Subgroups — #30854433

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 306 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	#22568637	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	#22560056	C₂	1	4	yes	maximal, complemented
6	#22560019	C₂	1	2	yes	maximal, complemented
7	#22559895	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	complemented
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×#14523954	—	2	2	no	—
18	C₂×#14523954	C₂²	1	1	yes	characteristic
19	C₂×#14523953	C₂²	1	2	yes	—
20	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
21	#16863242	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	C₂³	1	2	yes	—
39	#14523979	C₂×C₄	1	8	yes	complemented
40	#14523970	C₂×C₄	1	16	yes	complemented
41	#14523954	—	2	4	no	—
42	#14523954	—	2	2	no	—
43	#14523954	D₄	1	1	yes	characteristic
44	#14523954	D₄	1	1	yes	characteristic
45	#14523954	D₄	1	2	yes	—
46	#14523953	—	2	4	no	—
47	C₄×C₆₇⋊C₈	—	2	4	no	—
48	C₄×C₆₇⋊C₈	—	2	2	no	—
49	C₈×Dic₆₇	C₂×C₄	1	8	yes	complemented
50	C₆₇⋊#95	—	2	2	no	—
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	2	no	—
82	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
83	#41×C₆₇	—	2	4	no	—
84	#41×C₆₇	C₂²×C₄	1	8	yes	—
85	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	#41	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
92	Q₁₆×C₆₇	C₄²	1	4	yes	—
93	SD₁₆×C₆₇	C₄²	1	8	yes	—
94	D₈×C₆₇	C₄²	1	4	yes	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂²⋊C₄×C₆₇	—	2	2	no	—
103	C₂²⋊C₄×C₆₇	—	2	4	no	—
104	C₄²×C₆₇	—	2	4	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
107	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
111	Q₈×C₆₇	—	2	2	no	—
112	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
113	D₄×C₆₇	—	2	8	no	—
114	D₄×C₆₇	—	2	2	no	—
115	D₄×C₆₇	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	8	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	8	4	no	—
121	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
127	C₂×Dic₆₇	—	2	4	no	—
128	C₂×Dic₆₇	—	4	4	no	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	C₂×C₄²	1	2	yes	—
131	C₈×C₆₇	C₂×C₄²	1	2	yes	—
132	C₆₇⋊C₈	—	2	8	no	—
133	C₆₇⋊C₈	C₄×D₄	1	4	yes	—
134	C₆₇⋊C₈	C₄×D₄	1	4	yes	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	4	no	—
137	C₂²×C₆₇	—	4	4	no	—
138	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₆₇	C₄×#41	1	2	yes	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	4	no	—
143	C₄×C₆₇	—	4	4	no	—
144	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
145	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
146	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
147	Dic₆₇	—	4	4	no	—
148	#20514	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	—	4	4	no	—
151	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
152	C₂×C₆₇	#2957	1	2	yes	—
153	C₂×#443	—	67	2	no	—
154	C₄×#135	D₆₇	1	1	yes	characteristic
155	#2875	—	67	2	no	—
156	C₄×#108	—	67	1	no	—
157	#1772	—	67	2	no	—
158	#1765	—	67	4	no	—
159	#1759	—	67	2	no	—
160	#1726	—	67	1	no	—
161	C₆₇	#20514	1	1	yes	characteristic, complemented, Sylow
162	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
163	C₂×#131	—	67	4	no	—
164	C₄×#41	C₂×D₆₇	1	2	yes	—
165	#443	—	67	16	no	—
166	C₄×Q₁₆	Dic₆₇	1	2	yes	complemented
167	C₄×SD₁₆	Dic₆₇	1	4	yes	complemented
168	C₄×D₈	Dic₆₇	1	2	yes	complemented
169	C₂×#108	—	67	2	no	—
170	#427	C₂×D₆₇	1	1	yes	characteristic
171	C₂×#104	—	67	4	no	—
172	#416	C₂×D₆₇	1	2	yes	—
173	#405	—	67	4	no	—
174	#405	—	134	4	no	—
175	C₄×M₁₆	—	67	2	no	—
176	C₄×M₁₆	—	134	2	no	—
177	C₂×#97	—	67	1	no	—
178	C₂×#97	—	67	2	no	—
179	C₂×#97	—	134	2	no	—
180	C₂×C₄×C₈	—	67	2	no	—
181	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
182	#376	—	67	1	no	—
183	#341	—	67	2	no	—
184	#339	—	67	4	no	—
185	C₂×#41	C₂²×D₆₇	1	2	yes	—
186	#135	C₄×D₆₇	1	8	yes	—
187	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
