Subgroups — #30853853

Order 17152.

1018 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568799	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863354	C₂	1	2	yes	maximal
3	C₂×#16863354	C₂	1	1	yes	maximal, characteristic
4	#22560096	C₂	1	4	yes	maximal, complemented
5	#22560091	C₂	1	4	yes	maximal, complemented
6	#22559920	C₂	1	2	yes	maximal
7	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14524002	C₂²	1	2	yes	—
10	C₂×#14524002	C₂²	1	1	yes	characteristic
11	#16863438	C₄	1	8	yes	complemented
12	#16863434	C₂²	1	4	yes	—
13	#16863354	—	2	2	no	—
14	#16863354	—	2	2	no	—
15	#16863354	C₄	1	8	yes	complemented
16	#16863354	C₂²	1	4	yes	—
17	#16863354	C₂²	1	4	yes	—
18	C₂×#14523953	—	2	2	no	—
19	C₂×#14523953	C₂²	1	2	yes	—
20	C₂×C₈×Dic₆₇	—	2	2	no	—
21	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
22	#16863157	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
24	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	#430×C₆₇	C₄	1	8	yes	complemented
26	C₂×#108×C₆₇	C₂²	1	4	yes	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₂²×C₆₇⋊C₈	Q₈	1	4	yes	—
31	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
33	#14524002	—	2	2	no	—
34	#14524002	—	2	2	no	—
35	#14524002	—	2	2	no	—
36	#14524002	C₂×C₄	1	4	yes	—
37	#14524002	C₂×C₄	1	4	yes	—
38	#14524002	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₆₇	—	2	1	no	—
40	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
42	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
43	C₂×C₄×Dic₆₇	—	2	2	no	—
44	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
45	#14523977	C₂×C₄	1	8	yes	—
46	#14523975	C₂×C₄	1	8	yes	—
47	#14523953	—	2	4	no	—
48	#14523953	—	2	4	no	—
49	#14523953	C₂×C₄	1	8	yes	complemented
50	C₈×Dic₆₇	—	2	4	no	—
51	C₈×Dic₆₇	—	2	4	no	—
52	C₈×Dic₆₇	C₂×C₄	1	8	yes	complemented
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	4	no	—
56	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
57	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
59	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
60	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
61	C₂²×C₈×C₆₇	—	2	2	no	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	#108×C₆₇	C₂×C₄	1	16	yes	complemented
64	C₂²×Dic₆₇	—	2	4	no	—
65	C₂²×Dic₆₇	—	2	2	no	—
66	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	—
67	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	—
68	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
72	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
74	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
75	C₄⋊Dic₆₇	—	2	4	no	—
76	C₄⋊Dic₆₇	—	2	4	no	—
77	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
78	C₂²⋊Dic₆₇	—	2	4	no	—
79	C₂²⋊Dic₆₇	—	2	4	no	—
80	C₂²⋊Dic₆₇	—	2	4	no	—
81	C₄×Dic₆₇	—	2	4	no	—
82	C₄×Dic₆₇	—	2	4	no	—
83	C₄×Dic₆₇	—	2	2	no	—
84	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
85	C₂⁴×C₆₇	#41	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₆₇	#41	1	2	yes	—
92	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
94	M₁₆×C₆₇	—	2	8	no	—
95	M₁₆×C₆₇	—	2	4	no	—
96	M₁₆×C₆₇	C₄²	1	16	yes	—
97	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
98	C₂×C₈×C₆₇	—	2	8	no	—
99	C₂×C₈×C₆₇	—	2	4	no	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
103	C₂³×C₆₇	—	4	2	no	—
104	C₂³×C₆₇	#117	1	2	yes	—
105	C₂³×C₆₇	C₄×D₄	1	2	yes	—
106	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
107	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₆₇	#117	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₆₇	—	2	4	no	—
122	C₂×Dic₆₇	—	2	8	no	—
123	C₂×Dic₆₇	—	4	4	no	—
124	C₂×Dic₆₇	—	4	8	no	—
125	C₈×C₆₇	—	2	8	no	—
126	C₈×C₆₇	C₂×C₄²	1	8	yes	—
127	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
128	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
129	C₆₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	—	4	2	no	—
134	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
135	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
139	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
140	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
143	Dic₆₇	—	4	4	no	—
144	Dic₆₇	—	4	8	no	—
145	#20444	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	4	no	—
149	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₆₇	#2955	1	2	yes	—
151	C₂×#430	D₆₇	1	1	yes	characteristic
152	#2911	—	67	1	no	—
153	C₂×#405	—	67	1	no	—
154	C₂×#405	—	67	2	no	—
155	#1773	—	67	4	no	—
156	#1772	—	67	4	no	—
157	#1733	—	67	2	no	—
158	C₆₇	#20444	1	1	yes	characteristic, complemented, Sylow
159	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
