Subgroups — #30755666

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561171	C₂	1	2	yes	maximal
2	#22561012	C₂	1	2	yes	maximal
3	#22560366	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863117	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523975	—	2	2	no	—
7	C₂×#14523967	—	2	4	no	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#16863266	C₂²	1	2	yes	—
11	#16863158	—	2	4	no	—
12	#16863117	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	#500×C₆₇	—	2	2	no	—
15	#500×C₆₇	C₄	1	4	yes	complemented
16	C₂×#107×C₆₇	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
19	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	#14523975	—	4	4	no	—
24	#14523967	—	4	8	no	—
25	C₄×C₆₇⋊C₈	—	2	1	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₄×C₆₇⋊C₈	—	2	1	no	—
28	C₄×C₆₇⋊C₈	—	2	1	no	—
29	C₂²×Q₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	—	2	2	no	—
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#107×C₆₇	—	2	8	no	—
39	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₇	D₄	1	2	yes	—
41	C₂²×Dic₆₇	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	2	4	no	—
45	C₂×C₆₇⋊C₈	—	2	4	no	—
46	C₄⋊Dic₆₇	—	4	4	no	—
47	C₄⋊Dic₆₇	—	4	4	no	—
48	C₄⋊Dic₆₇	—	4	4	no	—
49	C₂×Q₈×C₆₇	—	2	4	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	4	no	—
52	C₂×Q₈×C₆₇	—	4	4	no	—
53	C₂²×C₄×C₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
58	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
59	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
60	C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₇	#41	1	1	yes	characteristic
67	C₄²×C₆₇	#41	1	1	yes	characteristic
68	C₂³×C₆₇	#127	1	1	yes	characteristic
69	Q₈×C₆₇	—	4	8	no	—
70	Q₈×C₆₇	—	8	4	no	—
71	C₂×C₄×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	—	8	2	no	—
73	C₂×C₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×C₄×C₆₇	#137	1	4	yes	—
84	C₂×Dic₆₇	—	8	2	no	—
85	C₂×Dic₆₇	—	8	2	no	—
86	C₂×Dic₆₇	—	8	2	no	—
87	C₈×C₆₇	#103	1	8	yes	complemented
88	C₆₇⋊C₈	—	4	2	no	—
89	C₆₇⋊C₈	—	4	2	no	—
90	C₂²×C₆₇	#390	1	1	yes	characteristic
91	C₂²×C₆₇	#492	1	1	yes	characteristic
92	C₂²×C₆₇	#494	1	1	yes	characteristic
93	C₂²×C₆₇	#496	1	2	yes	—
94	C₂²×C₆₇	#497	1	2	yes	—
95	C₄×C₆₇	—	8	4	no	—
96	C₄×C₆₇	C₂×#103	1	2	yes	—
97	C₄×C₆₇	#441	1	2	yes	—
98	C₄×C₆₇	#442	1	2	yes	—
99	C₄×C₆₇	#469	1	2	yes	—
100	C₄×C₆₇	#470	1	2	yes	—
101	C₄×C₆₇	#487	1	2	yes	—
102	Dic₆₇	—	16	2	no	—
103	#9836	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	#1714	1	2	yes	—
105	C₂×C₆₇	#1723	1	2	yes	—
106	C₂×C₆₇	#1956	1	1	yes	characteristic
107	C₂×C₆₇	#1971	1	1	yes	characteristic
108	C₂×C₆₇	#1972	1	1	yes	characteristic
109	C₂×#500	D₆₇	1	1	yes	characteristic
110	#1990	—	67	2	no	—
111	#1956	—	67	2	no	—
112	#1844	—	67	1	no	—
113	C₂×#322	—	67	1	no	—
114	C₆₇	#9836	1	1	yes	characteristic, complemented, Sylow
115	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
116	#500	—	2	2	no	—
117	#500	Dic₆₇	1	4	yes	complemented
118	C₂×#107	—	2	2	no	—
119	C₂×#106	—	134	2	no	—
120	C₂×#103	—	134	4	no	—
121	C₂×C₄×C₈	—	67	1	no	—
122	C₂×C₄×C₈	—	67	1	no	—
123	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
124	#384	—	67	2	no	—
125	#340	—	134	4	no	—
126	#322	—	134	2	no	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×C₈	—	67	2	no	—
129	C₂²×C₈	—	67	2	no	—
130	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	134	2	no	—
138	C₂×C₄⋊C₄	—	134	2	no	—
139	C₂×C₄⋊C₄	—	134	2	no	—
140	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
141	#107	—	2	8	no	—
142	#106	—	268	4	no	—
143	#103	—	268	8	no	—
144	C₄×C₈	—	134	1	no	—
145	C₄×C₈	—	134	1	no	—
146	C₄×C₈	—	134	1	no	—
147	C₄×C₈	—	134	1	no	—
148	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
149	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	268	2	no	—
156	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂×C₈	—	134	2	no	—
160	C₂×C₈	—	134	2	no	—
161	C₂×C₈	—	134	4	no	—
162	C₂×C₈	—	134	4	no	—
163	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
164	C₂×C₈	#11761615	1	4	yes	—
165	C₂×C₈	#11761618	1	4	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	268	4	no	—
171	C₄⋊C₄	—	268	4	no	—
172	C₄⋊C₄	—	268	4	no	—
173	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
175	C₄²	#11761627	1	1	yes	characteristic
176	C₄²	#11761627	1	1	yes	characteristic
177	C₂³	#14524039	1	1	yes	characteristic
178	Q₈	—	4	8	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	536	2	no	—
184	C₂×C₄	—	536	2	no	—
185	C₂×C₄	—	536	2	no	—
186	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
188	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
189	C₂×C₄	#14524014	1	1	yes	characteristic
190	C₂×C₄	#14524014	1	1	yes	characteristic
191	C₂×C₄	#14524030	1	1	yes	characteristic
192	C₂×C₄	C₂×#11761615	1	2	yes	—
193	C₂×C₄	C₂×#11761618	1	2	yes	—
194	C₂×C₄	#14524067	1	4	yes	—
195	C₂×C₄	#14524075	1	4	yes	—
196	C₈	—	268	2	no	—
197	C₈	—	268	2	no	—
198	C₈	#14523967	1	8	yes	complemented
199	C₂²	#16863299	1	1	yes	characteristic
200	C₂²	#16863770	1	1	yes	characteristic
201	C₂²	#16863775	1	1	yes	characteristic
202	C₂²	#16863784	1	2	yes	—
203	C₂²	#16863789	1	2	yes	—
204	C₄	—	8	4	no	—
205	C₄	—	1072	2	no	—
206	C₄	C₂×#14523967	1	2	yes	—
207	C₄	#16863503	1	2	yes	—
208	C₄	#16863508	1	2	yes	—
209	C₄	#16863646	1	2	yes	—
210	C₄	#16863651	1	2	yes	—
211	C₄	#16863746	1	2	yes	—
212	C₂	#22559832	1	2	yes	—
213	C₂	#22559887	1	2	yes	—
214	C₂	#22561012	1	1	yes	characteristic
215	C₂	#22561081	1	1	yes	characteristic
216	C₂	#22561086	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#9836
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
15. #500×C₆₇	C₄	semidirect
87. C₈×C₆₇	#103	semidirect
114. C₆₇	#9836	semidirect
117. #500	Dic₆₇	semidirect
198. C₈	#14523967	semidirect

Maximal subgroups

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

subgroup	index
#22561171	2
#22561012	2
#22560366	2
C₂×#16863117	2
C₂×#500×C₆₇	2
#9836	67