Subgroups — #30749638

Order 17152.

322 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560346	C₂	1	2	yes	maximal
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558964	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863160	—	2	4	no	—
10	#16863149	—	2	2	no	—
11	#16863146	—	2	4	no	—
12	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
16	#423×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#335×C₆₇	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
21	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	2	no	—
34	C₄⋊C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₇	D₄	1	2	yes	—
38	#11761612	—	4	4	no	—
39	C₂×C₆₇⋊C₈	—	2	2	no	—
40	C₂×C₆₇⋊C₈	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	4	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	M₁₆×C₆₇	—	4	4	no	—
51	C₂×C₈×C₆₇	—	2	4	no	—
52	C₂×C₈×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#135	1	2	yes	—
69	C₂×C₄×C₆₇	#135	1	2	yes	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	—	8	2	no	—
72	C₈×C₆₇	#108	1	4	yes	—
73	C₆₇⋊C₈	—	2	2	no	—
74	C₆₇⋊C₈	—	2	2	no	—
75	C₆₇⋊C₈	—	2	2	no	—
76	C₆₇⋊C₈	—	8	2	no	—
77	C₆₇⋊C₈	#108	1	4	yes	—
78	C₂²×C₆₇	—	2	1	no	—
79	C₂²×C₆₇	—	2	1	no	—
80	C₂²×C₆₇	#384	1	1	yes	characteristic
81	C₂²×C₆₇	#499	1	2	yes	—
82	C₄×C₆₇	—	2	1	no	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	#427	1	2	yes	—
85	C₄×C₆₇	C₂×#108	1	2	yes	—
86	C₄×C₆₇	C₂×#108	1	2	yes	—
87	C₄×C₆₇	#495	1	1	yes	characteristic
88	C₄×C₆₇	#495	1	1	yes	characteristic
89	#8824	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	#1846	1	1	yes	characteristic
92	C₂×C₆₇	#1848	1	1	yes	characteristic
93	C₂×C₆₇	#1848	1	1	yes	characteristic
94	#1837	—	67	2	no	—
95	#1567	—	67	1	no	—
96	#1567	D₆₇	1	1	yes	characteristic
97	#1566	—	67	2	no	—
98	C₂×C₈²	—	67	1	no	—
99	C₆₇	#8824	1	1	yes	characteristic, complemented, Sylow
100	#423	—	67	2	no	—
101	#423	C₂×D₆₇	1	2	yes	—
102	C₂×C₄×C₈	—	67	1	no	—
103	C₂×C₄×C₈	—	67	1	no	—
104	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
105	#341	—	134	4	no	—
106	#335	—	2	2	no	—
107	#335	—	134	2	no	—
108	#334	—	134	4	no	—
109	C₈²	—	67	1	no	—
110	C₈²	—	67	1	no	—
111	C₈²	—	67	1	no	—
112	C₈²	—	67	1	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	134	2	no	—
115	C₂²×C₈	—	67	2	no	—
116	C₂²×C₈	—	67	2	no	—
117	C₂²×C₈	Dic₁₃₄	1	2	yes	—
118	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	134	2	no	—
122	C₄⋊C₈	—	134	2	no	—
123	C₄×C₈	—	67	1	no	—
124	C₄×C₈	—	67	1	no	—
125	C₄×C₈	—	67	1	no	—
126	C₄×C₈	—	67	1	no	—
127	C₄×C₈	—	67	2	no	—
128	C₄×C₈	—	67	2	no	—
129	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
130	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
132	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
134	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
135	M₁₆	—	4	4	no	—
136	M₁₆	—	268	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	268	2	no	—
140	C₂×C₈	—	67	2	no	—
141	C₂×C₈	—	67	2	no	—
142	C₂×C₈	—	67	2	no	—
143	C₂×C₈	—	67	2	no	—
144	C₂×C₈	—	134	4	no	—
145	C₂×C₈	—	134	2	no	—
146	C₂×C₈	—	134	2	no	—
147	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
148	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
149	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
150	C₄²	D₄×D₆₇	1	1	yes	characteristic
151	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	C₄²	Q₈×D₆₇	1	1	yes	characteristic
153	C₂³	#14524042	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
158	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#14524039	1	1	yes	characteristic
160	C₂×C₄	#14524041	1	1	yes	characteristic
161	C₂×C₄	#14524042	1	1	yes	characteristic
162	C₂×C₄	#14524062	1	2	yes	—
163	C₂×C₄	#14524066	1	2	yes	—
164	C₈	—	2	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	536	2	no	—
167	C₈	—	67	4	no	—
168	C₈	—	134	2	no	—
169	C₈	—	134	2	no	—
170	C₈	—	134	2	no	—
171	C₈	#14523978	1	4	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#16863265	1	1	yes	characteristic
175	C₂²	#16863797	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#16863433	1	2	yes	—
179	C₄	#108×D₆₇	1	2	yes	—
180	C₄	C₂×#14523978	1	2	yes	—
181	C₄	#16863779	1	1	yes	characteristic
182	C₄	#16863781	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	#22560372	1	1	yes	characteristic
185	C₂	#22560379	1	1	yes	characteristic
186	C₂	#22560380	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#8824
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
99. C₆₇	#8824	semidirect

Maximal subgroups

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

subgroup	index
#22560346	2
#22558968	2
#22558964	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1567	2
#8824	67