Subgroups — C₄×#16863180 #30770879

Order 17152.

490 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×#16863180	C₂	1	2	yes	maximal
2	C₄×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558298	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	#16863180	C₄	1	8	yes	direct product factor
9	#16863172	—	2	1	no	—
10	C₄×C₆₇⋊C₁₆	—	2	1	no	—
11	C₄×C₆₇⋊C₁₆	—	2	1	no	—
12	C₄×C₆₇⋊C₁₆	—	2	1	no	—
13	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
16	#405×C₆₇	—	2	1	no	—
17	#405×C₆₇	—	2	1	no	—
18	#405×C₆₇	C₄	1	1	yes	characteristic
19	#405×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₆₇	—	2	1	no	—
21	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#14523983	—	2	2	no	—
25	#14523983	C₂×C₄	1	4	yes	complemented
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	C₂×C₆₇⋊C₁₆	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	—	2	2	no	—
29	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
30	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
31	#131×C₆₇	—	2	2	no	—
32	#131×C₆₇	—	2	2	no	—
33	#131×C₆₇	C₈	1	4	yes	—
34	#131×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₆₇	—	2	1	no	—
36	C₂×M₁₆×C₆₇	—	2	1	no	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	—	2	1	no	—
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₆₇	C₈	1	2	yes	—
44	C₄×D₄×C₆₇	—	2	2	no	—
45	C₄×D₄×C₆₇	C₈	1	2	yes	—
46	#117×C₆₇	—	2	1	no	—
47	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	—	2	1	no	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	#97×C₆₇	—	2	1	no	—
51	#97×C₆₇	—	2	1	no	—
52	#97×C₆₇	—	2	1	no	—
53	C₄×C₈×C₆₇	—	2	1	no	—
54	C₄×C₈×C₆₇	—	2	1	no	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₆₇⋊C₁₆	—	2	4	no	—
61	C₆₇⋊C₁₆	—	2	4	no	—
62	#41×C₆₇	—	2	1	no	—
63	#41×C₆₇	—	2	1	no	—
64	#41×C₆₇	C₄²	1	2	yes	—
65	#41×C₆₇	C₂×C₈	1	2	yes	—
66	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₆₇	—	2	1	no	—
68	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	M₁₆×C₆₇	—	2	1	no	—
74	M₁₆×C₆₇	—	2	1	no	—
75	M₁₆×C₆₇	—	2	4	no	—
76	M₁₆×C₆₇	—	2	4	no	—
77	M₁₆×C₆₇	C₄²	1	2	yes	—
78	M₁₆×C₆₇	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₆₇	—	2	1	no	—
80	C₂×C₈×C₆₇	—	2	1	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	#41	1	2	yes	—
91	C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₂²⋊C₄×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	—	2	1	no	—
97	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂×C₈	1	2	yes	—
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
102	Q₈×C₆₇	—	2	1	no	—
103	Q₈×C₆₇	C₄×C₈	1	2	yes	—
104	D₄×C₆₇	—	2	2	no	—
105	D₄×C₆₇	—	2	2	no	—
106	D₄×C₆₇	—	2	1	no	—
107	D₄×C₆₇	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
122	C₈×C₆₇	—	2	2	no	—
123	C₈×C₆₇	—	2	2	no	—
124	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
125	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
126	C₈×C₆₇	C₄×D₄	1	2	yes	—
127	C₈×C₆₇	C₄×D₄	1	2	yes	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	C₄×M₁₆	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₆₇	#431	1	1	yes	characteristic
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	4	2	no	—
137	C₄×C₆₇	#350	1	4	yes	complemented
138	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
140	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₄×C₆₇	#431	1	1	yes	characteristic
142	C₄×#350	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	—	2	1	no	—
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	—	4	2	no	—
146	C₂×C₆₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#350	1	2	yes	—
148	C₄×#131	D₆₇	1	1	yes	characteristic
149	C₂×#350	—	67	2	no	—
150	C₄×M₃₂	—	67	1	no	—
151	C₂×C₄×C₁₆	—	67	1	no	—
152	#1379	—	67	2	no	—
153	C₆₇	C₄×#350	1	1	yes	characteristic, complemented, Sylow
154	C₂×#131	Dic₆₇	1	1	yes	characteristic
155	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
156	C₄×#41	Dic₆₇	1	1	yes	characteristic
157	C₂×M₃₂	—	67	2	no	—
158	C₂²×C₁₆	—	67	2	no	—
159	#405	—	2	1	no	—
160	#405	—	2	1	no	—
161	#405	Dic₆₇	1	1	yes	characteristic
162	#405	C₂×D₆₇	1	1	yes	characteristic
163	C₄×M₁₆	—	2	1	no	—
164	C₄×M₁₆	Dic₆₇	1	1	yes	characteristic
165	C₂×C₄×C₈	—	2	1	no	—
166	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
167	#350	—	67	8	no	—
168	#346	—	134	1	no	—
169	C₄×C₁₆	—	134	1	no	—
170	C₄×C₁₆	—	134	1	no	—
171	C₄×C₁₆	—	134	1	no	—
172	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
173	#131	—	2	2	no	—
174	#131	—	2	2	no	—
175	#131	C₆₇⋊C₈	1	4	yes	—
176	#131	C₄×D₆₇	1	4	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
180	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
185	C₄×Q₈	C₆₇⋊C₈	1	2	yes	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	C₆₇⋊C₈	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	C₂×Dic₆₇	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
192	M₃₂	—	67	4	no	—
193	M₃₂	—	134	2	no	—
194	C₂×C₁₆	—	67	4	no	—
195	C₂×C₁₆	—	134	2	no	—
196	C₂×C₁₆	—	134	2	no	—
197	C₂×C₁₆	—	134	2	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	C₄×Dic₆₇	1	2	yes	—
211	#41	C₂×C₆₇⋊C₈	1	2	yes	—
212	C₂×Q₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	#11761612	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	C₄×Dic₆₇	1	2	yes	—
224	M₁₆	C₂×C₆₇⋊C₈	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
231	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
232	C₂×C₈	#11761612	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
234	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂×C₈	#11761628	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
244	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
245	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
246	C₁₆	—	134	4	no	—
247	C₁₆	—	134	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₂×#11761612	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×C₆₇⋊C₈	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×C₆₇⋊C₈	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
263	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
266	C₂×C₄	#14524002	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
273	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
274	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
275	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×#11761612	1	1	yes	characteristic
280	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
281	C₂²	#16863449	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	#16863180	1	4	yes	direct product factor
286	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
287	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
288	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
289	C₄	#16863449	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
294	C₂	C₂×#16863180	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	C₄×#350
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. C₄×#14523983	C₂	semidirect
3. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
4. #22558298	C₂	semidirect
8. #16863180	C₄	direct (N × H)
25. #14523983	C₂×C₄	semidirect
29. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
137. C₄×C₆₇	#350	semidirect
153. C₆₇	C₄×#350	semidirect
285. C₄	#16863180	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×#16863180	2
C₄×#14523983	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558298	2
C₄×#131×C₆₇	2
C₄×#350	67

Subgroups — C4×#16863180 #30770879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863180 #30770879