Subgroups — #30749600

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560368	C₂	1	2	yes	maximal
2	#22560366	C₂	1	1	yes	maximal, characteristic
3	#22560336	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523976	—	2	4	no	—
7	C₂×#14523975	—	2	2	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	C₈×C₆₇⋊C₈	C₄	1	4	yes	complemented
13	C₂×#106×C₆₇	—	2	2	no	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#384×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
17	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	#14523976	—	4	8	no	—
22	#14523975	—	4	4	no	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
26	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₆₇	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	#106×C₆₇	—	4	4	no	—
33	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₇	Q₈	1	2	yes	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	2	4	no	—
37	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
39	C₂×C₆₇⋊C₈	SD₁₆	1	8	yes	—
40	C₄⋊Dic₆₇	—	4	4	no	—
41	C₄⋊Dic₆₇	—	4	4	no	—
42	C₄⋊Dic₆₇	—	4	4	no	—
43	C₂²×C₄×C₆₇	—	4	2	no	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₆₇	D₈	1	4	yes	—
49	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
50	C₄⋊C₄×C₆₇	—	4	4	no	—
51	C₄⋊C₄×C₆₇	—	4	4	no	—
52	C₄⋊C₄×C₆₇	—	4	4	no	—
53	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₂³×C₆₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	8	2	no	—
59	C₂×C₄×C₆₇	—	8	2	no	—
60	C₂×C₄×C₆₇	—	8	2	no	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
69	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₆₇	#135	1	4	yes	—
71	C₂×Dic₆₇	—	8	2	no	—
72	C₂×Dic₆₇	—	8	2	no	—
73	C₂×Dic₆₇	—	8	2	no	—
74	C₈×C₆₇	#107	1	8	yes	complemented
75	C₆₇⋊C₈	—	2	4	no	—
76	C₆₇⋊C₈	#106	1	8	yes	complemented
77	C₂²×C₆₇	#384	1	1	yes	characteristic
78	C₂²×C₆₇	#492	1	2	yes	—
79	C₂²×C₆₇	#493	1	1	yes	characteristic
80	C₂²×C₆₇	#494	1	1	yes	characteristic
81	C₂²×C₆₇	#495	1	2	yes	—
82	C₄×C₆₇	—	16	2	no	—
83	C₄×C₆₇	C₂×#106	1	2	yes	—
84	C₄×C₆₇	C₂×#107	1	2	yes	—
85	C₄×C₆₇	#427	1	2	yes	—
86	C₄×C₆₇	#498	1	2	yes	—
87	C₄×C₆₇	#499	1	2	yes	—
88	C₄×C₆₇	#500	1	2	yes	—
89	Dic₆₇	—	16	2	no	—
90	#8814	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	#1706	1	2	yes	—
92	C₂×C₆₇	#1707	1	2	yes	—
93	C₂×C₆₇	#1844	1	1	yes	characteristic
94	C₂×C₆₇	#1845	1	1	yes	characteristic
95	C₂×C₆₇	#1846	1	1	yes	characteristic
96	#1845	—	67	2	no	—
97	#1844	—	67	1	no	—
98	#1844	D₆₇	1	1	yes	characteristic
99	#1834	—	67	2	no	—
100	C₂×C₈²	—	67	1	no	—
101	C₆₇	#8814	1	1	yes	characteristic, complemented, Sylow
102	C₂×#107	—	134	4	no	—
103	C₂×#106	—	2	2	no	—
104	C₂×#106	—	134	2	no	—
105	C₂×C₄×C₈	—	67	1	no	—
106	C₂×C₄×C₈	—	67	1	no	—
107	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
108	#384	—	67	2	no	—
109	#384	C₂×D₆₇	1	2	yes	—
110	#340	—	134	4	no	—
111	C₈²	—	67	4	no	—
112	C₂²×C₈	—	67	2	no	—
113	C₂²×C₈	—	67	2	no	—
114	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	134	2	no	—
119	C₂×C₄⋊C₄	—	134	2	no	—
120	C₂×C₄⋊C₄	—	134	2	no	—
121	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
122	#107	—	268	8	no	—
123	#106	—	4	4	no	—
124	#106	—	268	4	no	—
125	C₄×C₈	—	67	2	no	—
126	C₄×C₈	—	67	2	no	—
127	C₄×C₈	—	67	2	no	—
128	C₄×C₈	—	67	2	no	—
129	C₄×C₈	Dic₁₃₄	1	2	yes	—
130	C₄×C₈	C₄×D₆₇	1	2	yes	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	268	2	no	—
133	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
136	C₂×C₈	—	67	4	no	—
137	C₂×C₈	—	67	8	no	—
138	C₂×C₈	—	67	4	no	—
139	C₂×C₈	—	134	4	no	—
140	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
141	C₂×C₈	#11761613	1	4	yes	—
142	C₂×C₈	#11761618	1	4	yes	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	268	4	no	—
147	C₄⋊C₄	—	268	4	no	—
148	C₄⋊C₄	—	268	4	no	—
149	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
150	C₄²	Q₈×D₆₇	1	1	yes	characteristic
151	C₄²	Q₈×D₆₇	1	1	yes	characteristic
152	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
153	C₂³	#14524039	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	536	2	no	—
158	C₂×C₄	—	536	2	no	—
159	C₂×C₄	—	536	2	no	—
160	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
161	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#14524041	1	1	yes	characteristic
164	C₂×C₄	#14524042	1	1	yes	characteristic
165	C₂×C₄	#14524042	1	1	yes	characteristic
166	C₂×C₄	C₂×#11761613	1	2	yes	—
167	C₂×C₄	SD₁₆×D₆₇	1	4	yes	—
168	C₂×C₄	C₂×#11761618	1	2	yes	—
169	C₂×C₄	#14524064	1	4	yes	—
170	C₈	—	67	8	no	—
171	C₈	—	134	4	no	—
172	C₈	#14523976	1	8	yes	complemented
173	C₂²	#16863265	1	1	yes	characteristic
174	C₂²	#16863767	1	2	yes	—
175	C₂²	#16863772	1	1	yes	characteristic
176	C₂²	#16863775	1	1	yes	characteristic
177	C₂²	#16863777	1	2	yes	—
178	C₄	—	16	2	no	—
179	C₄	—	1072	2	no	—
180	C₄	#106×D₆₇	1	2	yes	—
181	C₄	C₂×#14523976	1	2	yes	—
182	C₄	#16863431	1	2	yes	—
183	C₄	#16863793	1	2	yes	—
184	C₄	#16863796	1	2	yes	—
185	C₄	#16863799	1	2	yes	—
186	C₂	#22559799	1	2	yes	—
187	C₂	#22559804	1	2	yes	—
188	C₂	#22560364	1	1	yes	characteristic
189	C₂	#22560368	1	1	yes	characteristic
190	C₂	#22560370	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1844	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1844	D₆₇
hypocenter	C₆₇	#8814
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
12. C₈×C₆₇⋊C₈	C₄	semidirect
74. C₈×C₆₇	#107	semidirect
76. C₆₇⋊C₈	#106	semidirect
101. C₆₇	#8814	semidirect
172. C₈	#14523976	semidirect

Maximal subgroups

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

subgroup	index
#22560368	2
#22560366	2
#22560336	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1844	2
#8814	67