	complemented
187	C₂⁴	—	134	4	no	—
188	C₂×D₄	—	268	8	no	—
189	C₂×D₄	—	268	8	no	—
190	C₂×D₄	—	67	8	no	—
191	C₂×D₄	—	134	16	no	—
192	C₂×D₄	—	134	8	no	—
193	C₂×D₄	—	134	4	no	—
194	C₂×D₄	—	134	8	no	—
195	C₂²×C₄	—	268	8	no	—
196	C₂²×C₄	—	268	16	no	—
197	C₂²×C₄	—	134	4	no	—
198	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
200	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
201	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
203	D₈	—	134	32	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
207	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
208	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
209	C₂×C₈	#11761628	1	4	yes	—
210	C₄⋊C₄	—	268	8	no	—
211	C₄⋊C₄	—	134	8	no	—
212	C₂²⋊C₄	—	268	8	no	—
213	C₂²⋊C₄	—	268	16	no	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	D₅₃₆	1	8	yes	complemented
217	C₄²	C₂×C₄×D₆₇	1	2	yes	—
218	C₄²	C₂×D₂₆₈	1	4	yes	—
219	C₄²	#11761628	1	2	yes	—
220	C₂³	—	268	8	no	—
221	C₂³	—	268	8	no	—
222	C₂³	—	268	4	no	—
223	C₂³	—	268	4	no	—
224	C₂³	—	134	8	no	—
225	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
226	D₄	—	268	8	no	—
227	D₄	—	268	32	no	—
228	D₄	—	134	16	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	268	8	no	—
233	C₂×C₄	—	268	4	no	—
234	C₂×C₄	—	536	16	no	—
235	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
236	C₂×C₄	#14524040	1	2	yes	—
237	C₂×C₄	#14524040	1	2	yes	—
238	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
239	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
240	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
241	C₂×C₄	#14524065	1	4	yes	—
242	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
244	C₂×C₄	C₂×#11761628	1	2	yes	—
245	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
246	C₈	—	2	4	no	—
247	C₈	C₄×D₂₆₈	1	8	yes	complemented
248	C₂²	—	268	16	no	—
249	C₂²	—	268	8	no	—
250	C₂²	—	536	8	no	—
251	C₂²	—	536	4	no	—
252	C₂²	—	536	4	no	—
253	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
254	C₂²	C₂×#14524040	1	1	yes	characteristic
255	C₂²	#16863946	1	1	yes	characteristic
256	C₂²	C₂²×D₅₃₆	1	2	yes	—
257	C₂²	C₂×#14524065	1	2	yes	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	536	4	no	—
261	C₄	C₄×D₅₃₆	1	8	yes	complemented
262	C₄	#16863773	1	4	yes	direct product factor
263	C₄	C₂×C₄×D₂₆₈	1	2	yes	—
264	C₄	#16863946	1	2	yes	—
265	C₂	—	536	8	no	—
266	C₂	C₄×#14524040	1	1	yes	characteristic
267	C₂	C₂×C₄×D₅₃₆	1	2	yes	—
268	C₂	C₂×#16863773	1	1	yes	characteristic
269	C₂	#22570637	1	1	yes	characteristic
270	C₂	#22570788	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂×#16863773	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	C₄×#493
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅₃₆	C₂	semidirect
3. C₄×#14524040	C₂	semidirect
4. #22561075	C₂	semidirect
5. #22561015	C₂	semidirect
6. #22560369	C₂	semidirect
7. C₄²×C₈×C₆₇	C₂	semidirect
12. #16863773	C₄	direct (N × H)
29. C₂×D₅₃₆	C₂×C₄	semidirect
31. #14524040	C₂×C₄	semidirect
49. C₄×C₈×C₆₇	C₂×C₄	semidirect
50. C₄×C₈×C₆₇	D₄	semidirect
80. C₄²×C₆₇	D₈	semidirect
111. C₈×C₆₇	C₄×D₄	semidirect
124. C₄×C₆₇	C₄×D₈	semidirect
125. C₄×C₆₇	#493	semidirect
142. C₄²×C₈	D₆₇	semidirect
143. C₆₇	C₄×#493	semidirect
185. C₄×C₈	C₄×D₆₇	semidirect
186. C₄×C₈	D₂₆₈	semidirect
216. C₄²	D₅₃₆	semidirect
247. C₈	C₄×D₂₆₈	semidirect
261. C₄	C₄×D₅₃₆	semidirect
262. C₄	#16863773	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×#16863773	2
C₂×C₄×D₅₃₆	2
C₄×#14524040	2
#22561075	2
#22561015	2
#22560369	2
C₄²×C₈×C₆₇	2
C₄×#493	67

Subgroups — C4×#16863773 #30864261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863773 #30864261