Subgroups — #30748798

Order 17152.

342 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558972	C₂	1	1	yes	maximal, characteristic
2	#22558972	C₂	1	1	yes	maximal, characteristic
3	#22558553	C₂	1	2	yes	maximal
4	#22558079	C₂	1	2	yes	maximal
5	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863453	C₂²	1	2	yes	—
7	#16863453	C₂²	1	2	yes	—
8	#16863149	—	2	1	no	—
9	#16863149	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863122	—	2	4	no	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#446×C₆₇	—	2	1	no	—
15	#446×C₆₇	—	2	1	no	—
16	#446×C₆₇	—	2	1	no	—
17	#446×C₆₇	—	2	1	no	—
18	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#97×C₆₇	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
26	#14523954	—	2	2	no	—
27	#14523954	—	2	2	no	—
28	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₇	—	2	1	no	—
33	C₄×Q₈×C₆₇	—	2	1	no	—
34	C₄×Q₈×C₆₇	—	2	1	no	—
35	C₄×Q₈×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₄⋊C₈×C₆₇	—	2	1	no	—
43	C₄⋊C₈×C₆₇	—	2	1	no	—
44	#97×C₆₇	—	2	2	no	—
45	#97×C₆₇	—	2	2	no	—
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×Q₈×C₆₇	—	2	1	no	—
53	C₂×Q₈×C₆₇	—	2	1	no	—
54	C₂×Q₈×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₆₇	—	2	2	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	4	1	no	—
63	C₂×C₈×C₆₇	—	4	2	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₄⋊C₄×C₆₇	—	2	1	no	—
71	C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₇	#41	1	1	yes	characteristic
77	C₄²×C₆₇	#41	1	1	yes	characteristic
78	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	Q₈×C₆₇	—	4	2	no	—
80	Q₈×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₈×C₆₇	—	2	1	no	—
96	C₈×C₆₇	—	2	1	no	—
97	C₈×C₆₇	—	2	1	no	—
98	C₈×C₆₇	—	4	2	no	—
99	C₈×C₆₇	#101	1	2	yes	—
100	C₆₇⋊C₈	—	8	2	no	—
101	C₆₇⋊C₈	—	8	2	no	—
102	C₂²×C₆₇	—	2	1	no	—
103	C₂²×C₆₇	—	2	1	no	—
104	C₂²×C₆₇	#380	1	1	yes	characteristic
105	C₂²×C₆₇	#430	1	2	yes	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	4	2	no	—
112	C₄×C₆₇	—	4	2	no	—
113	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
114	C₄×C₆₇	#413	1	1	yes	characteristic
115	#8625	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	—	2	2	no	—
117	C₂×C₆₇	#1801	1	1	yes	characteristic
118	C₂×C₆₇	#1811	1	1	yes	characteristic
119	C₂×C₆₇	#1812	1	1	yes	characteristic
120	C₂×#446	D₆₇	1	1	yes	characteristic
121	#1568	—	67	1	no	—
122	#1568	—	67	1	no	—
123	#1467	—	67	2	no	—
124	#1290	—	67	2	no	—
125	C₆₇	#8625	1	1	yes	characteristic, complemented, Sylow
126	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
127	#446	—	2	1	no	—
128	#446	—	2	1	no	—
129	#446	—	2	1	no	—
130	#446	—	2	1	no	—
131	#432	—	67	2	no	—
132	#432	—	67	2	no	—
133	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
134	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
135	C₂×#97	Dic₆₇	1	2	yes	—
136	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
137	#335	—	134	1	no	—
138	#335	—	134	1	no	—
139	#334	—	134	1	no	—
140	#334	—	134	1	no	—
141	#324	—	134	4	no	—
142	C₂²×Q₈	Dic₁₃₄	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
144	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
145	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	C₄×Q₈	—	2	1	no	—
150	C₂×C₄⋊C₄	Dic₁₃₄	1	1	yes	characteristic
151	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
152	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
153	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	134	2	no	—
159	C₄⋊C₈	—	134	2	no	—
160	C₂²⋊C₈	—	134	2	no	—
161	C₂²⋊C₈	—	134	2	no	—
162	#97	—	2	2	no	—
163	#97	—	2	2	no	—
164	#97	—	134	2	no	—
165	#97	—	134	2	no	—
166	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
174	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
177	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	268	2	no	—
183	C₂×C₈	—	268	2	no	—
184	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
196	C₄²	#11761628	1	1	yes	characteristic
197	C₄²	#11761628	1	1	yes	characteristic
198	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
210	C₂×C₄	#14524002	1	1	yes	characteristic
211	C₂×C₄	#14524017	1	1	yes	characteristic
212	C₂×C₄	#14524017	1	1	yes	characteristic
213	C₂×C₄	#14524022	1	1	yes	characteristic
214	C₂×C₄	#14524022	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	4	2	no	—
219	C₈	—	536	2	no	—
220	C₈	—	536	2	no	—
221	C₈	#14523962	1	2	yes	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	#16863256	1	1	yes	characteristic
225	C₂²	#16863445	1	2	yes	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×#14523962	1	1	yes	characteristic
234	C₄	#16863380	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	#22560205	1	1	yes	characteristic
237	C₂	#22560251	1	1	yes	characteristic
238	C₂	#22560255	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	#8625
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
125. C₆₇	#8625	semidirect

Maximal subgroups

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

subgroup	index
#22558972	2
#22558553	2
#22558079	2
C₂×#446×C₆₇	2
#8625	67