Subgroups — #30749081

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	#22569143	C₂	1	1	yes	maximal, characteristic
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22558897	C₂	1	2	yes	maximal
4	#22558084	C₂	1	2	yes	maximal
5	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	1	yes	characteristic
7	#16863526	—	2	1	no	—
8	#16863526	C₄	1	2	yes	—
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₈×Dic₁₃₄	C₄	1	2	yes	—
11	C₂×#14523979	—	2	1	no	—
12	C₂×#14523979	—	2	1	no	—
13	#16863422	—	2	1	no	—
14	#16863420	C₂²	1	2	yes	—
15	#16863420	C₂²	1	1	yes	characteristic
16	#16863351	—	2	1	no	—
17	#16863144	—	2	2	no	—
18	#16863143	—	2	2	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#334×C₆₇	—	2	1	no	—
22	#334×C₆₇	—	2	1	no	—
23	C₂×#11761612	—	2	1	no	—
24	C₂×#11761612	—	2	2	no	—
25	#14524043	D₄	1	1	yes	characteristic
26	#14524043	D₄	1	1	yes	characteristic
27	#14524035	—	2	1	no	—
28	#14524022	—	2	1	no	—
29	#14524022	Q₈	1	2	yes	—
30	C₄×Dic₁₃₄	—	2	1	no	—
31	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
32	#14524002	—	2	1	no	—
33	#14523979	—	4	2	no	—
34	#14523979	—	4	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
39	Dic₆₇⋊C₈	—	2	2	no	—
40	Dic₆₇⋊C₈	—	2	2	no	—
41	#14523953	—	2	2	no	—
42	C₈×Dic₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×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	C₂×Dic₁₃₄	—	2	1	no	—
54	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
55	#11761612	—	4	1	no	—
56	#11761612	—	4	1	no	—
57	#11761612	—	4	2	no	—
58	#11761612	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	2	1	no	—
62	C₄⋊Dic₆₇	—	2	1	no	—
63	C₄⋊Dic₆₇	—	2	1	no	—
64	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
65	Dic₆₇⋊C₄	—	2	2	no	—
66	Dic₆₇⋊C₄	—	2	2	no	—
67	C₂²⋊Dic₆₇	—	2	2	no	—
68	C₄×Dic₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	2	1	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	8	2	no	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
81	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
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	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
99	C₂×Dic₆₇	—	2	2	no	—
100	C₂×Dic₆₇	—	2	2	no	—
101	Dic₁₃₄	—	4	2	no	—
102	Dic₁₃₄	—	4	2	no	—
103	C₈×C₆₇	—	2	1	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	8	2	no	—
106	C₈×C₆₇	#103	1	2	yes	—
107	C₈×C₆₇	#103	1	2	yes	—
108	C₆₇⋊C₈	—	4	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	C₂×#103	1	1	yes	characteristic
119	C₄×C₆₇	#417	1	1	yes	characteristic
120	C₄×C₆₇	#460	1	1	yes	characteristic
121	C₄×C₆₇	#462	1	1	yes	characteristic
122	C₄×C₆₇	#471	1	2	yes	—
123	Dic₆₇	—	4	2	no	—
124	Dic₆₇	—	4	2	no	—
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	—	67	1	no	—
132	#1848	—	67	1	no	—
133	C₂×#334	D₆₇	1	1	yes	characteristic
134	#1550	—	67	2	no	—
135	#1292	—	67	2	no	—
136	C₆₇	#8683	1	1	yes	characteristic, complemented, Sylow
137	#533	—	67	1	no	—
138	#446	—	67	2	no	—
139	#446	—	134	1	no	—
140	C₈×Q₈	—	67	2	no	—
141	C₈×Q₈	—	134	1	no	—
142	C₂×#108	—	134	1	no	—
143	C₂×#108	—	134	1	no	—
144	#424	—	134	1	no	—
145	#423	—	67	1	no	—
146	#423	—	67	2	no	—
147	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
148	#405	—	134	1	no	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	#334	—	2	1	no	—
151	#334	—	2	1	no	—
152	#333	—	134	2	no	—
153	#332	—	134	2	no	—
154	C₂×M₁₆	—	134	2	no	—
