Subgroups — #30864702

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	#22569144	C₂	1	2	yes	maximal
2	#22561191	C₂	1	2	yes	maximal
3	#22560963	C₂	1	4	yes	maximal
4	#22560384	C₂	1	1	yes	maximal, characteristic
5	#22560090	C₂	1	4	yes	maximal
6	#22559897	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#3162	C₂	1	1	yes	maximal, characteristic
8	#16863527	—	2	4	no	—
9	Q₈×C₆₇⋊C₈	—	2	4	no	—
10	C₂×#14523979	C₂²	1	4	yes	—
11	#16863423	C₂²	1	4	yes	—
12	#16863420	C₂²	1	2	yes	—
13	#16863352	C₂²	1	4	yes	—
14	C₂×#14523953	C₂²	1	2	yes	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	#16863242	C₂²	1	1	yes	characteristic
18	#16863155	C₂²	1	8	yes	—
19	#533×C₆₇	C₂²	1	2	yes	—
20	#501×C₆₇	C₄	1	8	yes	complemented
21	#427×C₆₇	C₂²	1	4	yes	—
22	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	C₂³	1	4	yes	—
24	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
27	#14523979	—	2	8	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
30	#14523954	—	2	4	no	—
31	#14523953	—	2	4	no	—
32	C₄×C₆₇⋊C₈	—	2	4	no	—
33	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
34	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
35	C₈×Dic₆₇	Q₈	1	8	yes	complemented
36	C₆₇⋊#95	—	2	2	no	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
39	#125×C₆₇	C₂×C₄	1	4	yes	—
40	C₂²⋊Q₈×C₆₇	—	2	4	no	—
41	C₄×Q₈×C₆₇	—	2	4	no	—
42	#117×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	#107×C₆₇	C₂×C₄	1	8	yes	—
45	#106×C₆₇	C₂×C₄	1	8	yes	—
46	#97×C₆₇	C₂×C₄	1	4	yes	—
47	C₂²×Dic₆₇	—	2	2	no	—
48	#11761612	—	2	8	no	—
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₂×C₆₇⋊C₈	—	2	4	no	—
51	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
52	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
55	C₄×Dic₆₇	—	2	2	no	—
56	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
57	C₂×Q₈×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	#41	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₆₇	#41	1	4	yes	—
63	C₄⋊C₄×C₆₇	—	2	8	no	—
64	C₄⋊C₄×C₆₇	—	2	4	no	—
65	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
66	C₂²⋊C₄×C₆₇	—	2	4	no	—
67	C₄²×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₆₇	C₂×D₄	1	2	yes	—
70	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
71	Q₈×C₆₇	—	4	8	no	—
72	C₂×C₄×C₆₇	—	2	8	no	—
73	C₂×C₄×C₆₇	—	4	2	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	4	no	—
80	C₂×Dic₆₇	—	4	4	no	—
81	C₈×C₆₇	C₄×Q₈	1	8	yes	—
82	C₆₇⋊C₈	—	2	2	no	—
83	C₆₇⋊C₈	—	4	4	no	—
84	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
85	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
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	8	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	4	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	D₆₇	1	1	yes	characteristic
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	—	67	1	no	—
106	C₆₇	#21735	1	1	yes	characteristic, complemented, Sylow
107	#533	C₂×D₆₇	1	2	yes	—
108	#501	Dic₆₇	1	8	yes	complemented
109	#446	—	134	4	no	—
110	C₈×Q₈	—	134	4	no	—
111	C₂×#108	—	67	4	no	—
112	#427	C₂×D₆₇	1	4	yes	—
113	#424	—	67	4	no	—
114	#423	—	67	2	no	—
115	#405	—	67	4	no	—
116	C₂×#97	—	67	2	no	—
117	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄×C₈	—	67	2	no	—
119	C₂×C₄×C₈	—	67	1	no	—
120	#376	—	67	1	no	—
121	#339	—	67	8	no	—
122	C₂×M₁₆	—	67	4	no	—
123	C₂²×C₈	—	67	1	no	—
124	C₂²×C₈	—	67	1	no	—
125	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
126	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
127	#125	C₂×Dic₆₇	1	4	yes	—
128	C₂²⋊Q₈	—	2	4	no	—
129	C₄×Q₈	—	2	4	no	—
130	#117	C₂²×D₆₇	1	4	yes	—
131	C₂×C₄²	—	67	2	no	—
132	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
133	#108	—	134	8	no	—
134	#107	C₂×Dic₆₇	1	8	yes	—
135	#106	C₂×Dic₆₇	1	8	yes	—
136	C₄⋊C₈	—	134	8	no	—
137	C₄⋊C₈	—	134	4	no	—
138	#97	—	134	4	no	—
139	#97	—	134	4	no	—
140	#97	C₂×Dic₆₇	1	4	yes	—
141	C₄×C₈	—	67	8	no	—
142	C₄×C₈	—	67	2	no	—
143	C₄×C₈	—	67	2	no	—
144	C₄×C₈	—	134	4	no	—
145	#95	—	134	2	no	—
146	C₂×Q₈	—	2	4	no	—
147	C₂²×C₄	—	134	2	no	—
148	C₂²×C₄	#11761627	1	2	yes	—
149	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
150	M₁₆	—	134	8	no	—
151	C₂×C₈	—	67	1	no	—
152	C₂×C₈	—	67	1	no	—
153	C₂×C₈	—	67	4	no	—
154	C₂×C₈	—	67	2	no	—
155	C₂×C₈	—	134	4	no	—
156	C₂×C₈	—	134	2	no	—
157	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
158	C₂×C₈	Q₈×D₆₇	1	4	yes	—
159	C₂×C₈	#11761629	1	4	yes	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	67	4	no	—
166	C₄²	—	134	2	no	—
167	C₄²	C₂²×Dic₆₇	1	2	yes	—
168	C₄²	D₄×D₆₇	1	2	yes	—
169	C₂³	C₂×#11761627	1	1	yes	characteristic
170	Q₈	—	4	8	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	268	4	no	—
174	C₂×C₄	—	134	4	no	—
175	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
176	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
178	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
179	C₂×C₄	C₂×#11761629	1	2	yes	—
180	C₈	—	268	4	no	—
181	C₈	—	67	2	no	—
182	C₈	—	67	2	no	—
183	C₈	—	134	2	no	—
184	C₈	Q₈×Dic₆₇	1	8	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
187	C₂²	#16863951	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	4	8	no	—
190	C₄	—	268	4	no	—
191	C₄	C₂×Q₈×Dic₆₇	1	2	yes	—
192	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
193	C₄	#16863947	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
196	C₂	#22569112	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×#3162	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3162	D₆₇
hypocenter	C₆₇	#21735
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

Complements

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

N	H ≅ G/N	type
20. #501×C₆₇	C₄	semidirect
35. C₈×Dic₆₇	Q₈	semidirect
106. C₆₇	#21735	semidirect
108. #501	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569144	2
#22561191	2
#22560963	2
#22560384	2
#22560090	2
#22559897	2
C₆₇×#3162	2
#21735	67