Subgroups — #30856102

Order 17152.

684 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	#22560358	C₂	1	8	yes	maximal, complemented
2	C₂×#16863167	C₂	1	2	yes	maximal
3	#22559919	C₂	1	2	yes	maximal
4	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863153	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	2	yes	—
9	#16863167	C₂²	1	8	yes	—
10	#16863156	C₂²	1	8	yes	—
11	C₆₇⋊#338	C₂²	1	4	yes	—
12	#16863153	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	#405×C₆₇	C₄	1	4	yes	complemented
17	#405×C₆₇	C₄	1	4	yes	complemented
18	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	C₂³	1	4	yes	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	#14524002	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14524002	C₂³	1	4	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
41	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	#117×C₆₇	—	2	1	no	—
45	#117×C₆₇	—	2	1	no	—
46	#117×C₆₇	C₂×C₄	1	2	yes	—
47	#117×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
50	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
51	#97×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
53	C₂²×Dic₆₇	—	4	2	no	—
54	#11761612	—	2	4	no	—
55	#11761612	—	2	4	no	—
56	#11761612	—	4	2	no	—
57	#11761612	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	2	4	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₂²⋊Dic₆₇	—	2	4	no	—
64	C₂²⋊Dic₆₇	—	4	2	no	—
65	C₄×Dic₆₇	—	2	4	no	—
66	C₄×Dic₆₇	—	4	2	no	—
67	C₂⁴×C₆₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	#41	1	2	yes	—
73	C₂²×C₄×C₆₇	#41	1	2	yes	—
74	C₂²×C₄×C₆₇	#41	1	2	yes	—
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₆₇	—	2	4	no	—
80	M₁₆×C₆₇	—	2	2	no	—
81	M₁₆×C₆₇	—	2	2	no	—
82	M₁₆×C₆₇	—	2	4	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₂²⋊C₄×C₆₇	—	2	2	no	—
92	C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂³×C₆₇	—	8	1	no	—
96	C₂³×C₆₇	#117	1	1	yes	characteristic
97	C₂³×C₆₇	#117	1	1	yes	characteristic
98	C₂³×C₆₇	#126	1	2	yes	—
99	C₂³×C₆₇	#126	1	2	yes	—
100	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	8	2	no	—
103	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	2	yes	—
106	C₂×C₄×C₆₇	#117	1	2	yes	—
107	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#126	1	2	yes	—
112	C₂×C₄×C₆₇	#126	1	2	yes	—
113	C₂×C₄×C₆₇	#126	1	2	yes	—
114	C₂×C₄×C₆₇	#126	1	2	yes	—
115	C₂×C₄×C₆₇	#126	1	2	yes	—
116	C₂×C₄×C₆₇	#126	1	2	yes	—
117	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₆₇	—	4	4	no	—
122	C₂×Dic₆₇	—	8	4	no	—
123	C₈×C₆₇	—	2	4	no	—
124	C₈×C₆₇	—	2	4	no	—
125	C₆₇⋊C₈	—	4	4	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	—	4	1	no	—
129	C₂²×C₆₇	—	4	1	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	#383	1	1	yes	characteristic
132	C₂²×C₆₇	#383	1	2	yes	—
133	C₂²×C₆₇	#383	1	1	yes	characteristic
134	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#126	1	2	yes	—
136	C₄×C₆₇	—	4	4	no	—
137	C₄×C₆₇	#383	1	1	yes	characteristic
138	C₄×C₆₇	#383	1	2	yes	—
139	C₄×C₆₇	#383	1	1	yes	characteristic
140	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
141	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
142	C₄×C₆₇	C₂×#126	1	2	yes	—
143	Dic₆₇	—	8	4	no	—
144	#20776	—	67	1	no	maximal, Sylow
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	C₂×#383	1	1	yes	characteristic
151	#2873	D₆₇	1	1	yes	characteristic
152	#1841	—	67	8	no	—
153	C₂×#344	—	67	2	no	—
154	#1733	—	67	2	no	—
155	C₂×#338	—	67	1	no	—
156	C₂×H(3,ℤ/4)	—	67	1	no	—
157	C₆₇	#20776	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	67	2	no	—
159	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
160	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
161	C₂×#117	—	67	2	no	—
162	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
163	#405	Dic₆₇	1	4	yes	complemented
164	#405	Dic₆₇	1	4	yes	complemented
165	C₄×M₁₆	C₂×D₆₇	1	4	yes	—
166	#344	—	67	8	no	—
167	#339	—	67	8	no	—
168	#338	—	67	4	no	—
169	H(3,ℤ/4)	—	67	4	no	—
170	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	67	4	no	—
176	C₂×M₁₆	—	134	2	no	—
177	C₂×M₁₆	—	134	2	no	—
178	C₂×M₁₆	—	134	2	no	—
179	C₂×M₁₆	—	134	2	no	—
180	C₂×M₁₆	—	134	2	no	—
