Subgroups — #30854425

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863518	C₂	1	2	yes	maximal
2	#22568637	C₂	1	2	yes	maximal, complemented
3	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560095	C₂	1	2	yes	maximal, complemented
5	#22560049	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	#16863518	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	2	yes	—
44	#14523954	D₄	1	2	yes	—
45	#14523953	C₂×C₄	1	8	yes	complemented
46	C₄×C₆₇⋊C₈	—	2	4	no	—
47	C₄×C₆₇⋊C₈	—	2	2	no	—
48	C₈×Dic₆₇	—	2	4	no	—
49	C₆₇⋊#95	—	2	2	no	—
50	C₂×#41×C₆₇	C₂³	1	2	yes	—
51	#135×C₆₇	C₂×C₄	1	8	yes	—
52	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
58	C₄×D₄×C₆₇	—	2	4	no	—
59	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
60	#117×C₆₇	—	2	2	no	—
61	#117×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	C₂×C₄	1	2	yes	—
65	#106×C₆₇	C₂×C₄	1	2	yes	—
66	#103×C₆₇	C₂×C₄	1	4	yes	—
67	#102×C₆₇	C₂×C₄	1	4	yes	—
68	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₂²×Dic₆₇	—	2	2	no	—
71	#11761612	—	2	4	no	—
72	#11761612	—	2	16	no	—
73	#11761612	C₂²×C₄	1	8	yes	—
74	C₂×C₆₇⋊C₈	—	2	4	no	—
75	C₂×C₆₇⋊C₈	—	2	16	no	—
76	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
78	C₂×C₆₇⋊C₈	#41	1	4	yes	—
79	C₄×Dic₆₇	—	2	2	no	—
80	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
81	#41×C₆₇	—	2	4	no	—
82	#41×C₆₇	C₂²×C₄	1	8	yes	—
83	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	#41	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
90	Q₁₆×C₆₇	C₄²	1	4	yes	—
91	SD₁₆×C₆₇	C₄²	1	8	yes	—
92	D₈×C₆₇	C₄²	1	4	yes	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄×C₆₇	—	2	2	no	—
101	C₂²⋊C₄×C₆₇	—	2	4	no	—
102	C₄²×C₆₇	—	2	4	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
109	Q₈×C₆₇	—	2	2	no	—
110	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
111	D₄×C₆₇	—	2	8	no	—
112	D₄×C₆₇	—	2	2	no	—
113	D₄×C₆₇	C₂×C₄²	1	4	yes	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	8	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	8	4	no	—
119	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
125	C₂×Dic₆₇	—	2	4	no	—
126	C₂×Dic₆₇	—	4	4	no	—
127	C₈×C₆₇	—	2	2	no	—
128	C₈×C₆₇	C₂×C₄²	1	2	yes	—
129	C₈×C₆₇	C₂×C₄²	1	2	yes	—
130	C₆₇⋊C₈	—	2	8	no	—
131	C₆₇⋊C₈	C₄×D₄	1	8	yes	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	—	4	4	no	—
135	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₆₇	C₄×#41	1	2	yes	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	2	4	no	—
140	C₄×C₆₇	—	4	4	no	—
141	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
143	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
144	Dic₆₇	—	4	4	no	—
145	#20513	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	—	4	4	no	—
148	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
149	C₂×C₆₇	#2957	1	2	yes	—
150	C₂×#444	—	67	2	no	—
151	C₄×#135	D₆₇	1	1	yes	characteristic
152	#2875	—	67	2	no	—
153	C₄×#108	—	67	1	no	—
154	#1773	—	67	2	no	—
155	#1764	—	67	4	no	—
156	#1759	—	67	2	no	—
157	#1726	—	67	1	no	—
158	C₆₇	#20513	1	1	yes	characteristic, complemented, Sylow
159	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
160	C₂×#131	—	67	4	no	—
161	C₄×#41	C₂×D₆₇	1	2	yes	—
162	#444	—	67	16	no	—
163	C₄×Q₁₆	Dic₆₇	1	2	yes	complemented
164	C₄×SD₁₆	Dic₆₇	1	4	yes	complemented
165	C₄×D₈	Dic₆₇	1	2	yes	complemented
166	C₂×#108	—	67	2	no	—
167	#427	C₂×D₆₇	1	1	yes	characteristic
168	C₂×#104	—	67	4	no	—
169	#416	C₂×D₆₇	1	2	yes	—
170	#405	—	67	4	no	—
171	#405	—	134	4	no	—
172	C₄×M₁₆	—	67	2	no	—
173	C₄×M₁₆	—	134	2	no	—
174	C₂×#97	—	67	2	no	—
175	C₂×#97	—	67	1	no	—
176	C₂×#97	—	134	2	no	—
177	C₂×C₄×C₈	—	67	2	no	—
