Subgroups — #30754376

Order 17152.

382 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	#22560925	C₂	1	2	yes	maximal
2	C₄×#14523977	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863117	C₂	1	1	yes	maximal, characteristic
4	#22558048	C₂	1	2	yes	maximal
5	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	C₄	1	1	yes	characteristic
7	C₂×#14523977	C₄	1	1	yes	characteristic
8	C₂×Dic₆₇⋊C₈	—	2	2	no	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	#16863159	C₄	1	2	yes	—
13	#16863150	—	2	2	no	—
14	#16863117	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#334×C₆₇	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₆₇	—	2	2	no	—
23	#14523977	—	2	2	no	—
24	#14523977	C₈	1	4	yes	complemented
25	Dic₆₇⋊C₈	—	4	4	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₄×C₆₇⋊C₈	—	2	1	no	—
28	C₄×C₆₇⋊C₈	—	2	1	no	—
29	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
30	C₆₇⋊#95	—	2	2	no	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	—	2	2	no	—
36	C₄⋊C₈×C₆₇	—	2	2	no	—
37	C₄×C₈×C₆₇	D₄	1	2	yes	—
38	C₄×C₈×C₆₇	Q₈	1	2	yes	—
39	C₂²×Dic₆₇	—	2	2	no	—
40	C₂²×Dic₆₇	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	4	1	no	—
46	C₂×C₆₇⋊C₈	—	4	1	no	—
47	C₄⋊Dic₆₇	—	2	2	no	—
48	C₄⋊Dic₆₇	—	2	2	no	—
49	C₄⋊Dic₆₇	—	2	2	no	—
50	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
51	C₄×Dic₆₇	—	4	2	no	—
52	C₄×Dic₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	1	no	—
57	C₂×C₈×C₆₇	—	2	1	no	—
58	C₂×C₈×C₆₇	—	4	2	no	—
59	C₂×C₈×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	—	4	2	no	—
61	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₆₇	#41	1	1	yes	characteristic
69	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	2	1	no	—
72	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
82	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₆₇	#137	1	2	yes	—
84	C₂×Dic₆₇	—	2	4	no	—
85	C₂×Dic₆₇	—	4	2	no	—
86	C₂×Dic₆₇	—	4	2	no	—
87	C₂×Dic₆₇	—	4	2	no	—
88	C₂×Dic₆₇	—	4	2	no	—
89	C₂×Dic₆₇	—	4	2	no	—
90	C₈×C₆₇	—	2	2	no	—
91	C₈×C₆₇	—	8	2	no	—
92	C₈×C₆₇	C₄⋊C₈	1	4	yes	complemented
93	C₆₇⋊C₈	—	4	2	no	—
94	C₆₇⋊C₈	—	4	2	no	—
95	C₂²×C₆₇	#389	1	1	yes	characteristic
96	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
97	C₂²×C₆₇	#436	1	1	yes	characteristic
98	C₂²×C₆₇	#441	1	1	yes	characteristic
99	C₂²×C₆₇	#442	1	1	yes	characteristic
100	C₂²×C₆₇	#444	1	1	yes	characteristic
101	C₂²×C₆₇	#444	1	1	yes	characteristic
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄×C₆₇	#424	1	1	yes	characteristic
106	C₄×C₆₇	C₈×Q₈	1	1	yes	characteristic
107	C₄×C₆₇	#446	1	1	yes	characteristic
108	C₄×C₆₇	#468	1	2	yes	—
109	C₄×C₆₇	#501	1	2	yes	—
110	Dic₆₇	—	4	4	no	—
111	Dic₆₇	—	8	2	no	—
112	#9595	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	#1579	1	1	yes	characteristic
114	C₂×C₆₇	#1580	1	1	yes	characteristic
115	C₂×C₆₇	#1593	1	1	yes	characteristic
116	C₂×C₆₇	#1594	1	1	yes	characteristic
117	C₂×C₆₇	#1939	1	1	yes	characteristic
118	C₂×C₆₇	#1943	1	1	yes	characteristic
119	C₂×C₆₇	#1946	1	1	yes	characteristic
120	#1939	—	67	2	no	—
121	C₄×#107	—	67	1	no	—
122	C₂×#334	D₆₇	1	1	yes	characteristic
123	C₂×#322	—	67	1	no	—
124	#1279	—	67	2	no	—
125	C₆₇	#9595	1	1	yes	characteristic, complemented, Sylow
126	C₂×#107	—	67	1	no	—
127	C₂×#107	—	67	1	no	—
128	C₂×C₄⋊C₈	—	134	2	no	—
129	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
130	C₂×C₄×C₈	—	67	1	no	—
131	C₂×C₄×C₈	—	67	1	no	—
132	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	67	2	no	—
134	#340	—	67	2	no	—
135	#336	—	134	2	no	—
136	#334	—	2	2	no	—