181	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
182	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
183	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
184	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
185	C₂²×C₈	—	134	2	no	—
186	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
187	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	67	4	no	—
191	#117	—	134	2	no	—
192	#117	—	134	2	no	—
193	#117	—	134	2	no	—
194	#117	C₂×Dic₆₇	1	2	yes	—
195	#117	C₂×Dic₆₇	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	134	2	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	134	2	no	—
200	C₂×C₄²	—	134	2	no	—
201	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
202	#97	C₂×Dic₆₇	1	8	yes	complemented
203	C₄×C₈	C₂×Dic₆₇	1	8	yes	complemented
204	C₂⁴	#11761628	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	268	2	no	—
207	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
209	C₂²×C₄	#11761628	1	1	yes	characteristic
210	C₂²×C₄	#11761628	1	2	yes	—
211	C₂²×C₄	#11761628	1	2	yes	—
212	C₂²×C₄	#11761628	1	2	yes	—
213	C₂²×C₄	#11761628	1	2	yes	—
214	C₂²×C₄	#11761628	1	1	yes	characteristic
215	C₂²×C₄	#11761628	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	268	2	no	—
223	M₁₆	—	268	2	no	—
224	M₁₆	—	134	4	no	—
225	M₁₆	—	134	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	268	2	no	—
230	C₂×C₈	—	268	2	no	—
231	C₂×C₈	—	134	4	no	—
232	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
233	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	268	2	no	—
237	C₄⋊C₄	—	134	4	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	268	2	no	—
241	C₂²⋊C₄	—	134	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	268	2	no	—
244	C₄²	—	134	4	no	—
245	C₄²	C₂²×Dic₆₇	1	4	yes	—
246	C₂³	—	8	1	no	—
247	C₂³	#14524002	1	1	yes	characteristic
248	C₂³	#14524002	1	1	yes	characteristic
249	C₂³	#14524037	1	2	yes	—
250	C₂³	#14524037	1	2	yes	—
251	C₂³	C₂×#11761628	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	268	4	no	—
255	C₂×C₄	—	536	4	no	—
256	C₂×C₄	#14524002	1	1	yes	characteristic
257	C₂×C₄	#14524002	1	1	yes	characteristic
258	C₂×C₄	#14524002	1	1	yes	characteristic
259	C₂×C₄	#14524002	1	1	yes	characteristic
260	C₂×C₄	#14524002	1	2	yes	—
261	C₂×C₄	#14524002	1	2	yes	—
262	C₂×C₄	#14524002	1	1	yes	characteristic
263	C₂×C₄	#14524002	1	1	yes	characteristic
264	C₂×C₄	#14524037	1	2	yes	—
265	C₂×C₄	#14524037	1	2	yes	—
266	C₂×C₄	#14524037	1	2	yes	—
267	C₂×C₄	#14524037	1	2	yes	—
268	C₂×C₄	#14524037	1	2	yes	—
269	C₂×C₄	#14524037	1	2	yes	—
270	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#11761628	1	2	yes	—
272	C₂×C₄	C₂×#11761628	1	2	yes	—
273	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	—	268	4	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#16863264	1	1	yes	characteristic
283	C₂²	#16863264	1	1	yes	characteristic
284	C₂²	#16863264	1	2	yes	—
285	C₂²	C₂×#14524002	1	1	yes	characteristic
286	C₂²	C₂×#14524037	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	536	4	no	—
289	C₄	#16863264	1	1	yes	characteristic
290	C₄	#16863264	1	1	yes	characteristic
291	C₄	#16863264	1	2	yes	—
292	C₄	C₂×#14524002	1	1	yes	characteristic
293	C₄	C₂×#14524002	1	1	yes	characteristic
294	C₄	C₂×#14524037	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	1	no	—
300	C₂	C₂×#16863264	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524002
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2873	C₂
socle	C₂×C₆₇	C₂×#383
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2873	D₆₇
hypocenter	C₆₇	#20776
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ #2873

Complements

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

N	H ≅ G/N	type
1. #22560358	C₂	semidirect
16. #405×C₆₇	C₄	semidirect
17. #405×C₆₇	C₄	semidirect
51. #97×C₆₇	C₂×C₄	semidirect
52. C₄×C₈×C₆₇	C₂×C₄	semidirect
157. C₆₇	#20776	semidirect
163. #405	Dic₆₇	semidirect
164. #405	Dic₆₇	semidirect
202. #97	C₂×Dic₆₇	semidirect
203. C₄×C₈	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22560358	2
C₂×#16863167	2
#22559919	2
C₂×C₆₇⋊#338	2
C₂×#16863153	2
C₆₇×#2873	2
#20776	67