Subgroups — #30864689

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22569139	C₂	1	2	yes	maximal
3	#22561185	C₂	1	2	yes	maximal
4	#22560960	C₂	1	4	yes	maximal
5	#22560096	C₂	1	4	yes	maximal
6	C₄×#14523953	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₂²	1	2	yes	—
9	#16863801	C₄	1	2	yes	complemented
10	#16863798	C₄	1	4	yes	complemented
11	#16863794	C₄	1	2	yes	complemented
12	#16863523	—	2	4	no	—
13	#16863520	—	2	4	no	—
14	#16863434	C₂²	1	4	yes	—
15	#16863424	C₂²	1	4	yes	—
16	#16863354	C₂²	1	4	yes	—
17	C₂×#14523953	C₂²	1	2	yes	—
18	C₂×#14523953	C₂²	1	2	yes	—
19	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
21	#16863157	C₂²	1	8	yes	—
22	C₂×#108×C₆₇	C₂²	1	4	yes	—
23	#423×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
27	#14524043	C₂×C₄	1	4	yes	—
28	#14524035	C₂×C₄	1	4	yes	—
29	#14524022	—	2	4	no	—
30	C₄×Dic₁₃₄	—	2	4	no	—
31	#14524002	C₂³	1	4	yes	—
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
34	#14523977	C₂×C₄	1	8	yes	—
35	#14523975	C₂×C₄	1	8	yes	—
36	Dic₆₇⋊C₈	—	2	8	no	—
37	#14523953	—	2	4	no	—
38	#14523953	—	2	4	no	—
39	#14523953	Q₈	1	8	yes	complemented
40	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
41	C₈×Dic₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	#108×C₆₇	—	2	8	no	—
46	C₄⋊C₈×C₆₇	—	2	4	no	—
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×Dic₁₃₄	—	2	4	no	—
51	C₂²×Dic₆₇	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
54	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
55	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
56	C₄⋊Dic₆₇	—	2	4	no	—
57	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
58	Dic₆₇⋊C₄	—	2	8	no	—
59	C₂²⋊Dic₆₇	—	2	4	no	—
60	C₄×Dic₆₇	—	2	4	no	—
61	C₄×Dic₆₇	—	2	2	no	—
62	C₄×Dic₆₇	—	4	4	no	—
63	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₆₇	#41	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₆₇	—	2	8	no	—
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₆₇	#41	1	4	yes	—
72	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×D₄	1	2	yes	—
75	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
83	C₂×Dic₆₇	—	2	8	no	—
84	C₂×Dic₆₇	—	2	4	no	—
85	C₂×Dic₆₇	—	4	4	no	—
86	Dic₁₃₄	—	4	8	no	—
87	C₈×C₆₇	—	4	4	no	—
88	C₈×C₆₇	C₄×Q₈	1	4	yes	—
89	C₈×C₆₇	C₄×Q₈	1	4	yes	—
90	C₆₇⋊C₈	—	2	2	no	—
91	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	—
92	C₂²×C₆₇	—	2	2	no	—
93	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
94	C₂²×C₆₇	#533	1	2	yes	—
95	C₄×C₆₇	—	2	2	no	—
96	C₄×C₆₇	#444	1	4	yes	—
97	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
98	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₄×C₆₇	#532	1	1	yes	characteristic
100	Dic₆₇	—	4	4	no	—
101	Dic₆₇	—	4	8	no	—
102	#21734	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	—	2	2	no	—
104	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#444	1	2	yes	—
106	#3160	—	67	1	no	—
107	#2960	—	67	2	no	—
108	#1994	—	67	2	no	—
109	#1946	—	67	4	no	—
110	#1848	D₆₇	1	1	yes	characteristic
111	#1773	—	67	4	no	—
112	C₄×#97	—	67	1	no	—
113	C₆₇	#21734	1	1	yes	characteristic, complemented, Sylow
114	#533	—	67	2	no	—
115	#500	—	67	2	no	—
116	#499	—	67	4	no	—
117	#498	—	67	2	no	—
118	#446	—	134	4	no	—
119	C₈×Q₈	—	134	4	no	—