188	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
189	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
190	#131	—	67	16	no	—
191	#131	—	134	8	no	—
192	C₂×M₁₆	—	67	4	no	—
193	C₂×M₁₆	—	134	4	no	—
194	C₂²×C₈	—	67	2	no	—
195	C₂²×C₈	—	134	4	no	—
196	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
198	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	C₂×Dic₆₇	1	4	yes	—
201	#117	—	2	2	no	—
202	#117	C₂²×D₆₇	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	67	2	no	—
205	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
206	#108	—	67	8	no	—
207	#107	C₂×Dic₆₇	1	2	yes	—
208	#106	C₂×Dic₆₇	1	2	yes	—
209	#104	—	67	16	no	—
210	#103	C₂×Dic₆₇	1	4	yes	—
211	#102	C₂×Dic₆₇	1	4	yes	—
212	#97	—	67	1	no	—
213	#97	—	67	1	no	—
214	#97	—	67	2	no	—
215	#97	—	134	4	no	—
216	#97	—	134	4	no	—
217	#97	—	134	2	no	—
218	C₄×C₈	—	67	8	no	—
219	C₄×C₈	—	134	4	no	—
220	C₄×C₈	—	134	2	no	—
221	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
222	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
223	#95	—	134	2	no	—
224	#41	—	2	4	no	—
225	#41	C₂×C₄×D₆₇	1	8	yes	—
226	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	134	2	no	—
232	C₂²×C₄	#11761627	1	2	yes	—
233	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
234	Q₁₆	C₄×Dic₆₇	1	4	yes	—
235	SD₁₆	C₄×Dic₆₇	1	8	yes	—
236	D₈	C₄×Dic₆₇	1	4	yes	—
237	M₁₆	—	67	8	no	—
238	M₁₆	—	134	16	no	—
239	M₁₆	—	134	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	67	4	no	—
242	C₂×C₈	—	67	2	no	—
243	C₂×C₈	—	67	2	no	—
244	C₂×C₈	—	67	4	no	—
245	C₂×C₈	—	134	16	no	—
246	C₂×C₈	—	134	4	no	—
247	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
248	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
249	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
250	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	67	4	no	—
258	C₄²	—	134	2	no	—
259	C₄²	C₂²×Dic₆₇	1	2	yes	—
260	C₄²	D₄×D₆₇	1	1	yes	characteristic
261	C₄²	D₄×D₆₇	1	1	yes	characteristic
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#11761627	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	C₂×C₄×Dic₆₇	1	4	yes	—
266	D₄	—	2	8	no	—
267	D₄	—	2	2	no	—
268	D₄	C₂×C₄×Dic₆₇	1	4	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	—	268	4	no	—
275	C₂×C₄	—	134	4	no	—
276	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
277	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
279	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
281	C₂×C₄	#41×D₆₇	1	2	yes	—
282	C₈	—	2	2	no	—
283	C₈	—	67	4	no	—
284	C₈	—	67	4	no	—
285	C₈	—	134	8	no	—
286	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
287	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
292	C₂²	C₄×#11761627	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	4	4	no	—
297	C₄	—	268	4	no	—
298	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
299	C₄	C₄×D₄×D₆₇	1	2	yes	—
300	C₄	#41×Dic₆₇	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	4	no	—
303	C₂	C₄×D₄×Dic₆₇	1	1	yes	characteristic
304	C₂	#22569126	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×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	#20514
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. #22568637	C₂	semidirect
3. C₄×#14523979	C₂	semidirect
4. #22560090	C₂	semidirect
5. #22560056	C₂	semidirect
6. #22560019	C₂	semidirect
7. #22559895	C₂	semidirect
10. #16863513	C₄	semidirect
26. C₄×Q₁₆×C₆₇	C₄	semidirect
27. C₄×SD₁₆×C₆₇	C₄	semidirect
28. C₄×D₈×C₆₇	C₄	semidirect
33. #14524050	C₂×C₄	semidirect
39. #14523979	C₂×C₄	semidirect
40. #14523970	C₂×C₄	semidirect
49. C₈×Dic₆₇	C₂×C₄	semidirect
161. C₆₇	#20514	semidirect
166. C₄×Q₁₆	Dic₆₇	semidirect
167. C₄×SD₁₆	Dic₆₇	semidirect
168. C₄×D₈	Dic₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16863513	2
#22568637	2
C₄×#14523979	2
#22560090	2
#22560056	2
#22560019	2
#22559895	2
C₄×#135×C₆₇	2
#20514	67