178	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
179	#376	—	67	1	no	—
180	#341	—	67	2	no	—
181	#339	—	67	4	no	—
182	C₂×#41	C₂²×D₆₇	1	2	yes	—
183	#135	C₄×D₆₇	1	8	yes	—
184	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
185	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
186	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
187	#131	—	67	16	no	—
188	#131	—	134	8	no	—
189	C₂×M₁₆	—	67	4	no	—
190	C₂×M₁₆	—	134	4	no	—
191	C₂²×C₈	—	67	2	no	—
192	C₂²×C₈	—	134	4	no	—
193	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	C₂×Dic₆₇	1	4	yes	—
198	#117	—	2	2	no	—
199	#117	C₂²×D₆₇	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	67	2	no	—
202	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
203	#108	—	67	8	no	—
204	#107	C₂×Dic₆₇	1	2	yes	—
205	#106	C₂×Dic₆₇	1	2	yes	—
206	#104	—	67	16	no	—
207	#103	C₂×Dic₆₇	1	4	yes	—
208	#102	C₂×Dic₆₇	1	4	yes	—
209	#97	—	67	8	no	—
210	#97	—	67	2	no	—
211	#97	—	67	2	no	—
212	#97	—	134	4	no	—
213	#97	—	134	2	no	—
214	C₄×C₈	—	134	4	no	—
215	C₄×C₈	—	134	4	no	—
216	C₄×C₈	—	134	2	no	—
217	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
218	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
219	#95	—	134	2	no	—
220	#41	—	2	4	no	—
221	#41	C₂×C₄×D₆₇	1	8	yes	—
222	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	134	2	no	—
228	C₂²×C₄	#11761627	1	2	yes	—
229	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
230	Q₁₆	C₄×Dic₆₇	1	4	yes	—
231	SD₁₆	C₄×Dic₆₇	1	8	yes	—
232	D₈	C₄×Dic₆₇	1	4	yes	—
233	M₁₆	—	67	8	no	—
234	M₁₆	—	134	16	no	—
235	M₁₆	—	134	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	67	4	no	—
238	C₂×C₈	—	67	4	no	—
239	C₂×C₈	—	67	4	no	—
240	C₂×C₈	—	134	16	no	—
241	C₂×C₈	—	134	4	no	—
242	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
243	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
244	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
245	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	67	4	no	—
253	C₄²	—	134	2	no	—
254	C₄²	C₂²×Dic₆₇	1	2	yes	—
255	C₄²	D₄×D₆₇	1	1	yes	characteristic
256	C₄²	D₄×D₆₇	1	1	yes	characteristic
257	C₂³	—	2	2	no	—
258	C₂³	C₂×#11761627	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	Q₈	C₂×C₄×Dic₆₇	1	4	yes	—
261	D₄	—	2	8	no	—
262	D₄	—	2	2	no	—
263	D₄	C₂×C₄×Dic₆₇	1	4	yes	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	268	4	no	—
270	C₂×C₄	—	134	4	no	—
271	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
272	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
274	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
276	C₂×C₄	#41×D₆₇	1	2	yes	—
277	C₈	—	2	2	no	—
278	C₈	—	67	8	no	—
279	C₈	—	134	8	no	—
280	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
281	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
286	C₂²	C₄×#11761627	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	268	4	no	—
292	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
293	C₄	C₄×D₄×D₆₇	1	2	yes	—
294	C₄	#41×Dic₆₇	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	C₄×D₄×Dic₆₇	1	1	yes	characteristic
298	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	#20513
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. #22560095	C₂	semidirect
5. #22560049	C₂	semidirect
6. #22560019	C₂	semidirect
7. #22559895	C₂	semidirect
10. #16863518	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
45. #14523953	C₂×C₄	semidirect
158. C₆₇	#20513	semidirect
163. C₄×Q₁₆	Dic₆₇	semidirect
164. C₄×SD₁₆	Dic₆₇	semidirect
165. 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×#16863518	2
#22568637	2
C₄×#14523979	2
#22560095	2
#22560049	2
#22560019	2
#22559895	2
C₄×#135×C₆₇	2
#20513	67