Subgroups — C₄×#16863513 #30854387

Order 17152.

1046 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863513	C₂	1	2	yes	maximal
2	C₄×#14524050	C₂	1	2	yes	maximal, complemented
3	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560090	C₂	1	2	yes	maximal, complemented
5	#22560049	C₂	1	4	yes	maximal, complemented
6	C₄×#14523970	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524050	C₂²	1	4	yes	—
10	#16863513	C₄	1	16	yes	direct product factor
11	C₂×#14523979	C₂²	1	2	yes	—
12	C₂×#14523970	C₂²	1	4	yes	—
13	#16863352	—	2	4	no	—
14	#16863352	C₂²	1	4	yes	—
15	C₄×#11761612	—	2	2	no	—
16	C₄×#11761612	C₂²	1	2	yes	—
17	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
20	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
21	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
22	#16863161	C₂²	1	2	yes	—
23	#16863155	C₂²	1	4	yes	—
24	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×#41×C₆₇	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₆₇	C₄	1	2	yes	complemented
27	C₄×SD₁₆×C₆₇	C₄	1	4	yes	complemented
28	C₄×D₈×C₆₇	C₄	1	2	yes	complemented
29	#427×C₆₇	C₂²	1	1	yes	characteristic
30	#416×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#14524050	—	2	8	no	—
33	#14524050	C₂×C₄	1	16	yes	complemented
34	C₂×#11761612	—	2	4	no	—
35	C₂×#11761612	C₂³	1	4	yes	—
36	C₂²×C₆₇⋊C₈	—	2	4	no	—
37	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
38	C₂×C₄×Dic₆₇	—	2	2	no	—
39	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
40	#14523979	C₂×C₄	1	8	yes	complemented
41	#14523970	C₂×C₄	1	16	yes	complemented
42	#14523954	—	2	4	no	—
43	#14523954	—	2	2	no	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₄×C₆₇⋊C₈	—	2	2	no	—
46	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
47	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
48	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
49	C₈×Dic₆₇	—	2	4	no	—
50	C₈×Dic₆₇	C₂×C₄	1	8	yes	complemented
51	C₂×#41×C₆₇	C₂³	1	2	yes	—
52	#135×C₆₇	C₂×C₄	1	8	yes	—
53	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₆₇	—	2	4	no	—
60	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
61	#117×C₆₇	—	2	2	no	—
62	#117×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	#107×C₆₇	C₂×C₄	1	2	yes	—
66	#106×C₆₇	C₂×C₄	1	2	yes	—
67	#103×C₆₇	C₂×C₄	1	4	yes	—
68	#102×C₆₇	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
71	C₂²×Dic₆₇	—	2	2	no	—
72	#11761612	—	2	4	no	—
73	#11761612	—	2	16	no	—
74	#11761612	C₂²×C₄	1	8	yes	—
75	C₂×C₆₇⋊C₈	—	2	4	no	—
76	C₂×C₆₇⋊C₈	—	2	16	no	—
77	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
78	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
79	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
80	C₂×C₆₇⋊C₈	#41	1	4	yes	—
81	C₄×Dic₆₇	—	2	8	no	—
82	C₄×Dic₆₇	—	2	2	no	—
83	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
84	#41×C₆₇	—	2	4	no	—
85	#41×C₆₇	C₂²×C₄	1	8	yes	—
86	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	#41	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₆₇	C₄²	1	4	yes	complemented
94	SD₁₆×C₆₇	C₄²	1	8	yes	complemented
95	D₈×C₆₇	C₄²	1	4	yes	complemented
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₆₇	—	2	4	no	—
102	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
103	C₂²⋊C₄×C₆₇	—	2	2	no	—
104	C₂²⋊C₄×C₆₇	—	2	4	no	—