120	C₂×#108	C₂×D₆₇	1	4	yes	—
121	#427	—	67	4	no	—
122	#424	—	67	4	no	—
123	#423	C₂×D₆₇	1	2	yes	—
124	#405	—	67	4	no	—
125	C₂×#97	—	67	2	no	—
126	C₂×#97	—	67	2	no	—
127	C₂×C₄×C₈	—	67	1	no	—
128	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₄³	—	67	1	no	—
130	#339	—	67	8	no	—
131	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
132	C₂²×C₈	—	67	1	no	—
133	C₂²×C₈	—	67	1	no	—
134	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
135	C₄⋊Q₈	—	67	4	no	—
136	#125	—	67	4	no	—
137	C₂²⋊Q₈	—	134	4	no	—
138	C₄×Q₈	—	134	4	no	—
139	#117	—	67	4	no	—
140	C₂×C₄²	—	67	2	no	—
141	C₂×C₄²	—	134	2	no	—
142	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
143	#108	—	2	8	no	—
144	#107	—	67	8	no	—
145	#106	—	67	8	no	—
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	134	8	no	—
148	#97	—	67	8	no	—
149	#97	—	134	4	no	—
150	#97	—	134	4	no	—
151	C₄×C₈	—	67	4	no	—
152	C₄×C₈	—	134	4	no	—
153	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
154	C₄×C₈	C₄×D₆₇	1	2	yes	—
155	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
156	C₂×Q₈	—	134	4	no	—
157	C₂²×C₄	—	134	2	no	—
158	C₂²×C₄	#11761629	1	2	yes	—
159	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
160	M₁₆	—	2	8	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	67	2	no	—
163	C₂×C₈	—	67	4	no	—
164	C₂×C₈	—	67	2	no	—
165	C₂×C₈	—	134	2	no	—
166	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
167	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
168	C₂×C₈	Q₈×D₆₇	1	4	yes	—
169	C₂×C₈	#11761627	1	4	yes	—
170	C₄⋊C₄	—	67	8	no	—
171	C₄⋊C₄	—	134	8	no	—
172	C₄⋊C₄	—	134	4	no	—
173	C₂²⋊C₄	—	134	4	no	—
174	C₄²	—	268	4	no	—
175	C₄²	—	67	4	no	—
176	C₄²	—	134	4	no	—
177	C₄²	—	134	2	no	—
178	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
179	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
180	C₄²	D₄×D₆₇	1	2	yes	—
181	C₂³	C₂×#11761629	1	1	yes	characteristic
182	Q₈	—	268	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	268	4	no	—
185	C₂×C₄	—	134	8	no	—
186	C₂×C₄	—	134	4	no	—
187	C₂×C₄	#14524005	1	2	yes	—
188	C₂×C₄	#14524005	1	2	yes	—
189	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
190	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
191	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
192	C₂×C₄	C₂×#11761627	1	2	yes	—
193	C₈	—	4	4	no	—
194	C₈	—	67	4	no	—
195	C₈	—	134	2	no	—
196	C₈	#14524008	1	4	yes	—
197	C₈	#14524008	1	4	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×#14524005	1	1	yes	characteristic
200	C₂²	#16863951	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	268	4	no	—
203	C₄	—	268	8	no	—
204	C₄	#16863517	1	4	yes	—
205	C₄	C₂×#14524008	1	2	yes	—
206	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
207	C₄	#16863944	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	#22562776	1	1	yes	characteristic
210	C₂	C₂×#16863517	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#21734
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
9. #16863801	C₄	semidirect
10. #16863798	C₄	semidirect
11. #16863794	C₄	semidirect
39. #14523953	Q₈	semidirect
113. C₆₇	#21734	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22569139	2
#22561185	2
#22560960	2
#22560096	2
C₄×#14523953	2
C₆₇×#1848	2
#21734	67