Subgroups — #30749609

Order 17152.

390 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560337	C₂	1	1	yes	maximal, characteristic
2	#22560336	C₂	1	2	yes	maximal
3	#22560336	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	#16863266	C₂²	1	2	yes	—
9	#16863159	—	2	2	no	—
10	#16863158	—	2	4	no	—
11	#16863158	—	2	4	no	—
12	C₈×C₆₇⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#384×C₆₇	C₂²	1	2	yes	—
15	#340×C₆₇	—	2	2	no	—
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	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
24	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
25	C₂²×C₈×C₆₇	D₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₇	Q₈	1	2	yes	—
32	C₂²×Dic₆₇	—	4	2	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	4	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₄⋊Dic₆₇	—	4	4	no	—
39	C₄⋊Dic₆₇	—	4	4	no	—
40	C₄⋊Dic₆₇	—	4	4	no	—
41	C₂²×C₄×C₆₇	—	4	2	no	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	—	2	4	no	—
46	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
47	C₄⋊C₄×C₆₇	—	4	4	no	—
48	C₄⋊C₄×C₆₇	—	4	4	no	—
49	C₄⋊C₄×C₆₇	—	4	4	no	—
50	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₆₇	#127	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	8	2	no	—
56	C₂×C₄×C₆₇	—	8	2	no	—
57	C₂×C₄×C₆₇	—	8	2	no	—
58	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#135	1	2	yes	—
65	C₂×C₄×C₆₇	#135	1	4	yes	—
66	C₂×C₄×C₆₇	#135	1	4	yes	—
67	C₂×C₄×C₆₇	#135	1	2	yes	—
68	C₂×Dic₆₇	—	8	2	no	—
69	C₂×Dic₆₇	—	8	2	no	—
70	C₂×Dic₆₇	—	8	2	no	—
71	C₈×C₆₇	—	2	4	no	—
72	C₆₇⋊C₈	—	2	4	no	—
73	C₆₇⋊C₈	—	2	4	no	—
74	C₂²×C₆₇	#384	1	1	yes	characteristic
75	C₂²×C₆₇	#495	1	2	yes	—
76	C₂²×C₆₇	#495	1	2	yes	—
77	C₂²×C₆₇	#495	1	1	yes	characteristic
78	C₂²×C₆₇	#495	1	1	yes	characteristic
79	C₄×C₆₇	—	16	2	no	—
80	C₄×C₆₇	#427	1	2	yes	—
81	C₄×C₆₇	#427	1	2	yes	—
82	C₄×C₆₇	#427	1	2	yes	—
83	C₄×C₆₇	#499	1	2	yes	—
84	C₄×C₆₇	#499	1	2	yes	—
85	C₄×C₆₇	#499	1	2	yes	—
86	Dic₆₇	—	16	2	no	—
87	#8817	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	#1711	1	2	yes	—
89	C₂×C₆₇	#1711	1	2	yes	—
90	C₂×C₆₇	#1846	1	1	yes	characteristic
91	C₂×C₆₇	#1846	1	1	yes	characteristic
92	C₂×C₆₇	#1846	1	1	yes	characteristic
93	#1834	—	67	1	no	—
94	#1834	—	67	2	no	—
95	#1834	—	67	2	no	—
96	#1834	D₆₇	1	1	yes	characteristic
97	C₂×C₈²	—	67	1	no	—
98	C₆₇	#8817	1	1	yes	characteristic, complemented, Sylow
99	C₂×C₄×C₈	—	67	1	no	—
100	C₂×C₄×C₈	—	67	1	no	—
101	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
102	#384	—	67	2	no	—
103	#384	C₂×D₆₇	1	2	yes	—
104	#340	—	2	2	no	—
105	#340	—	134	4	no	—
106	#340	—	134	4	no	—
107	#340	—	134	2	no	—
108	C₈²	—	67	4	no	—
109	C₂²×C₈	—	67	2	no	—
110	C₂²×C₈	—	67	2	no	—
111	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	134	2	no	—
116	C₂×C₄⋊C₄	—	134	2	no	—
117	C₂×C₄⋊C₄	—	134	2	no	—
118	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
119	C₄×C₈	—	67	2	no	—
120	C₄×C₈	—	67	2	no	—
121	C₄×C₈	—	67	2	no	—
122	C₄×C₈	—	67	2	no	—
123	C₄×C₈	Dic₁₃₄	1	2	yes	—
124	C₄×C₈	C₄×D₆₇	1	2	yes	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	—	268	2	no	—
127	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
128	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	67	4	no	—
132	C₂×C₈	—	67	4	no	—
133	C₂×C₈	—	134	2	no	—
134	C₂×C₈	—	134	4	no	—
135	C₂×C₈	—	134	2	no	—
136	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	268	4	no	—
141	C₄⋊C₄	—	268	4	no	—
142	C₄⋊C₄	—	268	4	no	—
143	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
144	C₄²	Q₈×D₆₇	1	1	yes	characteristic
145	C₄²	Q₈×D₆₇	1	1	yes	characteristic
146	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
147	C₂³	#14524039	1	1	yes	characteristic
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	536	2	no	—
152	C₂×C₄	—	536	2	no	—
153	C₂×C₄	—	536	2	no	—
154	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
155	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#14524041	1	1	yes	characteristic
158	C₂×C₄	#14524042	1	1	yes	characteristic
159	C₂×C₄	#14524042	1	1	yes	characteristic
160	C₂×C₄	#14524062	1	2	yes	—
161	C₂×C₄	#14524063	1	4	yes	—
162	C₂×C₄	#14524064	1	4	yes	—
163	C₂×C₄	#14524066	1	2	yes	—
164	C₈	—	2	4	no	—
165	C₈	—	134	4	no	—
166	C₈	—	134	4	no	—
167	C₂²	#16863265	1	1	yes	characteristic
168	C₂²	#16863777	1	2	yes	—
169	C₂²	#16863778	1	2	yes	—
170	C₂²	#16863779	1	1	yes	characteristic
171	C₂²	#16863781	1	1	yes	characteristic
172	C₄	—	16	2	no	—
173	C₄	—	1072	2	no	—
174	C₄	#16863431	1	2	yes	—
175	C₄	#16863432	1	2	yes	—
176	C₄	#16863433	1	2	yes	—
177	C₄	#16863795	1	2	yes	—
178	C₄	#16863796	1	2	yes	—
179	C₄	#16863797	1	2	yes	—
180	C₂	#22559820	1	2	yes	—
181	C₂	#22559822	1	2	yes	—
182	C₂	#22560370	1	1	yes	characteristic
183	C₂	#22560371	1	1	yes	characteristic
184	C₂	#22560372	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1834	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1834	D₆₇
hypocenter	C₆₇	#8817
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
12. C₈×C₆₇⋊C₈	C₄	semidirect
98. C₆₇	#8817	semidirect

Maximal subgroups

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

subgroup	index
#22560337	2
#22560336	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1834	2
#8817	67