105	C₄²×C₆₇	—	2	4	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₆₇	—	2	2	no	—
111	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
112	Q₈×C₆₇	—	2	2	no	—
113	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
114	D₄×C₆₇	—	2	8	no	—
115	D₄×C₆₇	—	2	2	no	—
116	D₄×C₆₇	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	8	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	4	8	no	—
122	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
128	C₂×Dic₆₇	—	2	8	no	—
129	C₂×Dic₆₇	—	2	4	no	—
130	C₈×C₆₇	—	2	2	no	—
131	C₈×C₆₇	C₂×C₄²	1	2	yes	—
132	C₈×C₆₇	C₂×C₄²	1	2	yes	—
133	C₆₇⋊C₈	—	2	8	no	—
134	C₆₇⋊C₈	C₄×D₄	1	4	yes	—
135	C₆₇⋊C₈	C₄×D₄	1	4	yes	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	4	no	—
138	C₂²×C₆₇	—	4	4	no	—
139	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₆₇	C₄×#41	1	2	yes	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	4	no	—
143	C₄×C₆₇	—	4	4	no	—
144	C₄×C₆₇	#443	1	4	yes	complemented
145	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
147	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
148	Dic₆₇	—	2	8	no	—
149	C₄×#443	—	67	1	no	maximal, Sylow
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	—	4	4	no	—
152	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
153	C₂×C₆₇	C₂×#443	1	2	yes	—
154	C₂×#443	—	67	2	no	—
155	C₄×#135	D₆₇	1	1	yes	characteristic
156	C₄×#131	—	67	2	no	—
157	C₄×#108	—	67	1	no	—
158	#1772	—	67	2	no	—
159	#1764	—	67	4	no	—
160	C₄×#104	—	67	2	no	—
161	C₄²×C₈	—	67	1	no	—
162	C₆₇	C₄×#443	1	1	yes	characteristic, complemented, Sylow
163	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#131	—	67	4	no	—
165	C₄×#41	C₂×D₆₇	1	2	yes	—
166	#443	—	67	16	no	—
167	C₄×Q₁₆	Dic₆₇	1	2	yes	complemented
168	C₄×SD₁₆	Dic₆₇	1	4	yes	complemented
169	C₄×D₈	Dic₆₇	1	2	yes	complemented
170	C₂×#108	—	67	2	no	—
171	#427	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#104	—	67	4	no	—
173	#416	C₂×D₆₇	1	2	yes	—
174	#405	—	67	4	no	—
175	#405	—	134	4	no	—
176	C₄×M₁₆	—	67	2	no	—
177	C₄×M₁₆	—	134	2	no	—
178	C₂×C₄×C₈	—	67	2	no	—
179	C₂×C₄×C₈	—	67	1	no	—
180	C₂×C₄×C₈	—	67	2	no	—
181	C₂×C₄×C₈	—	134	2	no	—
182	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
183	C₄³	—	67	1	no	—
184	#341	—	67	2	no	—
185	#339	—	67	4	no	—
186	C₂×#41	C₂²×D₆₇	1	2	yes	—
187	#135	C₄×D₆₇	1	8	yes	—
188	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
189	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
190	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
191	#131	—	67	16	no	—
192	#131	—	134	8	no	—
193	C₂×M₁₆	—	67	4	no	—
194	C₂×M₁₆	—	134	4	no	—
195	C₂²×C₈	—	67	2	no	—
196	C₂²×C₈	—	134	4	no	—
197	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
199	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	C₂×Dic₆₇	1	4	yes	—
202	#117	—	2	2	no	—
203	#117	C₂²×D₆₇	1	2	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	67	2	no	—
206	C₂×C₄²	—	134	2	no	—
207	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
208	#108	—	67	8	no	—
209	#107	C₂×Dic₆₇	1	2	yes	—
210	#106	C₂×Dic₆₇	1	2	yes	—
211	#104	—	67	16	no	—
212	#103	C₂×Dic₆₇	1	4	yes	—
213	#102	C₂×Dic₆₇	1	4	yes	—
214	#97	—	134	4	no	—
215	#97	—	134	2	no	—
216	C₄×C₈	—	67	8	no	—
217	C₄×C₈	—	67	2	no	—
218	C₄×C₈	—	67	1	no	—
219	C₄×C₈	—	67	1	no	—
220	C₄×C₈	—	134	4	no	—
221	C₄×C₈	—	134	4	no	—
222	C₄×C₈	—	134	2	no	—
