Subgroups — #30749639

Order 17152.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560347	C₂	1	2	yes	maximal
2	#22558968	C₂	1	2	yes	maximal
3	#22558965	C₂	1	2	yes	maximal
4	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863420	C₂²	1	4	yes	—
6	#16863161	—	2	4	no	—
7	#16863149	—	2	4	no	—
8	#16863147	—	2	4	no	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₈²×C₆₇	C₄	1	1	yes	characteristic
13	C₈²×C₆₇	C₄	1	1	yes	characteristic
14	C₈²×C₆₇	C₄	1	1	yes	characteristic
15	C₈²×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#11761612	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₂²×C₈×C₆₇	D₄	1	2	yes	—
20	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
21	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
22	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	D₄	1	2	yes	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	C₄×C₈×C₆₇	Q₈	1	2	yes	—
32	#11761612	—	4	8	no	—
33	C₂×C₆₇⋊C₈	—	4	4	no	—
34	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₆₇	—	2	4	no	—
38	C₂×C₈×C₆₇	—	2	4	no	—
39	C₂×C₈×C₆₇	—	2	4	no	—
40	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#135	1	4	yes	—
60	C₈×C₆₇	—	2	2	no	—
61	C₈×C₆₇	—	2	2	no	—
62	C₈×C₆₇	—	2	2	no	—
63	C₈×C₆₇	—	2	2	no	—
64	C₈×C₆₇	#108	1	4	yes	—
65	C₈×C₆₇	#108	1	4	yes	—
66	C₆₇⋊C₈	—	8	4	no	—
67	C₂²×C₆₇	—	2	1	no	—
68	C₂²×C₆₇	—	2	1	no	—
69	C₂²×C₆₇	#384	1	1	yes	characteristic
70	C₂²×C₆₇	#499	1	2	yes	—
71	C₄×C₆₇	—	2	1	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	#427	1	2	yes	—
74	C₄×C₆₇	C₂×#108	1	2	yes	—
75	C₄×C₆₇	C₂×#108	1	2	yes	—
76	C₄×C₆₇	#495	1	2	yes	—
77	#8824	—	67	1	no	maximal, Sylow
78	C₂×C₆₇	—	2	2	no	—
79	C₂×C₆₇	#1846	1	1	yes	characteristic
80	C₂×C₆₇	#1848	1	1	yes	characteristic
81	C₂×C₆₇	#1848	1	1	yes	characteristic
82	#1837	—	67	2	no	—
83	#1567	—	67	2	no	—
84	#1566	—	67	2	no	—
85	C₂×C₈²	D₆₇	1	1	yes	characteristic
86	C₆₇	#8824	1	1	yes	characteristic, complemented, Sylow
87	#423	—	67	4	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
91	#341	—	134	4	no	—
92	#335	—	134	4	no	—
93	#334	—	134	4	no	—
94	C₈²	Dic₆₇	1	1	yes	characteristic
95	C₈²	Dic₆₇	1	1	yes	characteristic
96	C₈²	Dic₆₇	1	1	yes	characteristic
97	C₈²	Dic₆₇	1	1	yes	characteristic
98	C₂×M₁₆	—	134	4	no	—
99	C₂²×C₈	Dic₁₃₄	1	2	yes	—
100	C₂²×C₈	Dic₁₃₄	1	2	yes	—
101	C₂²×C₈	D₂₆₈	1	2	yes	—
102	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
103	C₄⋊C₈	—	134	4	no	—
104	C₄⋊C₈	—	134	4	no	—
105	C₄×C₈	Dic₁₃₄	1	2	yes	—
106	C₄×C₈	D₂₆₈	1	2	yes	—
107	C₄×C₈	D₂₆₈	1	2	yes	—
108	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
117	M₁₆	—	268	8	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	268	4	no	—
122	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
123	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
124	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
125	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
126	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
128	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
129	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
130	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
131	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
132	C₂³	#14524043	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
138	C₂×C₄	#14524040	1	1	yes	characteristic
139	C₂×C₄	#14524043	1	1	yes	characteristic
140	C₂×C₄	#14524043	1	1	yes	characteristic
141	C₂×C₄	#14524065	1	4	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	536	4	no	—
147	C₈	#14523979	1	4	yes	—
148	C₈	#14523979	1	4	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#16863266	1	1	yes	characteristic
152	C₂²	#16863798	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#16863434	1	2	yes	—
156	C₄	C₂×#14523979	1	2	yes	—
157	C₄	C₂×#14523979	1	2	yes	—
158	C₄	#16863780	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#22560373	1	1	yes	characteristic
161	C₂	#22560381	1	1	yes	characteristic
162	C₂	#22560381	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#8824
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
86. C₆₇	#8824	semidirect

Maximal subgroups

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

subgroup	index
#22560347	2
#22558968	2
#22558965	2
C₂×C₈²×C₆₇	2
#8824	67