Subgroups — #30827647

Order 17152.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569158	C₂	1	2	yes	maximal
2	#22559118	C₂	1	2	yes	maximal, complemented
3	#22559081	C₂	1	4	yes	maximal, complemented
4	#22559028	C₂	1	2	yes	maximal, complemented
5	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
6	#22558612	C₂	1	2	yes	maximal
7	#22558498	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic
9	#16863527	—	2	4	no	—
10	#16863527	C₂²	1	4	yes	—
11	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	—
12	#16863449	C₂²	1	1	yes	characteristic
13	#16863423	—	2	2	no	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	#16863352	—	2	2	no	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	#16863147	C₂²	1	1	yes	characteristic
18	#16863147	C₂²	1	2	yes	—
19	#16863147	C₂²	1	1	yes	characteristic
20	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
21	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
22	#16863118	C₂²	1	2	yes	—
23	#16863117	C₂²	1	2	yes	—
24	#533×C₆₇	C₂²	1	2	yes	—
25	#500×C₆₇	C₄	1	1	yes	characteristic
26	#494×C₆₇	C₄	1	1	yes	characteristic
27	#487×C₆₇	C₄	1	2	yes	—
28	#467×C₆₇	C₄	1	2	yes	—
29	#467×C₆₇	C₄	1	2	yes	—
30	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
31	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#11761612	D₄	1	2	yes	—
34	C₂²×C₆₇⋊C₈	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
39	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
42	#14523954	—	2	4	no	—
43	#14523954	D₄	1	2	yes	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
53	#125×C₆₇	—	2	2	no	—
54	C₂²⋊Q₈×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
58	#117×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	#107×C₆₇	C₂×C₄	1	2	yes	—
61	#107×C₆₇	C₂×C₄	1	2	yes	—
62	#103×C₆₇	C₂×C₄	1	4	yes	—
63	#103×C₆₇	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	#11761612	—	2	4	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	4	4	no	—
73	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
74	C₂×Q₈×C₆₇	—	2	2	no	—
75	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	Q₁₆×C₆₇	—	2	4	no	—
80	Q₁₆×C₆₇	—	4	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	4	1	no	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
93	C₂²⋊C₄×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	2	4	no	—
95	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₇	#41	1	1	yes	characteristic
98	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₆₇	—	2	4	no	—
101	Q₈×C₆₇	—	2	4	no	—
102	Q₈×C₆₇	—	8	2	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	4	1	no	—
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#137	1	4	yes	—
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₈×C₆₇	—	2	1	no	—
116	C₈×C₆₇	—	2	1	no	—
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	#131	1	2	yes	—
119	C₆₇⋊C₈	—	2	4	no	—
120	C₆₇⋊C₈	—	4	2	no	—
121	C₆₇⋊C₈	—	4	2	no	—
122	C₆₇⋊C₈	—	4	2	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	#441	1	2	yes	—
126	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	4	4	no	—
131	C₄×C₆₇	—	4	4	no	—
132	C₄×C₆₇	#441	1	2	yes	—
133	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
134	C₄×C₆₇	#568	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#137	1	2	yes	—
136	#18072	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	#2935	1	1	yes	characteristic
139	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
140	C₂×C₆₇	#2957	1	1	yes	characteristic
141	#3170	D₆₇	1	1	yes	characteristic
142	#2962	—	67	2	no	—
143	#1593	—	67	2	no	—
144	#1587	—	67	4	no	—
145	#1579	—	67	2	no	—
146	C₂×#334	—	67	1	no	—
147	#1481	—	67	2	no	—
148	#1451	—	67	1	no	—
149	C₆₇	#18072	1	1	yes	characteristic, complemented, Sylow
150	#533	C₂×D₆₇	1	2	yes	—
151	#500	Dic₆₇	1	1	yes	characteristic
152	#494	Dic₆₇	1	1	yes	characteristic
153	#487	Dic₆₇	1	2	yes	—
154	#467	Dic₆₇	1	2	yes	—
155	#467	Dic₆₇	1	2	yes	—
156	#446	—	67	4	no	—
157	#446	—	134	4	no	—
158	C₈×Q₈	—	67	4	no	—
159	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
160	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
161	#431	—	67	1	no	—
162	#424	—	134	2	no	—
163	C₂×C₄⋊C₈	—	67	2	no	—
164	#405	—	134	2	no	—
165	C₄×M₁₆	—	67	1	no	—
166	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
167	#334	—	67	1	no	—
168	#334	—	67	2	no	—
169	#334	—	67	1	no	—
170	Q₈⋊C₈	—	67	4	no	—
171	Q₈⋊C₈	—	67	4	no	—
172	#322	—	67	2	no	—
173	#322	—	67	2	no	—
174	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
175	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
176	C₂×M₁₆	—	67	2	no	—
177	C₂²×C₈	—	134	2	no	—
178	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
180	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
181	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
182	#125	—	2	2	no	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	C₂×Dic₆₇	1	4	yes	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
187	#117	—	2	2	no	—
188	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	#107	C₂×Dic₆₇	1	2	yes	—
190	#107	C₂×Dic₆₇	1	2	yes	—
191	C₄⋊C₈	—	67	2	no	—
192	C₄⋊C₈	—	67	2	no	—
193	C₄⋊C₈	—	134	4	no	—
194	C₄⋊C₈	—	134	4	no	—
195	C₄⋊C₈	—	134	2	no	—
196	C₄⋊C₈	—	134	1	no	—
197	#103	C₂×Dic₆₇	1	4	yes	—
198	#103	C₂×Dic₆₇	1	4	yes	—
199	C₂²⋊C₈	—	134	1	no	—
200	#97	—	67	2	no	—
201	#97	—	134	4	no	—
202	C₄×C₈	—	67	2	no	—
203	C₄×C₈	—	67	2	no	—
204	C₄×C₈	—	134	4	no	—
205	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
211	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	#11761627	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	4	2	no	—
216	M₁₆	—	134	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	268	4	no	—
220	C₂×C₈	—	67	4	no	—
221	C₂×C₈	—	134	2	no	—
222	C₂×C₈	—	134	2	no	—
223	C₂×C₈	—	134	2	no	—
224	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
237	C₄²	D₄×D₆₇	1	1	yes	characteristic
238	C₄²	#11761627	1	1	yes	characteristic
239	C₄²	C₂³×D₆₇	1	1	yes	characteristic
240	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	8	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
251	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
252	C₂×C₄	#14524075	1	4	yes	—
253	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
255	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	268	2	no	—
260	C₈	—	268	2	no	—
261	C₈	—	268	2	no	—
262	C₈	—	134	4	no	—
263	C₈	#14524050	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#16863503	1	2	yes	—
267	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	—	4	4	no	—
273	C₄	#16863503	1	2	yes	—
274	C₄	C₂×#14524050	1	1	yes	characteristic
275	C₄	#16864163	1	1	yes	characteristic
276	C₄	C₂×#14524075	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#22568953	1	1	yes	characteristic
279	C₂	C₂×#16863503	1	1	yes	characteristic
280	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3170	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3170	D₆₇
hypocenter	C₆₇	#18072
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #22559118	C₂	semidirect
3. #22559081	C₂	semidirect
4. #22559028	C₂	semidirect
149. C₆₇	#18072	semidirect

Maximal subgroups

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

subgroup	index
#22569158	2
#22559118	2
#22559081	2
#22559028	2
C₂×#16863147	2
#22558612	2
#22558498	2
C₆₇×#3170	2
#18072	67