Subgroups — #30748772

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560205	C₂	1	2	yes	maximal
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558965	C₂	1	1	yes	maximal, characteristic
4	#22558084	C₂	1	2	yes	maximal
5	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	#16863420	C₂²	1	2	yes	—
8	C₂×#14523962	—	2	2	no	—
9	#16863165	—	2	2	no	—
10	#16863149	—	2	1	no	—
11	#16863149	—	2	1	no	—
12	#16863147	—	2	1	no	—
13	#16863147	—	2	1	no	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	—	2	1	no	—
16	#446×C₆₇	—	2	1	no	—
17	#446×C₆₇	—	2	1	no	—
18	#446×C₆₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₇	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	#14523962	—	4	4	no	—
30	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₆₇	—	2	1	no	—
35	C₄×Q₈×C₆₇	—	2	1	no	—
36	C₄×Q₈×C₆₇	—	2	1	no	—
37	C₄×Q₈×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	—	2	1	no	—
45	C₄⋊C₈×C₆₇	—	2	1	no	—
46	#97×C₆₇	—	2	2	no	—
47	#97×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	D₄	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	D₄	1	1	yes	characteristic
52	#11761612	—	4	2	no	—
53	#11761612	—	4	2	no	—
54	#11761612	—	4	2	no	—
55	#11761612	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×Q₈×C₆₇	—	2	1	no	—
59	C₂×Q₈×C₆₇	—	2	1	no	—
60	C₂×Q₈×C₆₇	—	2	2	no	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	4	1	no	—
70	C₂×C₈×C₆₇	—	4	2	no	—
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₄⋊C₄×C₆₇	—	2	1	no	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	—	2	2	no	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	#41	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	Q₈×C₆₇	—	4	2	no	—
87	Q₈×C₆₇	—	4	1	no	—
88	Q₈×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	4	no	—
94	C₂×C₄×C₆₇	—	4	1	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₆₇	#137	1	2	yes	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	—	4	2	no	—
108	C₈×C₆₇	#101	1	2	yes	complemented
109	C₆₇⋊C₈	—	8	2	no	—
110	C₆₇⋊C₈	—	8	2	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	#380	1	1	yes	characteristic
114	C₂²×C₆₇	#428	1	2	yes	—
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×#101	1	1	yes	characteristic
121	C₄×C₆₇	#413	1	1	yes	characteristic
122	C₄×C₆₇	#468	1	2	yes	—
123	C₄×C₆₇	#469	1	1	yes	characteristic
124	C₄×C₆₇	#470	1	1	yes	characteristic
125	#8619	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	—	2	2	no	—
127	C₂×C₆₇	#1801	1	1	yes	characteristic
128	C₂×C₆₇	#1809	1	1	yes	characteristic
129	C₂×C₆₇	#1813	1	1	yes	characteristic
130	C₂×#446	D₆₇	1	1	yes	characteristic
131	#1801	—	67	2	no	—
132	#1567	—	67	1	no	—
133	#1566	—	67	1	no	—
134	#1292	—	67	2	no	—
135	C₆₇	#8619	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
137	#446	—	2	1	no	—
138	#446	—	2	1	no	—
139	#446	—	2	1	no	—
140	#446	—	2	1	no	—
141	#423	—	67	2	no	—
142	#423	—	67	2	no	—
143	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
144	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
145	C₂×#101	—	134	2	no	—
146	C₂×#97	Dic₆₇	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
148	#343	—	134	2	no	—
149	#335	—	134	1	no	—
150	#335	—	134	1	no	—
151	#334	—	134	1	no	—
152	#334	—	134	1	no	—
153	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
154	C₂×M₁₆	—	134	2	no	—
155	C₂×M₁₆	—	134	2	no	—
156	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
158	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	—	2	1	no	—
163	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
164	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	134	2	no	—
172	C₄⋊C₈	—	134	2	no	—
173	C₄⋊C₈	—	134	2	no	—
174	C₄⋊C₈	—	134	2	no	—
175	#101	—	268	4	no	—
176	#97	—	2	2	no	—
177	#97	—	2	2	no	—
178	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
191	M₁₆	—	268	2	no	—
192	M₁₆	—	268	2	no	—
193	M₁₆	—	268	2	no	—
194	M₁₆	—	268	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	268	2	no	—
200	C₂×C₈	—	268	2	no	—
201	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₄²	#11761628	1	1	yes	characteristic
214	C₄²	#11761628	1	1	yes	characteristic
215	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
216	Q₈	—	4	2	no	—
217	Q₈	—	4	1	no	—
218	Q₈	—	4	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
227	C₂×C₄	#14524002	1	1	yes	characteristic
228	C₂×C₄	#14524017	1	1	yes	characteristic
229	C₂×C₄	#14524017	1	1	yes	characteristic
230	C₂×C₄	#14524022	1	1	yes	characteristic
231	C₂×C₄	#14524022	1	1	yes	characteristic
232	C₂×C₄	#14524071	1	2	yes	—
233	C₂×C₄	#14524076	1	2	yes	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	—	536	2	no	—
239	C₈	—	536	2	no	—
240	C₈	#14523962	1	2	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#16863256	1	1	yes	characteristic
244	C₂²	#16863438	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂×#14523962	1	1	yes	characteristic
251	C₄	#16863380	1	1	yes	characteristic
252	C₄	#16863642	1	2	yes	—
253	C₄	#16863649	1	1	yes	characteristic
254	C₄	#16863654	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#22560205	1	1	yes	characteristic
257	C₂	#22560242	1	1	yes	characteristic
258	C₂	#22560259	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#446×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#446	D₆₇
hypocenter	C₆₇	#8619
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
108. C₈×C₆₇	#101	semidirect
135. C₆₇	#8619	semidirect
240. C₈	#14523962	semidirect

Maximal subgroups

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

subgroup	index
#22560205	2
#22558968	2
#22558965	2
#22558084	2
C₂×#446×C₆₇	2
#8619	67