137	#322	—	134	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	67	2	no	—
140	C₂²×C₈	—	134	1	no	—
141	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
142	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	67	2	no	—
144	C₂×C₄⋊C₄	—	67	2	no	—
145	C₂×C₄²	—	134	2	no	—
146	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
147	#107	—	67	4	no	—
148	#107	—	134	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	268	4	no	—
152	C₄×C₈	—	67	2	no	—
153	C₄×C₈	—	134	1	no	—
154	C₄×C₈	—	134	1	no	—
155	C₄×C₈	—	134	1	no	—
156	C₄×C₈	Dic₁₃₄	1	2	yes	—
157	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
158	#95	—	134	2	no	—
159	C₂²×C₄	—	134	2	no	—
160	C₂²×C₄	—	134	2	no	—
161	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
162	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
163	C₂²×C₄	#11761629	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	268	1	no	—
170	C₂×C₈	—	268	1	no	—
171	C₂×C₈	—	134	2	no	—
172	C₂×C₈	—	134	2	no	—
173	C₂×C₈	—	134	2	no	—
174	C₂×C₈	—	134	2	no	—
175	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
176	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
177	C₂×C₈	C₈×D₆₇	1	2	yes	—
178	C₂×C₈	#11761611	1	2	yes	—
179	C₄⋊C₄	—	67	4	no	—
180	C₄⋊C₄	—	134	2	no	—
181	C₄⋊C₄	—	134	2	no	—
182	C₄⋊C₄	—	134	2	no	—
183	C₄²	—	268	2	no	—
184	C₄²	—	268	2	no	—
185	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
186	C₄²	Q₈×D₆₇	1	1	yes	characteristic
187	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
188	C₄²	#11761627	1	1	yes	characteristic
189	C₂³	#14524005	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	268	2	no	—
193	C₂×C₄	—	268	2	no	—
194	C₂×C₄	—	268	2	no	—
195	C₂×C₄	—	268	2	no	—
196	C₂×C₄	—	268	2	no	—
197	C₂×C₄	—	134	4	no	—
198	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
199	C₂×C₄	#14524008	1	1	yes	characteristic
200	C₂×C₄	#14524014	1	1	yes	characteristic
201	C₂×C₄	#14524019	1	1	yes	characteristic
202	C₂×C₄	#14524034	1	1	yes	characteristic
203	C₂×C₄	#14524042	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
205	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
206	C₂×C₄	#14524051	1	1	yes	characteristic
207	C₂×C₄	#14524051	1	1	yes	characteristic
208	C₂×C₄	#14524070	1	2	yes	—
209	C₂×C₄	#14524072	1	2	yes	—
210	C₈	—	2	2	no	—
211	C₈	—	8	2	no	—
212	C₈	—	268	2	no	—
213	C₈	—	268	2	no	—
214	C₈	Dic₆₇⋊C₈	1	4	yes	complemented
215	C₂²	#16863292	1	1	yes	characteristic
216	C₂²	#16863461	1	1	yes	characteristic
217	C₂²	#16863473	1	1	yes	characteristic
218	C₂²	#16863502	1	1	yes	characteristic
219	C₂²	#16863507	1	1	yes	characteristic
220	C₂²	#16863517	1	1	yes	characteristic
221	C₂²	#16863517	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	268	4	no	—
225	C₄	—	536	2	no	—
226	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
227	C₄	#16863424	1	1	yes	characteristic
228	C₄	#16863520	1	1	yes	characteristic
229	C₄	#16863523	1	1	yes	characteristic
230	C₄	#16863639	1	2	yes	—
231	C₄	#16863803	1	2	yes	—
232	C₂	#22559027	1	1	yes	characteristic
233	C₂	#22559032	1	1	yes	characteristic
234	C₂	#22559117	1	1	yes	characteristic
235	C₂	#22559122	1	1	yes	characteristic
236	C₂	#22560925	1	1	yes	characteristic
237	C₂	#22560945	1	1	yes	characteristic
238	C₂	#22560960	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#9595
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
24. #14523977	C₈	semidirect
92. C₈×C₆₇	C₄⋊C₈	semidirect
125. C₆₇	#9595	semidirect
214. C₈	Dic₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22560925	2
C₄×#14523977	2
C₂×#16863117	2
#22558048	2
C₂×#334×C₆₇	2
#9595	67