Subgroups — #30749492

Order 17152.

422 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	2	yes	maximal
2	#22558044	C₂	1	4	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
11	C₄³×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₈	—	2	4	no	—
13	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
14	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
15	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₆₇	—	2	4	no	—
26	C₄×C₈×C₆₇	—	2	2	no	—
27	C₄×C₈×C₆₇	—	2	4	no	—
28	C₄×C₈×C₆₇	C₈	1	4	yes	complemented
29	C₂×C₆₇⋊C₈	—	4	4	no	—
30	C₂×C₆₇⋊C₈	—	4	4	no	—
31	C₂×C₆₇⋊C₈	—	4	4	no	—
32	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	—	2	2	no	—
43	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
45	C₄²×C₆₇	—	2	2	no	—
46	C₄²×C₆₇	—	2	2	no	—
47	C₄²×C₆₇	—	2	2	no	—
48	C₄²×C₆₇	C₂×C₈	1	2	yes	—
49	C₄²×C₆₇	M₁₆	1	2	yes	—
50	C₄²×C₆₇	D₈	1	2	yes	—
51	C₄²×C₆₇	SD₁₆	1	4	yes	—
52	C₄²×C₆₇	Q₁₆	1	2	yes	—
53	C₄²×C₆₇	#41	1	4	yes	—
54	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	2	1	no	—
56	C₂×C₄×C₆₇	—	2	1	no	—
57	C₂×C₄×C₆₇	—	2	1	no	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	—	2	1	no	—
61	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
63	C₂×C₄×C₆₇	#106	1	2	yes	—
64	C₂×C₄×C₆₇	#106	1	2	yes	—
65	C₂×C₄×C₆₇	#107	1	2	yes	—
66	C₂×C₄×C₆₇	#107	1	2	yes	—
67	C₂×C₄×C₆₇	#108	1	2	yes	—
68	C₂×C₄×C₆₇	#108	1	2	yes	—
69	C₂×C₄×C₆₇	#117	1	2	yes	—
70	C₂×C₄×C₆₇	#124	1	2	yes	—
71	C₂×C₄×C₆₇	#125	1	2	yes	—
72	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₈×C₆₇	—	2	4	no	—
79	C₈×C₆₇	—	2	4	no	—
80	C₆₇⋊C₈	—	8	4	no	—
81	C₂²×C₆₇	#382	1	1	yes	characteristic
82	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₂²×C₆₇	#423	1	1	yes	characteristic
84	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
85	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
86	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
87	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
88	C₄×C₆₇	—	2	2	no	—
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	#334	1	4	yes	—
92	C₄×C₆₇	#335	1	4	yes	—
93	C₄×C₆₇	#431	1	2	yes	—
94	C₄×C₆₇	#432	1	2	yes	—
95	C₄×C₆₇	#486	1	2	yes	—
96	C₄×C₆₇	#487	1	2	yes	—
97	#8784	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	C₂×#334	1	1	yes	characteristic
99	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
100	C₂×C₆₇	#1566	1	1	yes	characteristic
101	C₂×C₆₇	#1567	1	1	yes	characteristic
102	C₂×C₆₇	#1831	1	1	yes	characteristic
103	C₂×C₆₇	#1834	1	1	yes	characteristic
104	C₂×C₆₇	#1837	1	1	yes	characteristic
105	#1842	—	67	2	no	—
106	C₄²×C₈	D₆₇	1	1	yes	characteristic
107	#1277	—	67	4	no	—
108	C₆₇	#8784	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄⋊C₈	—	67	4	no	—
110	C₂×C₄⋊C₈	—	134	2	no	—
111	C₂×C₄⋊C₈	—	134	2	no	—
112	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
113	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
114	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
116	C₄³	C₂×D₆₇	1	1	yes	characteristic
117	C₂²×C₈	—	134	4	no	—
118	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
119	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
120	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
121	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
122	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
123	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
127	C₄⋊C₈	—	268	4	no	—
128	C₄⋊C₈	—	268	4	no	—
129	C₄⋊C₈	—	134	8	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	4	no	—
133	C₄×C₈	C₆₇⋊C₈	1	4	yes	complemented
134	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
136	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
137	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	268	4	no	—
146	C₂×C₈	—	268	4	no	—
147	C₂×C₈	—	268	4	no	—
148	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
149	C₂×C₈	#11761612	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
154	C₄²	#11761612	1	2	yes	—
155	C₄²	D₅₃₆	1	2	yes	—
156	C₄²	#11761617	1	4	yes	—
157	C₄²	Dic₂₆₈	1	2	yes	—
158	C₄²	#11761628	1	4	yes	—
159	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
167	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
168	C₂×C₄	#14523975	1	2	yes	—
169	C₂×C₄	#14523975	1	2	yes	—
170	C₂×C₄	#14523977	1	2	yes	—
171	C₂×C₄	#14523977	1	2	yes	—
172	C₂×C₄	#14523979	1	2	yes	—
173	C₂×C₄	#14523979	1	2	yes	—
174	C₂×C₄	#14524002	1	2	yes	—
175	C₂×C₄	#14524031	1	2	yes	—
176	C₂×C₄	#14524035	1	2	yes	—
177	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
178	C₂×C₄	C₂×#11761612	1	2	yes	—
179	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
180	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
181	C₂×C₄	C₂×#11761617	1	2	yes	—
182	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
183	C₈	—	2	4	no	—
184	C₈	—	2	4	no	—
185	C₈	—	536	4	no	—
186	C₂²	#16863262	1	1	yes	characteristic
187	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
188	C₂²	#16863420	1	1	yes	characteristic
189	C₂²	C₂×#14523975	1	1	yes	characteristic
190	C₂²	C₂×#14523977	1	1	yes	characteristic
191	C₂²	C₂×#14523979	1	1	yes	characteristic
192	C₂²	C₂×#14523979	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	#16863147	1	4	yes	—
197	C₄	#16863149	1	4	yes	—
198	C₄	#16863449	1	2	yes	—
199	C₄	#16863453	1	2	yes	—
200	C₄	#16863743	1	2	yes	—
201	C₄	#16863747	1	2	yes	—
202	C₂	C₂×#16863147	1	1	yes	characteristic
203	C₂	C₂×#16863149	1	1	yes	characteristic
204	C₂	#22558965	1	1	yes	characteristic
205	C₂	#22558968	1	1	yes	characteristic
206	C₂	#22560326	1	1	yes	characteristic
207	C₂	#22560337	1	1	yes	characteristic
208	C₂	#22560347	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#8784
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
28. C₄×C₈×C₆₇	C₈	semidirect
108. C₆₇	#8784	semidirect
133. C₄×C₈	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
#22558044	2
C₄²×C₈×C₆₇	2
#8784	67