Subgroups — #30773842

Order 17152.

298 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863217	C₂	1	1	yes	maximal, characteristic
2	#22561968	C₂	1	2	yes	maximal
3	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#22558298	C₂	1	2	yes	maximal
5	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	#16863217	—	2	2	no	—
10	C₄⋊(C₆₇⋊C₁₆)	—	2	4	no	—
11	C₄×C₆₇⋊C₁₆	—	2	2	no	—
12	C₂×#108×C₆₇	C₄	1	1	yes	characteristic
13	#427×C₆₇	C₄	1	1	yes	characteristic
14	#405×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#339×C₆₇	C₄	1	2	yes	—
17	#14523983	—	2	2	no	—
18	#14523983	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	1	no	—
20	C₂×C₆₇⋊C₁₆	—	2	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	1	no	—
23	C₂×C₆₇⋊C₁₆	—	2	1	no	—
24	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
26	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	#117×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	#108×C₆₇	—	2	2	no	—
32	#107×C₆₇	C₈	1	2	yes	—
33	#106×C₆₇	C₈	1	2	yes	—
34	#97×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	—	2	2	no	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₄×C₈×C₆₇	Q₈	1	2	yes	—
38	C₆₇⋊C₁₆	—	2	2	no	—
39	C₆₇⋊C₁₆	—	2	2	no	—
40	C₆₇⋊C₁₆	—	4	2	no	—
41	C₂²×C₄×C₆₇	—	2	1	no	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	M₁₆×C₆₇	—	2	2	no	—
44	M₁₆×C₆₇	—	2	2	no	—
45	C₂×C₈×C₆₇	—	2	2	no	—
46	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
55	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₄²×C₆₇	—	2	2	no	—
59	C₄²×C₆₇	—	2	1	no	—
60	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
61	C₂³×C₆₇	#131	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	4	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	4	no	—
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
71	C₈×C₆₇	—	4	2	no	—
72	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
73	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
74	C₈×C₆₇	#121	1	1	yes	characteristic
75	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₆₇	#125	1	1	yes	characteristic
77	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₆₇	—	2	2	no	—
79	C₂²×C₆₇	#424	1	1	yes	characteristic
80	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
81	C₂²×C₆₇	#446	1	1	yes	characteristic
82	C₄×C₆₇	—	2	4	no	—
83	C₄×C₆₇	—	4	4	no	—
84	C₄×C₆₇	#389	1	1	yes	characteristic
85	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₆₇	#436	1	1	yes	characteristic
87	C₄×C₆₇	C₈×Q₈	1	1	yes	characteristic
88	#13161	—	67	1	no	maximal, Sylow
89	C₂×C₆₇	—	2	1	no	—
90	C₂×C₆₇	—	2	1	no	—
91	C₂×C₆₇	#1939	1	1	yes	characteristic
92	C₂×C₆₇	#2170	1	2	yes	—
93	C₂×#364	—	67	1	no	—
94	#2151	—	67	2	no	—
95	C₂×C₄×C₁₆	—	67	1	no	—
96	#1772	D₆₇	1	1	yes	characteristic
97	#1379	—	67	2	no	—
98	C₆₇	#13161	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	—	67	2	no	—
100	C₂²×C₁₆	—	67	1	no	—
101	C₂²×C₁₆	—	67	1	no	—
102	C₂×#108	Dic₆₇	1	1	yes	characteristic
103	#427	Dic₆₇	1	1	yes	characteristic
104	#405	C₂×D₆₇	1	2	yes	—
105	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
106	#364	—	134	2	no	—
107	C₄⋊C₁₆	—	134	4	no	—
108	C₄×C₁₆	—	134	2	no	—
109	#339	Dic₆₇	1	2	yes	—
110	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
111	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
113	#117	C₂×Dic₆₇	1	2	yes	—
114	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
115	M₃₂	—	134	2	no	—
116	M₃₂	—	134	2	no	—
117	C₂×C₁₆	—	67	2	no	—
118	C₂×C₁₆	—	67	2	no	—
119	C₂×C₁₆	—	134	2	no	—
120	C₂×C₁₆	—	134	1	no	—
121	C₂×C₁₆	—	134	1	no	—
122	C₂×C₁₆	—	134	1	no	—
123	C₂×C₁₆	—	134	1	no	—
124	#108	—	2	2	no	—
125	#107	C₆₇⋊C₈	1	2	yes	—
126	#106	C₆₇⋊C₈	1	2	yes	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	Dic₁₃₄	1	2	yes	—
130	C₄×C₈	D₂₆₈	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
137	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
138	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
139	C₂×C₈	#11761612	1	2	yes	—
140	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
141	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
144	C₂×C₈	#11761627	1	1	yes	characteristic
145	C₂×C₈	#11761629	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄⋊Dic₆₇	1	2	yes	—
151	C₁₆	—	268	2	no	—
152	C₁₆	—	134	2	no	—
153	C₁₆	—	134	2	no	—
154	C₂³	#14524050	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
159	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
160	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
163	C₂×C₄	#14524050	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
166	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
167	C₈	#14524018	1	1	yes	characteristic
168	C₈	#14524023	1	1	yes	characteristic
169	C₈	#14524034	1	1	yes	characteristic
170	C₈	#14524042	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#16863423	1	1	yes	characteristic
173	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
174	C₂²	#16863527	1	1	yes	characteristic
175	C₄	—	2	4	no	—
176	C₄	—	4	4	no	—
177	C₄	#16863295	1	1	yes	characteristic
178	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
179	C₄	#16863474	1	1	yes	characteristic
180	C₄	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#22560928	1	1	yes	characteristic
184	C₂	#22562046	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1772	C₂
socle	C₂²×C₆₇	C₈×D₄
prime radical	C₂³×C₆₇	#131
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1772	D₆₇
hypocenter	C₆₇	#13161
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
98. C₆₇	#13161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22561968	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558298	2
C₆₇×#1772	2
#13161	67