Subgroups — #30765618

Order 17152.

380 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	C₂×#16863217	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558182	C₂	1	4	yes	maximal, complemented
4	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	—	2	1	no	—
6	#16863449	—	2	1	no	—
7	#16863217	—	2	1	no	—
8	#16863217	C₂²	1	2	yes	complemented
9	#16863117	—	2	1	no	—
10	C₈×C₆₇⋊C₈	C₂²	1	2	yes	complemented
11	C₂×#135×C₆₇	C₄	1	1	yes	characteristic
12	C₂×#131×C₆₇	—	2	1	no	—
13	#443×C₆₇	—	2	1	no	—
14	#443×C₆₇	—	2	1	no	—
15	#443×C₆₇	C₄	1	2	yes	—
16	#443×C₆₇	C₂²	1	2	yes	—
17	C₂×#108×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#104×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#14523983	—	2	2	no	—
21	#14523983	—	4	1	no	—
22	C₂×C₆₇⋊C₁₆	—	4	1	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₂×#41×C₆₇	—	2	1	no	—
27	#135×C₆₇	—	2	1	no	—
28	#135×C₆₇	—	2	1	no	—
29	#135×C₆₇	C₈	1	2	yes	complemented
30	#135×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₆₇	C₈	1	1	yes	characteristic, complemented
32	C₂×SD₁₆×C₆₇	—	2	1	no	—
33	C₂×D₈×C₆₇	C₈	1	1	yes	characteristic, complemented
34	#131×C₆₇	—	2	2	no	—
35	#131×C₆₇	—	2	2	no	—
36	#131×C₆₇	—	4	1	no	—
37	#131×C₆₇	—	4	1	no	—
38	C₂×M₁₆×C₆₇	—	2	1	no	—
39	C₂×M₁₆×C₆₇	—	4	1	no	—
40	C₂²×C₈×C₆₇	—	4	1	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	#108×C₆₇	—	2	1	no	—
45	#108×C₆₇	C₂×C₄	1	2	yes	—
46	#104×C₆₇	—	2	2	no	—
47	#104×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₆₇⋊C₁₆	—	4	2	no	—
54	#41×C₆₇	—	2	2	no	—
55	#41×C₆₇	—	2	2	no	—
56	#41×C₆₇	—	4	1	no	—
57	#41×C₆₇	—	4	1	no	—
58	C₂×Q₈×C₆₇	—	2	1	no	—
59	C₂×D₄×C₆₇	—	2	1	no	—
60	C₂×D₄×C₆₇	—	4	1	no	—
61	C₂²×C₄×C₆₇	—	2	1	no	—
62	C₂²×C₄×C₆₇	—	4	1	no	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	Q₁₆×C₆₇	—	2	1	no	—
65	Q₁₆×C₆₇	C₂×C₈	1	2	yes	—
66	SD₁₆×C₆₇	—	2	2	no	—
67	SD₁₆×C₆₇	—	2	2	no	—
68	D₈×C₆₇	—	2	1	no	—
69	D₈×C₆₇	C₂×C₈	1	2	yes	—
70	M₁₆×C₆₇	—	2	2	no	—
71	M₁₆×C₆₇	—	4	1	no	—
72	M₁₆×C₆₇	—	4	2	no	—
73	M₁₆×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	—	4	1	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	—	4	2	no	—
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
81	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
82	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
83	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
84	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
85	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
86	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	—	2	1	no	—
90	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₆₇	—	4	1	no	—
93	C₂³×C₆₇	#104	1	1	yes	characteristic
94	Q₈×C₆₇	—	2	2	no	—
95	Q₈×C₆₇	—	4	1	no	—
96	D₄×C₆₇	—	2	2	no	—
97	D₄×C₆₇	—	4	2	no	—
98	D₄×C₆₇	—	4	2	no	—
99	D₄×C₆₇	—	4	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	4	1	no	—
104	C₂×C₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#104	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₈×C₆₇	—	4	2	no	—
113	C₈×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
114	C₈×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₆₇	#102	1	1	yes	characteristic
116	C₈×C₆₇	#102	1	1	yes	characteristic
117	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
118	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
119	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
120	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
121	C₆₇⋊C₈	—	4	4	no	—
122	C₂²×C₆₇	—	4	1	no	—
123	C₂²×C₆₇	—	4	2	no	—
124	C₂²×C₆₇	—	8	2	no	—
125	C₂²×C₆₇	D₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₆₇	D₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	—	4	1	no	—
130	C₄×C₆₇	—	4	2	no	—
131	C₄×C₆₇	D₄⋊C₈	1	1	yes	characteristic
132	C₄×C₆₇	D₄⋊C₈	1	1	yes	characteristic
133	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
135	#11602	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	—	2	1	no	—
137	C₂×C₆₇	—	2	1	no	—
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	—	8	2	no	—
140	C₂×C₆₇	C₂×D₄⋊C₈	1	1	yes	characteristic
141	C₂×#443	D₆₇	1	1	yes	characteristic
142	C₂×#364	—	67	1	no	—
143	#1450	—	67	1	no	—
144	#1335	—	67	4	no	—
145	C₆₇	#11602	1	1	yes	characteristic, complemented, Sylow
