Subgroups — #30864701

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570733	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	2	yes	maximal
3	#22561190	C₂	1	2	yes	maximal
4	#22560962	C₂	1	4	yes	maximal
5	#22560384	C₂	1	1	yes	maximal, characteristic
6	#22560088	C₂	1	4	yes	maximal
7	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₂²	1	2	yes	—
9	#16863805	C₄	1	8	yes	complemented
10	#16863526	—	2	4	no	—
11	C₈×Dic₁₃₄	—	2	4	no	—
12	C₂×#14523979	C₂²	1	4	yes	—
13	#16863434	C₂²	1	4	yes	—
14	#16863422	C₂²	1	4	yes	—
15	#16863420	C₂²	1	2	yes	—
16	#16863351	C₂²	1	4	yes	—
17	#16863156	C₂²	1	8	yes	—
18	C₂×#97×C₆₇	C₂²	1	2	yes	—
19	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#376×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	C₂³	1	4	yes	—
24	#14524043	C₂×C₄	1	4	yes	—
25	#14524035	C₂×C₄	1	4	yes	—
26	#14524022	—	2	4	no	—
27	C₄×Dic₁₃₄	—	2	4	no	—
28	#14524002	C₂³	1	4	yes	—
29	#14523979	—	2	8	no	—
30	#14523977	C₂×C₄	1	8	yes	—
31	#14523975	C₂×C₄	1	8	yes	—
32	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
33	Dic₆₇⋊C₈	—	2	8	no	—
34	#14523953	—	2	4	no	—
35	C₈×Dic₆₇	—	2	4	no	—
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	#97×C₆₇	—	2	4	no	—
42	#97×C₆₇	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₆₇	D₄	1	2	yes	—
44	C₄×C₈×C₆₇	D₄	1	2	yes	—
45	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
46	#95×C₆₇	—	2	2	no	—
47	C₂×Dic₁₃₄	—	2	4	no	—
48	#11761612	—	2	8	no	—
49	C₂×C₆₇⋊C₈	—	2	4	no	—
50	C₄⋊Dic₆₇	—	2	4	no	—
51	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
52	Dic₆₇⋊C₄	—	2	8	no	—
53	C₂²⋊Dic₆₇	—	2	4	no	—
54	C₄×Dic₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	#41	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
64	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₆₇	#41	1	4	yes	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₆₇	C₂×D₄	1	2	yes	—
69	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
70	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	—	4	4	no	—
74	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
75	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
79	C₂×Dic₆₇	—	2	8	no	—
80	Dic₁₃₄	—	4	8	no	—
81	C₈×C₆₇	—	2	2	no	—
82	C₈×C₆₇	C₄×Q₈	1	8	yes	—
83	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
84	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
85	C₆₇⋊C₈	—	4	4	no	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
88	C₂²×C₆₇	#533	1	2	yes	—
89	C₄×C₆₇	—	2	4	no	—
90	C₄×C₆₇	—	4	4	no	—
91	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
92	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₆₇	#532	1	1	yes	characteristic
94	Dic₆₇	—	4	8	no	—
95	#21735	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₆₇	#2955	1	2	yes	—
99	#3162	—	67	1	no	—
100	#2960	—	67	2	no	—
101	#1995	—	67	2	no	—
102	#1946	—	67	4	no	—
103	#1849	—	67	1	no	—
104	#1772	—	67	4	no	—
105	#1726	D₆₇	1	1	yes	characteristic
106	C₆₇	#21735	1	1	yes	characteristic, complemented, Sylow
107	#533	—	67	2	no	—
108	#501	—	67	8	no	—
109	#446	—	134	4	no	—
110	C₈×Q₈	—	134	4	no	—
111	C₂×#108	—	67	4	no	—
112	#427	—	67	4	no	—
113	#424	—	67	4	no	—
114	#423	—	67	2	no	—
115	#405	—	67	4	no	—
116	C₂×#97	C₂×D₆₇	1	2	yes	—
117	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
119	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
120	#376	C₂×D₆₇	1	1	yes	characteristic
121	#339	—	67	8	no	—
122	C₂×M₁₆	—	67	4	no	—
123	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
125	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
126	C₄⋊Q₈	—	67	4	no	—
127	#125	—	67	4	no	—
128	C₂²⋊Q₈	—	134	4	no	—
129	C₄×Q₈	—	134	4	no	—
130	#117	—	67	4	no	—
131	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
132	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
133	#108	—	134	8	no	—
134	#107	—	67	8	no	—
135	#106	—	67	8	no	—
136	C₄⋊C₈	—	134	8	no	—
137	C₄⋊C₈	—	134	4	no	—
138	#97	—	2	4	no	—
139	#97	—	134	4	no	—
140	#97	C₄×D₆₇	1	4	yes	—
141	C₄×C₈	—	134	4	no	—
142	C₄×C₈	Dic₁₃₄	1	8	yes	complemented
143	C₄×C₈	D₂₆₈	1	2	yes	—
144	C₄×C₈	D₂₆₈	1	2	yes	—
145	#95	—	2	2	no	—
146	C₂×Q₈	—	134	4	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	#11761628	1	2	yes	—
149	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
150	M₁₆	—	134	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	134	4	no	—
153	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
154	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
155	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
156	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
157	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
158	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
159	C₂×C₈	#11761628	1	4	yes	—
160	C₄⋊C₄	—	67	8	no	—
161	C₄⋊C₄	—	134	8	no	—
162	C₄⋊C₄	—	134	4	no	—
163	C₂²⋊C₄	—	134	4	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	134	4	no	—
166	C₄²	C₂×C₄×D₆₇	1	2	yes	—
167	C₄²	C₂×D₂₆₈	1	2	yes	—
168	C₄²	C₂×Dic₁₃₄	1	4	yes	—
169	C₂³	C₂×#11761628	1	1	yes	characteristic
170	Q₈	—	268	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	134	8	no	—
175	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
176	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
179	C₂×C₄	C₂×#11761628	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	268	4	no	—
182	C₈	C₄×Dic₁₃₄	1	8	yes	—
183	C₈	#14524043	1	2	yes	—
184	C₈	#14524043	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
187	C₂²	#16863953	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	4	4	no	—
190	C₄	—	268	8	no	—
191	C₄	C₂×C₄×Dic₁₃₄	1	2	yes	—
192	C₄	C₂×#14524043	1	1	yes	characteristic
193	C₄	#16863946	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₄×#14524043	1	1	yes	characteristic
196	C₂	#22569113	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#21735
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
9. #16863805	C₄	semidirect
45. C₄×C₈×C₆₇	Q₈	semidirect
106. C₆₇	#21735	semidirect
142. C₄×C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570733	2
#22569143	2
#22561190	2
#22560962	2
#22560384	2
#22560088	2
C₆₇×#1726	2
#21735	67