Subgroups — #30749080

Order 17152.

362 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560381	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
3	#22558899	C₂	1	2	yes	maximal
4	#22558083	C₂	1	2	yes	maximal
5	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	—	2	1	no	—
7	C₂×#14523979	—	2	1	no	—
8	#16863420	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863145	—	2	2	no	—
13	#16863142	—	2	2	no	—
14	#533×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	—	2	1	no	—
16	#446×C₆₇	C₄	1	2	yes	—
17	C₈×Q₈×C₆₇	—	2	1	no	—
18	C₈×Q₈×C₆₇	C₄	1	2	yes	—
19	#424×C₆₇	—	2	1	no	—
20	#423×C₆₇	C₂²	1	1	yes	characteristic
21	#405×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	#14523979	—	4	2	no	—
26	#14523979	—	4	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₂×M₁₆×C₆₇	—	2	1	no	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
36	#125×C₆₇	—	2	1	no	—
37	C₂²⋊Q₈×C₆₇	—	2	1	no	—
38	C₂²⋊Q₈×C₆₇	Q₈	1	2	yes	—
39	C₄×Q₈×C₆₇	—	2	1	no	—
40	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
41	#117×C₆₇	—	2	1	no	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	—	2	2	no	—
45	C₄⋊C₈×C₆₇	—	2	1	no	—
46	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
47	#97×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	—	2	2	no	—
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
53	#11761612	—	4	2	no	—
54	#11761612	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	8	2	no	—
58	C₂×Q₈×C₆₇	—	2	1	no	—
59	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
63	M₁₆×C₆₇	—	4	1	no	—
64	M₁₆×C₆₇	—	4	1	no	—
65	C₂×C₈×C₆₇	—	2	1	no	—
66	C₂×C₈×C₆₇	—	2	1	no	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
73	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₄⋊C₄×C₆₇	—	2	1	no	—
77	C₄⋊C₄×C₆₇	—	2	1	no	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂²⋊C₄×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₇	#41	1	1	yes	characteristic
86	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
87	Q₈×C₆₇	—	4	2	no	—
88	Q₈×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	4	1	no	—
93	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
103	C₈×C₆₇	—	2	1	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	4	2	no	—
106	C₈×C₆₇	#103	1	2	yes	—
107	C₈×C₆₇	#103	1	2	yes	—
108	C₆₇⋊C₈	—	8	2	no	—
109	C₆₇⋊C₈	—	8	2	no	—
110	C₂²×C₆₇	—	2	1	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	#380	1	1	yes	characteristic
113	C₂²×C₆₇	#475	1	1	yes	characteristic
114	C₂²×C₆₇	#477	1	1	yes	characteristic
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	4	2	no	—
119	C₄×C₆₇	—	4	2	no	—
120	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
121	C₄×C₆₇	#417	1	1	yes	characteristic
122	C₄×C₆₇	#460	1	1	yes	characteristic
123	C₄×C₆₇	#462	1	1	yes	characteristic
124	C₄×C₆₇	#471	1	2	yes	—
125	#8683	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	—	2	1	no	—
127	C₂×C₆₇	—	2	1	no	—
128	C₂×C₆₇	#1803	1	1	yes	characteristic
129	C₂×C₆₇	#1813	1	1	yes	characteristic
130	C₂×C₆₇	#1815	1	1	yes	characteristic
131	#2960	D₆₇	1	1	yes	characteristic
132	#1848	—	67	1	no	—
133	C₂×#334	—	67	1	no	—
134	#1550	—	67	2	no	—
135	#1292	—	67	2	no	—
136	C₆₇	#8683	1	1	yes	characteristic, complemented, Sylow
137	#533	C₂×D₆₇	1	1	yes	characteristic
138	#446	—	2	1	no	—
139	#446	Dic₆₇	1	2	yes	—
140	C₈×Q₈	—	2	1	no	—
141	C₈×Q₈	Dic₆₇	1	2	yes	—
142	C₂×#108	—	134	1	no	—
143	C₂×#108	—	134	1	no	—
144	#424	—	2	1	no	—
145	#423	—	67	2	no	—
146	#423	C₂×D₆₇	1	1	yes	characteristic
147	C₂×C₄⋊C₈	—	67	2	no	—
148	#405	—	2	1	no	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	#334	—	134	1	no	—
151	#334	—	134	1	no	—
152	#333	—	134	2	no	—
153	#332	—	134	2	no	—
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	134	2	no	—
156	C₂²×C₈	—	134	2	no	—
157	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
160	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
161	#125	—	2	1	no	—
162	C₂²⋊Q₈	—	2	1	no	—
163	C₂²⋊Q₈	Dic₁₃₄	1	2	yes	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
166	#117	—	2	1	no	—
167	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
168	#108	—	268	2	no	—
169	#108	—	268	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	134	2	no	—
174	C₄⋊C₈	—	134	2	no	—
175	C₄⋊C₈	—	134	2	no	—
176	C₄⋊C₈	—	134	2	no	—
177	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
178	#97	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
181	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
186	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
188	C₂²×C₄	#11761628	1	1	yes	characteristic
189	M₁₆	—	4	1	no	—
190	M₁₆	—	4	1	no	—
191	M₁₆	—	268	2	no	—
192	M₁₆	—	268	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	268	2	no	—
197	C₂×C₈	—	268	2	no	—
198	C₂×C₈	—	536	2	no	—
199	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
203	C₂×C₈	#11761615	1	2	yes	—
204	C₂×C₈	#11761618	1	2	yes	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
216	C₄²	#11761628	1	1	yes	characteristic
217	C₂³	#14524022	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
226	C₂×C₄	#14524002	1	1	yes	characteristic
227	C₂×C₄	#14524017	1	1	yes	characteristic
228	C₂×C₄	#14524017	1	1	yes	characteristic
229	C₂×C₄	#14524022	1	1	yes	characteristic
230	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
231	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
232	C₂×C₄	#14524068	1	1	yes	characteristic
233	C₂×C₄	#14524073	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	4	2	no	—
237	C₈	—	536	2	no	—
238	C₈	—	536	2	no	—
239	C₈	#14523967	1	2	yes	—
240	C₈	#14523967	1	2	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#16863256	1	1	yes	characteristic
244	C₂²	#16863681	1	1	yes	characteristic
245	C₂²	#16863691	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	C₂×#14523967	1	1	yes	characteristic
252	C₄	#16863397	1	1	yes	characteristic
253	C₄	#16863596	1	1	yes	characteristic
254	C₄	#16863606	1	1	yes	characteristic
255	C₄	#16863659	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#22560214	1	1	yes	characteristic
259	C₂	#22560259	1	1	yes	characteristic
260	C₂	#22560268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#8683
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
136. C₆₇	#8683	semidirect

Maximal subgroups

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

subgroup	index
#22560381	2
C₂×#16863147	2
#22558899	2
#22558083	2
C₆₇×#2960	2
#8683	67