Subgroups — #30856033

Order 17152.

910 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562616	C₂	1	2	yes	maximal
2	#22560343	C₂	1	8	yes	maximal, complemented
3	#22559926	C₂	1	4	yes	maximal
4	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	C₂²	1	4	yes	—
6	#16863266	C₂²	1	4	yes	—
7	#16863262	C₂²	1	4	yes	—
8	#16863256	—	2	4	no	—
9	C₄×C₄⋊Dic₆₇	—	2	4	no	—
10	#16863159	C₂²	1	16	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#97×C₆₇	C₄	1	8	yes	complemented
14	C₂×#97×C₆₇	C₂²	1	4	yes	—
15	#14524002	—	2	16	no	—
16	C₂×C₄⋊Dic₆₇	—	2	8	no	—
17	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
18	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
19	C₂×C₄×Dic₆₇	—	2	4	no	—
20	C₆₇⋊#95	—	2	8	no	—
21	C₂³×C₄×C₆₇	D₄	1	2	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	—	2	8	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₆₇	D₄	1	2	yes	—
28	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
29	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
30	#97×C₆₇	C₂×C₄	1	16	yes	—
31	C₂²×Dic₆₇	—	2	8	no	—
32	C₄⋊Dic₆₇	—	2	16	no	—
33	C₄⋊Dic₆₇	—	4	16	no	—
34	C₂²⋊Dic₆₇	—	4	8	no	—
35	C₄×Dic₆₇	—	4	8	no	—
36	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	—	4	4	no	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
43	C₂²×C₄×C₆₇	#41	1	8	yes	—
44	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₆₇	—	2	16	no	—
46	C₂×C₈×C₆₇	—	2	8	no	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
48	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
49	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
50	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
51	C₂³×C₆₇	—	2	4	no	—
52	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₂³×C₆₇	#124	1	4	yes	—
54	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	2	8	no	—
56	C₂×C₄×C₆₇	—	4	8	no	—
57	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
58	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₆₇	#117	1	8	yes	—
60	C₂×C₄×C₆₇	#124	1	4	yes	—
61	C₂×C₄×C₆₇	#125	1	8	yes	—
62	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
65	C₂×Dic₆₇	—	4	8	no	—
66	C₂×Dic₆₇	—	4	16	no	—
67	C₈×C₆₇	—	2	8	no	—
68	C₂²×C₆₇	—	2	8	no	—
69	C₂²×C₆₇	—	2	4	no	—
70	C₂²×C₆₇	#382	1	4	yes	—
71	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₆₇	C₂×#124	1	2	yes	—
73	C₂²×C₆₇	#575	1	4	yes	—
74	C₄×C₆₇	—	2	8	no	—
75	C₄×C₆₇	#382	1	4	yes	—
76	C₄×C₆₇	C₂×#117	1	2	yes	—
77	C₄×C₆₇	C₂×#125	1	2	yes	—
78	Dic₆₇	—	8	8	no	—
79	#20762	—	67	1	no	maximal, Sylow
80	C₂×C₆₇	—	2	4	no	—
81	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
82	C₂×C₆₇	#2911	1	2	yes	—
83	C₂×C₆₇	#3135	1	4	yes	—
84	C₂²×#97	D₆₇	1	1	yes	characteristic
85	#2285	—	67	2	no	—
86	#1836	—	67	8	no	—
87	#1735	—	67	4	no	—
88	C₆₇	#20762	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₈	C₂×D₆₇	1	2	yes	—
90	C₂×#117	—	67	4	no	—
91	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
92	C₂×#97	Dic₆₇	1	8	yes	complemented
93	C₂×#97	C₂×D₆₇	1	4	yes	—
94	#384	—	67	4	no	—
95	#382	—	67	4	no	—
96	#380	—	134	4	no	—
97	C₄×C₄⋊C₄	—	134	4	no	—
98	#340	—	67	16	no	—
99	C₂³×C₄	D₂₆₈	1	2	yes	—
100	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
101	C₂²×C₈	—	2	8	no	—
102	C₂²×C₈	C₂×Dic₆₇	1	8	yes	—
103	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
104	#117	—	134	16	no	—
105	C₂×C₄⋊C₄	—	67	8	no	—
106	C₂×C₄⋊C₄	—	134	8	no	—
107	C₂×C₂²⋊C₄	—	134	4	no	—
108	C₂×C₄²	—	134	4	no	—
109	C₂×C₄²	Dic₁₃₄	1	4	yes	—
110	C₂×C₄²	D₂₆₈	1	2	yes	—
111	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
112	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
113	#97	C₂×Dic₆₇	1	16	yes	—
114	#95	—	134	8	no	—
115	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
116	C₂²×C₄	—	4	4	no	—
117	C₂²×C₄	—	134	8	no	—
118	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
119	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
120	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
121	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
122	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
123	C₂²×C₄	#11761628	1	8	yes	—
124	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
125	C₂×C₈	—	2	16	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
128	C₄⋊C₄	—	268	16	no	—
129	C₄⋊C₄	—	134	16	no	—
130	C₂²⋊C₄	—	268	8	no	—
131	C₄²	—	268	8	no	—
132	C₄²	C₄⋊Dic₆₇	1	8	yes	—
133	C₄²	C₂²×Dic₆₇	1	4	yes	—
134	C₄²	C₂×Dic₁₃₄	1	4	yes	—
135	C₂³	—	2	4	no	—
136	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
137	C₂³	#14524031	1	4	yes	—
138	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	—	268	8	no	—
142	C₂×C₄	—	268	16	no	—
143	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
145	C₂×C₄	#14524002	1	8	yes	—
146	C₂×C₄	#14524031	1	4	yes	—
147	C₂×C₄	#14524035	1	8	yes	—
148	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
149	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
150	C₂×C₄	C₂×#11761628	1	4	yes	—
151	C₈	—	2	8	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	2	4	no	—
154	C₂²	#16863262	1	4	yes	—
155	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
156	C₂²	C₂×#14524031	1	2	yes	—
157	C₂²	#16864196	1	4	yes	—
158	C₄	—	2	8	no	—
159	C₄	—	536	8	no	—
160	C₄	#16863262	1	4	yes	—
161	C₄	C₂×#14524002	1	2	yes	—
162	C₄	C₂×#14524035	1	2	yes	—
163	C₂	—	2	4	no	—
164	C₂	C₂×#16863262	1	1	yes	characteristic
165	C₂	#22568799	1	2	yes	—
166	C₂	#22570555	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	#20762
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #22560343	C₂	semidirect
13. C₂×#97×C₆₇	C₄	semidirect
88. C₆₇	#20762	semidirect
92. C₂×#97	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22562616	2
#22560343	2
#22559926	2
C₂²×#97×C₆₇	2
#20762	67