155	C₂×M₁₆	—	134	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
158	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
159	C₄⋊Q₈	—	67	1	no	—
160	C₄⋊Q₈	—	67	1	no	—
161	#125	—	134	1	no	—
162	C₂²⋊Q₈	—	67	2	no	—
163	C₂²⋊Q₈	—	134	1	no	—
164	C₄×Q₈	—	67	2	no	—
165	C₄×Q₈	—	134	1	no	—
166	#117	—	134	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	—	67	2	no	—
173	C₄⋊C₈	—	134	2	no	—
174	C₄⋊C₈	—	134	2	no	—
175	C₄⋊C₈	—	134	2	no	—
176	C₄⋊C₈	—	134	1	no	—
177	C₄⋊C₈	—	134	2	no	—
178	#97	—	134	2	no	—
179	C₄×C₈	—	134	2	no	—
180	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
181	C₄×C₈	C₄×D₆₇	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	—	67	2	no	—
185	C₂×Q₈	—	134	1	no	—
186	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
187	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
188	C₂²×C₄	#11761627	1	1	yes	characteristic
189	M₁₆	—	268	1	no	—
190	M₁₆	—	268	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	—	4	2	no	—
196	C₂×C₈	—	8	2	no	—
197	C₂×C₈	—	268	2	no	—
198	C₂×C₈	—	134	2	no	—
199	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
200	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
201	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
202	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	#11761617	1	2	yes	—
204	C₂×C₈	Dic₂₆₈	1	2	yes	—
205	C₄⋊C₄	—	67	2	no	—
206	C₄⋊C₄	—	134	2	no	—
207	C₄⋊C₄	—	134	1	no	—
208	C₄⋊C₄	—	134	1	no	—
209	C₄⋊C₄	—	134	1	no	—
210	C₄⋊C₄	—	134	2	no	—
211	C₂²⋊C₄	—	134	2	no	—
212	C₄²	—	134	2	no	—
213	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₄²	D₄×D₆₇	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
216	C₄²	#11761629	1	1	yes	characteristic
217	C₂³	#14524023	1	1	yes	characteristic
218	Q₈	—	268	2	no	—
219	Q₈	—	268	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	134	2	no	—
223	C₂×C₄	—	134	2	no	—
224	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
225	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
226	C₂×C₄	#14524001	1	1	yes	characteristic
227	C₂×C₄	#14524014	1	1	yes	characteristic
228	C₂×C₄	#14524018	1	1	yes	characteristic
229	C₂×C₄	#14524019	1	1	yes	characteristic
230	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
231	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
232	C₂×C₄	#14524071	1	1	yes	characteristic
233	C₂×C₄	#14524076	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	—	268	2	no	—
238	C₈	—	536	2	no	—
239	C₈	#14523968	1	2	yes	—
240	C₈	#14523968	1	2	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#16863253	1	1	yes	characteristic
244	C₂²	#16863682	1	1	yes	characteristic
245	C₂²	#16863692	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	268	2	no	—
250	C₄	—	268	2	no	—
251	C₄	C₂×#14523968	1	1	yes	characteristic
252	C₄	#16863398	1	1	yes	characteristic
253	C₄	#16863597	1	1	yes	characteristic
254	C₄	#16863607	1	1	yes	characteristic
255	C₄	#16863656	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#22560215	1	1	yes	characteristic
259	C₂	#22560256	1	1	yes	characteristic
260	C₂	#22560269	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863253
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#8683
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#334

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
#22569143	2
#22560381	2
#22558897	2
#22558084	2
C₂×#334×C₆₇	2
#8683	67