223	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
224	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
225	#41	—	2	4	no	—
226	#41	C₂×C₄×D₆₇	1	8	yes	—
227	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	134	2	no	—
233	C₂²×C₄	#11761627	1	2	yes	—
234	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
235	Q₁₆	C₄×Dic₆₇	1	4	yes	complemented
236	SD₁₆	C₄×Dic₆₇	1	8	yes	complemented
237	D₈	C₄×Dic₆₇	1	4	yes	complemented
238	M₁₆	—	67	8	no	—
239	M₁₆	—	134	16	no	—
240	M₁₆	—	134	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	67	4	no	—
243	C₂×C₈	—	67	2	no	—
244	C₂×C₈	—	67	2	no	—
245	C₂×C₈	—	67	4	no	—
246	C₂×C₈	—	134	16	no	—
247	C₂×C₈	—	134	4	no	—
248	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
249	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
250	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	67	4	no	—
259	C₄²	—	134	8	no	—
260	C₄²	—	134	2	no	—
261	C₄²	C₂²×Dic₆₇	1	2	yes	—
262	C₄²	D₄×D₆₇	1	1	yes	characteristic
263	C₄²	D₄×D₆₇	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	C₂×#11761627	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	C₂×C₄×Dic₆₇	1	4	yes	—
268	D₄	—	2	8	no	—
269	D₄	—	2	2	no	—
270	D₄	C₂×C₄×Dic₆₇	1	4	yes	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	—	134	8	no	—
277	C₂×C₄	—	134	4	no	—
278	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
279	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
281	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
283	C₂×C₄	#41×D₆₇	1	2	yes	—
284	C₈	—	2	2	no	—
285	C₈	—	67	4	no	—
286	C₈	—	67	4	no	—
287	C₈	—	134	8	no	—
288	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
289	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
294	C₂²	C₄×#11761627	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	4	no	—
298	C₄	—	134	8	no	—
299	C₄	#16863513	1	4	yes	direct product factor
300	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
301	C₄	C₄×D₄×D₆₇	1	2	yes	—
302	C₄	#41×Dic₆₇	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	4	4	no	—
305	C₂	C₄×D₄×Dic₆₇	1	1	yes	characteristic
306	C₂	C₂×#16863513	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#135×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#135	D₆₇
hypocenter	C₆₇	C₄×#443
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₄×#14524050	C₂	semidirect
3. C₄×#14523979	C₂	semidirect
4. #22560090	C₂	semidirect
5. #22560049	C₂	semidirect
6. C₄×#14523970	C₂	semidirect
7. C₄×C₈×Dic₆₇	C₂	semidirect
10. #16863513	C₄	direct (N × H)
26. C₄×Q₁₆×C₆₇	C₄	semidirect
27. C₄×SD₁₆×C₆₇	C₄	semidirect
28. C₄×D₈×C₆₇	C₄	semidirect
33. #14524050	C₂×C₄	semidirect
40. #14523979	C₂×C₄	semidirect
41. #14523970	C₂×C₄	semidirect
50. C₈×Dic₆₇	C₂×C₄	semidirect
93. Q₁₆×C₆₇	C₄²	semidirect
94. SD₁₆×C₆₇	C₄²	semidirect
95. D₈×C₆₇	C₄²	semidirect
144. C₄×C₆₇	#443	semidirect
162. C₆₇	C₄×#443	semidirect
167. C₄×Q₁₆	Dic₆₇	semidirect
168. C₄×SD₁₆	Dic₆₇	semidirect
169. C₄×D₈	Dic₆₇	semidirect
235. Q₁₆	C₄×Dic₆₇	semidirect
236. SD₁₆	C₄×Dic₆₇	semidirect
237. D₈	C₄×Dic₆₇	semidirect
299. C₄	#16863513	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×#16863513	2
C₄×#14524050	2
C₄×#14523979	2
#22560090	2
#22560049	2
C₄×#14523970	2
C₄×C₈×Dic₆₇	2
C₄×#135×C₆₇	2
C₄×#443	67

Subgroups — C4×#16863513 #30854387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863513 #30854387