160	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
161	C₂³×C₈	—	67	1	no	—
162	C₂×#117	—	67	1	no	—
163	C₂×#117	—	67	2	no	—
164	#430	Dic₆₇	1	8	yes	complemented
165	C₂×#108	C₂×D₆₇	1	4	yes	—
166	#428	—	67	8	no	—
167	#427	—	67	4	no	—
168	#405	—	67	8	no	—
169	#405	—	67	4	no	—
170	#405	—	67	4	no	—
171	#405	—	134	2	no	—
172	#405	—	134	2	no	—
173	C₂×#97	—	67	2	no	—
174	C₂×#97	—	134	2	no	—
175	C₂×C₄×C₈	—	67	2	no	—
176	C₂×C₄×C₈	—	134	2	no	—
177	#339	—	67	8	no	—
178	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
182	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
183	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
184	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
185	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	67	4	no	—
188	C₂²×C₈	—	67	2	no	—
189	C₂²×C₈	—	67	2	no	—
190	C₂²×C₈	—	67	4	no	—
191	C₂²×C₈	—	67	1	no	—
192	C₂²×C₈	—	67	1	no	—
193	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
194	#117	—	67	4	no	—
195	#117	—	67	4	no	—
196	#117	—	67	4	no	—
197	#117	—	134	2	no	—
198	#117	—	134	2	no	—
199	#117	—	134	2	no	—
200	C₂×C₄⋊C₄	—	67	4	no	—
201	C₂×C₄⋊C₄	—	134	1	no	—
202	C₂×C₂²⋊C₄	—	134	1	no	—
203	C₂×C₂²⋊C₄	—	134	2	no	—
204	C₂×C₄²	—	67	2	no	—
205	C₂×C₄²	—	134	2	no	—
206	#108	C₂×Dic₆₇	1	16	yes	complemented
207	#107	—	67	8	no	—
208	#106	—	67	8	no	—
209	#97	—	67	8	no	—
210	#97	—	134	4	no	—
211	#97	—	134	4	no	—
212	C₄×C₈	—	67	8	no	—
213	C₄×C₈	—	134	4	no	—
214	C₄×C₈	—	134	4	no	—
215	C₂⁴	#11761629	1	1	yes	characteristic
216	C₂²×C₄	—	134	4	no	—
217	C₂²×C₄	—	134	2	no	—
218	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
221	C₂²×C₄	D₄×D₆₇	1	2	yes	—
222	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
223	C₂²×C₄	#11761627	1	2	yes	—
224	C₂²×C₄	#11761629	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
226	M₁₆	—	2	8	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	C₄×Dic₆₇	1	16	yes	—
229	M₁₆	C₂²×Dic₆₇	1	8	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	67	8	no	—
233	C₂×C₈	—	67	8	no	—
234	C₂×C₈	—	67	2	no	—
235	C₂×C₈	—	67	2	no	—
236	C₂×C₈	—	67	1	no	—
237	C₂×C₈	—	67	2	no	—
238	C₂×C₈	—	67	1	no	—
239	C₂×C₈	—	67	2	no	—
240	C₂×C₈	—	67	2	no	—
241	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
242	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
243	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
244	C₄⋊C₄	—	67	8	no	—
245	C₄⋊C₄	—	134	4	no	—
246	C₄⋊C₄	—	134	4	no	—
247	C₂²⋊C₄	—	134	4	no	—
248	C₂²⋊C₄	—	134	4	no	—
249	C₂²⋊C₄	—	134	4	no	—
250	C₄²	—	67	4	no	—
251	C₄²	—	134	4	no	—
252	C₄²	—	134	4	no	—
253	C₄²	—	134	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	#14524001	1	2	yes	—
256	C₂³	#14524005	1	2	yes	—
257	C₂³	Q₈×Dic₆₇	1	2	yes	—
258	C₂³	C₂×#11761629	1	1	yes	characteristic
259	C₂×C₄	—	268	4	no	—
260	C₂×C₄	—	268	8	no	—
261	C₂×C₄	—	134	4	no	—
262	C₂×C₄	—	134	8	no	—
263	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
264	C₂×C₄	C₄⋊C₄×D₆₇	1	4	yes	—
265	C₂×C₄	#14524001	1	2	yes	—
266	C₂×C₄	#14524005	1	2	yes	—
267	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
268	C₂×C₄	#14524008	1	4	yes	—
269	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
276	C₈	—	2	8	no	—
277	C₈	—	67	2	no	—
278	C₈	—	67	2	no	—
279	C₈	—	67	4	no	—
280	C₈	C₂×C₄×Dic₆₇	1	8	yes	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₄⋊C₄×Dic₆₇	1	4	yes	—
286	C₂²	C₂×#14524001	1	1	yes	characteristic
287	C₂²	C₂×#14524005	1	1	yes	characteristic
288	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
289	C₄	—	268	4	no	—
290	C₄	—	268	8	no	—
291	C₄	C₄⋊C₄×Dic₆₇	1	4	yes	—
292	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
293	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
294	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
295	C₄	C₂×#14524008	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	4	no	—
298	C₂	—	2	1	no	—
299	C₂	C₂×C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
300	C₂	#22569111	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#430×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#430	D₆₇
hypocenter	C₆₇	#20444
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
4. #22560096	C₂	semidirect
5. #22560091	C₂	semidirect
11. #16863438	C₄	semidirect
15. #16863354	C₄	semidirect
25. #430×C₆₇	C₄	semidirect
49. #14523953	C₂×C₄	semidirect
52. C₈×Dic₆₇	C₂×C₄	semidirect
63. #108×C₆₇	C₂×C₄	semidirect
158. C₆₇	#20444	semidirect
164. #430	Dic₆₇	semidirect
206. #108	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568799	2
C₂×#16863354	2
#22560096	2
#22560091	2
#22559920	2
C₂×#430×C₆₇	2
#20444	67