Subgroups — #30774390

Order 17152.

366 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	C₂×#16863794	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863217	C₂	1	1	yes	maximal, characteristic
3	#22558314	C₂	1	2	yes	maximal
4	#22558309	C₂	1	2	yes	maximal
5	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	—	2	1	no	—
7	C₂×#14523983	C₂²	1	2	yes	—
8	#16863794	—	2	2	no	—
9	#16863794	C₄	1	4	yes	complemented
10	C₂×#14523975	—	2	1	no	—
11	C₂×#14523975	C₂²	1	2	yes	—
12	#16863217	—	2	2	no	—
13	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
14	#427×C₆₇	C₂²	1	1	yes	characteristic
15	#405×C₆₇	—	2	1	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#339×C₆₇	—	2	1	no	—
18	C₂²×Dic₁₃₄	—	4	1	no	—
19	#14524043	—	2	4	no	—
20	#14524043	—	4	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	—	4	1	no	—
23	#14523983	—	2	2	no	—
24	#14523983	—	2	2	no	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	#14523975	—	2	4	no	—
27	#14523975	—	2	2	no	—
28	#14523975	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₆₇	—	2	1	no	—
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	#117×C₆₇	—	2	1	no	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	#108×C₆₇	—	2	1	no	—
35	#108×C₆₇	Q₈	1	2	yes	complemented
36	#107×C₆₇	Q₈	1	2	yes	complemented
37	#106×C₆₇	—	2	1	no	—
38	#97×C₆₇	—	4	1	no	—
39	C₄×C₈×C₆₇	—	4	1	no	—
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	D₄	1	2	yes	—
42	C₂×Dic₁₃₄	—	4	4	no	—
43	C₂×Dic₁₃₄	—	8	4	no	—
44	C₂²×Dic₆₇	—	4	2	no	—
45	C₄⋊Dic₆₇	—	2	4	no	—
46	C₄⋊Dic₆₇	—	4	4	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	C₆₇⋊C₁₆	—	4	2	no	—
49	C₂²×C₄×C₆₇	—	2	1	no	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₆₇	—	2	2	no	—
52	M₁₆×C₆₇	—	4	1	no	—
53	C₂×C₈×C₆₇	—	4	1	no	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
63	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
64	C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂²⋊C₄×C₆₇	—	4	1	no	—
66	C₄²×C₆₇	—	2	1	no	—
67	C₄²×C₆₇	—	4	1	no	—
68	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₆₇	—	4	4	no	—
80	C₂×Dic₆₇	—	8	4	no	—
81	Dic₁₃₄	—	8	8	no	—
82	C₈×C₆₇	—	4	2	no	—
83	C₈×C₆₇	#103	1	2	yes	—
84	C₈×C₆₇	#103	1	2	yes	—
85	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₆₇	#124	1	1	yes	characteristic
88	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
89	C₂²×C₆₇	—	2	2	no	—
90	C₂²×C₆₇	#418	1	1	yes	characteristic
91	C₂²×C₆₇	#476	1	1	yes	characteristic
92	C₂²×C₆₇	#477	1	1	yes	characteristic
93	C₄×C₆₇	—	2	4	no	—
94	C₄×C₆₇	—	8	2	no	—
95	C₄×C₆₇	#391	1	1	yes	characteristic
96	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
97	C₄×C₆₇	#476	1	1	yes	characteristic
98	C₄×C₆₇	#477	1	1	yes	characteristic
99	Dic₆₇	—	8	4	no	—
100	#13272	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	—	2	1	no	—
103	C₂×C₆₇	#1987	1	1	yes	characteristic
104	C₂×C₆₇	#2230	1	2	yes	—
105	C₂×#498	—	67	1	no	—
106	C₂×#364	—	67	1	no	—
107	#1772	D₆₇	1	1	yes	characteristic
108	#1386	—	67	2	no	—
109	#1383	—	67	2	no	—
110	C₆₇	#13272	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊Q₈	—	134	1	no	—
112	C₂×M₃₂	—	67	2	no	—
113	#498	—	67	4	no	—
114	#498	—	134	2	no	—
115	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
116	#427	C₂×D₆₇	1	1	yes	characteristic
117	C₂×#106	—	67	2	no	—
118	C₂×#106	—	134	1	no	—
119	#405	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
121	#364	—	134	2	no	—
122	#339	—	2	1	no	—
123	C₂²×Q₈	—	268	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	268	2	no	—
128	C₄⋊Q₈	—	134	4	no	—
129	#117	—	2	1	no	—
130	C₂×C₄⋊C₄	—	268	1	no	—
131	C₂×C₄⋊C₄	—	134	2	no	—
132	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
133	M₃₂	—	134	2	no	—
134	M₃₂	—	134	2	no	—
135	C₂×C₁₆	—	134	2	no	—
136	#108	—	2	1	no	—
137	#108	Dic₁₃₄	1	2	yes	complemented
138	#107	Dic₁₃₄	1	2	yes	complemented
139	#106	—	2	1	no	—
140	#106	—	67	4	no	—
141	#106	—	134	4	no	—
142	#106	—	134	2	no	—
143	#97	—	4	1	no	—
144	C₄×C₈	—	4	1	no	—
145	C₄×C₈	C₄×D₆₇	1	2	yes	—
146	C₄×C₈	D₂₆₈	1	2	yes	—
147	C₂×Q₈	—	268	4	no	—
148	C₂×Q₈	—	536	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	268	2	no	—
151	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	4	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
156	C₂×C₈	#11761616	1	1	yes	characteristic
157	C₂×C₈	#11761616	1	1	yes	characteristic
158	C₂×C₈	#11761618	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
160	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
161	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
162	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
163	C₂×C₈	#11761628	1	1	yes	characteristic
164	C₂×C₈	#11761628	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	268	4	no	—
167	C₄⋊C₄	—	268	2	no	—
168	C₄⋊C₄	—	134	4	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
173	C₁₆	—	268	2	no	—
174	C₂³	#14524068	1	1	yes	characteristic
175	Q₈	—	536	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	268	4	no	—
180	C₂×C₄	—	536	4	no	—
181	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
182	C₂×C₄	#14524022	1	1	yes	characteristic
183	C₂×C₄	#14524022	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
185	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
186	C₂×C₄	#14524068	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	#14523966	1	2	yes	—
189	C₈	#14523966	1	2	yes	—
190	C₈	C₄×Dic₁₃₄	1	1	yes	characteristic
191	C₈	C₄×Dic₁₃₄	1	1	yes	characteristic
192	C₈	#14524031	1	1	yes	characteristic
193	C₈	#14524043	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#16863400	1	1	yes	characteristic
196	C₂²	#16863686	1	1	yes	characteristic
197	C₂²	#16863691	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	2	no	—
200	C₄	—	536	4	no	—
201	C₄	#16863302	1	1	yes	characteristic
202	C₄	C₂×#14523966	1	1	yes	characteristic
203	C₄	#16863686	1	1	yes	characteristic
204	C₄	#16863691	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#22561157	1	1	yes	characteristic
208	C₂	#22562380	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863686
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1772	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1772	D₆₇
hypocenter	C₆₇	#13272
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
9. #16863794	C₄	semidirect
35. #108×C₆₇	Q₈	semidirect
36. #107×C₆₇	Q₈	semidirect
110. C₆₇	#13272	semidirect
137. #108	Dic₁₃₄	semidirect
138. #107	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863794	2
C₂×#16863217	2
#22558314	2
#22558309	2
C₆₇×#1772	2
#13272	67