146	C₂×#135	Dic₆₇	1	1	yes	characteristic
147	C₂×#131	—	2	1	no	—
148	C₂×M₃₂	—	134	1	no	—
149	#443	—	2	1	no	—
150	#443	—	2	1	no	—
151	#443	Dic₆₇	1	2	yes	—
152	#443	C₂×D₆₇	1	2	yes	—
153	#431	—	134	1	no	—
154	C₂×#108	Dic₆₇	1	1	yes	characteristic
155	C₂×#104	—	2	1	no	—
156	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	#364	—	67	2	no	—
158	#364	—	134	1	no	—
159	#322	—	134	1	no	—
160	C₈²	—	67	2	no	—
161	C₂×#41	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	C₆₇⋊C₈	1	2	yes	complemented
165	#135	C₂×Dic₆₇	1	2	yes	—
166	C₂×Q₁₆	C₆₇⋊C₈	1	1	yes	characteristic, complemented
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	C₆₇⋊C₈	1	1	yes	characteristic, complemented
169	#131	—	2	2	no	—
170	#131	—	2	2	no	—
171	#131	—	4	1	no	—
172	#131	—	4	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	4	1	no	—
175	C₂²×C₈	—	4	1	no	—
176	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
177	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
179	M₃₂	—	268	1	no	—
180	M₃₂	—	134	2	no	—
181	C₂×C₁₆	—	268	1	no	—
182	#108	—	2	1	no	—
183	#108	C₂×Dic₆₇	1	2	yes	—
184	C₄⋊C₈	—	134	2	no	—
185	#104	—	2	2	no	—
186	#104	—	2	2	no	—
187	C₂²⋊C₈	—	134	2	no	—
188	C₄×C₈	—	134	2	no	—
189	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
190	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
191	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
203	Q₁₆	—	2	1	no	—
204	Q₁₆	C₂×C₆₇⋊C₈	1	2	yes	—
205	SD₁₆	—	2	2	no	—
206	SD₁₆	—	2	2	no	—
207	D₈	—	2	1	no	—
208	D₈	C₂×C₆₇⋊C₈	1	2	yes	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	134	4	no	—
217	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
219	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
220	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
221	C₂×C₈	#11761612	1	1	yes	characteristic
222	C₂×C₈	#11761612	1	1	yes	characteristic
223	C₂×C₈	#11761613	1	1	yes	characteristic
224	C₂×C₈	#11761613	1	1	yes	characteristic
225	C₂×C₈	#11761616	1	1	yes	characteristic
226	C₂×C₈	#11761616	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
229	C₄²	—	2	1	no	—
230	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
232	C₁₆	—	268	2	no	—
233	C₂³	—	4	1	no	—
234	C₂³	#14523970	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	4	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
248	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
249	C₂×C₄	#14523964	1	1	yes	characteristic
250	C₂×C₄	#14523964	1	1	yes	characteristic
251	C₂×C₄	#14523970	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
253	C₈	—	4	2	no	—
254	C₈	—	268	4	no	—
255	C₈	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
256	C₈	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
257	C₈	#14523964	1	1	yes	characteristic
258	C₈	#14523964	1	1	yes	characteristic
259	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
260	C₈	C₂×#11761612	1	1	yes	characteristic
261	C₈	C₂×#11761613	1	1	yes	characteristic
262	C₈	C₂×#11761616	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	D₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
267	C₂²	D₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
268	C₂²	C₂×#14523970	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	D₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
273	C₄	D₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
274	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
275	C₄	C₂×#14523964	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₂×D₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#11761613
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×#443×C₆₇	C₂
socle	C₂×C₆₇	C₂×D₄⋊C₈
prime radical	C₂×#135×C₆₇	C₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×#443	D₆₇
hypocenter	C₆₇	#11602
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16863217	C₂	semidirect
2. #22558492	C₂	semidirect
3. #22558182	C₂	semidirect
8. #16863217	C₂²	semidirect
10. C₈×C₆₇⋊C₈	C₂²	semidirect
29. #135×C₆₇	C₈	semidirect
31. C₂×Q₁₆×C₆₇	C₈	semidirect
33. C₂×D₈×C₆₇	C₈	semidirect
145. C₆₇	#11602	semidirect
164. #135	C₆₇⋊C₈	semidirect
166. C₂×Q₁₆	C₆₇⋊C₈	semidirect
168. C₂×D₈	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22558492	2
#22558182	2
C₂×#443×C₆₇	2
#11602	67