Subgroups — #30767261

Order 17152.

330 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562008	C₂	1	1	yes	maximal, characteristic
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558264	C₂	1	4	yes	maximal, complemented
4	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863420	—	2	1	no	—
6	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
7	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₄×C₆₇⋊C₁₆	—	2	1	no	—
9	#16863149	—	2	1	no	—
10	#16863149	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	#500×C₆₇	—	2	1	no	—
13	#500×C₆₇	—	2	1	no	—
14	#500×C₆₇	C₄	1	2	yes	—
15	#500×C₆₇	C₂²	1	2	yes	—
16	C₂×#107×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	—	4	1	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
23	C₂²×Q₈×C₆₇	—	4	1	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₄⋊Q₈×C₆₇	—	2	2	no	—
27	C₄⋊Q₈×C₆₇	—	2	2	no	—
28	C₄⋊Q₈×C₆₇	—	4	1	no	—
29	C₄⋊Q₈×C₆₇	—	4	1	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	#107×C₆₇	—	2	2	no	—
34	#107×C₆₇	—	2	2	no	—
35	#107×C₆₇	—	2	2	no	—
36	#107×C₆₇	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	#11761612	—	8	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₆₇⋊C₁₆	—	4	2	no	—
44	C₂×Q₈×C₆₇	—	4	2	no	—
45	C₂×Q₈×C₆₇	—	4	2	no	—
46	C₂×Q₈×C₆₇	—	8	2	no	—
47	C₂×Q₈×C₆₇	—	8	2	no	—
48	C₂²×C₄×C₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₆₇	—	2	4	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₆₇	D₈	1	1	yes	characteristic
64	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₆₇	#102	1	1	yes	characteristic
67	Q₈×C₆₇	—	8	4	no	—
68	Q₈×C₆₇	—	8	2	no	—
69	Q₈×C₆₇	—	8	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	1	no	—
72	C₂×C₄×C₆₇	—	2	1	no	—
73	C₂×C₄×C₆₇	—	4	4	no	—
74	C₂×C₄×C₆₇	—	8	4	no	—
75	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
78	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
79	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₆₇	—	2	1	no	—
84	C₈×C₆₇	—	2	1	no	—
85	C₈×C₆₇	#101	1	2	yes	—
86	C₈×C₆₇	#137	1	2	yes	—
87	C₆₇⋊C₈	—	8	2	no	—
88	C₂²×C₆₇	—	2	1	no	—
89	C₂²×C₆₇	—	2	1	no	—
90	C₂²×C₆₇	#357	1	2	yes	—
91	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
92	C₄×C₆₇	—	2	1	no	—
93	C₄×C₆₇	—	2	1	no	—
94	C₄×C₆₇	—	8	4	no	—
95	C₄×C₆₇	#357	1	2	yes	—
96	C₄×C₆₇	#362	1	2	yes	—
97	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
98	C₄×C₆₇	#419	1	1	yes	characteristic
99	C₄×C₆₇	C₂×D₁₆	1	1	yes	characteristic
100	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
101	#11971	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	—	2	2	no	—
103	C₂×C₆₇	#1546	1	1	yes	characteristic
104	C₂×C₆₇	C₂×#357	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
106	C₂×#500	D₆₇	1	1	yes	characteristic
107	#2162	—	67	1	no	—
108	#1567	—	67	1	no	—
109	#1364	—	67	4	no	—
110	C₆₇	#11971	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊Q₈	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	—	2	1	no	—
114	#500	Dic₆₇	1	2	yes	—
115	#500	C₂×D₆₇	1	2	yes	—
116	C₂×#107	—	2	1	no	—
117	C₂×#107	Dic₆₇	1	2	yes	—
118	#423	—	134	1	no	—
119	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
120	C₄⋊C₁₆	—	67	2	no	—
121	C₂²⋊C₁₆	—	134	1	no	—
122	C₄×C₁₆	—	134	1	no	—
123	#335	—	67	2	no	—
124	#335	—	134	1	no	—
125	C₂²×Q₈	—	4	1	no	—
126	C₂×M₁₆	—	268	1	no	—
127	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
128	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	4	1	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
136	C₂×C₁₆	—	134	2	no	—
137	#107	—	2	2	no	—
138	#107	—	2	2	no	—
139	#107	—	2	2	no	—
140	#107	C₂×Dic₆₇	1	4	yes	—
141	C₄⋊C₈	—	268	1	no	—
142	C₄⋊C₈	—	134	2	no	—
143	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	8	2	no	—
150	C₂×Q₈	—	8	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
153	C₂²×C₄	#11761613	1	1	yes	characteristic
154	C₂²×C₄	#11761616	1	1	yes	characteristic
155	M₁₆	—	536	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	268	2	no	—
159	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
161	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
168	C₄²	#11761613	1	1	yes	characteristic
169	C₄²	#11761616	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
171	C₁₆	—	268	2	no	—
172	C₂³	#14523964	1	1	yes	characteristic
173	Q₈	—	8	2	no	—
174	Q₈	—	8	2	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	#14523964	1	1	yes	characteristic
182	C₂×C₄	#14523964	1	1	yes	characteristic
183	C₂×C₄	#14523964	1	1	yes	characteristic
184	C₂×C₄	#14523984	1	2	yes	—
185	C₂×C₄	#14523987	1	2	yes	—
186	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
187	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
188	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	536	2	no	—
192	C₈	#14523962	1	2	yes	—
193	C₈	#14524074	1	2	yes	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#16863197	1	2	yes	—
197	C₂²	C₂×#14523964	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	#16863197	1	2	yes	—
202	C₄	#16863212	1	2	yes	—
203	C₄	C₂×#14523962	1	1	yes	characteristic
204	C₄	#16863405	1	1	yes	characteristic
205	C₄	C₂×#14523984	1	1	yes	characteristic
206	C₄	C₂×#14523987	1	1	yes	characteristic
207	C₂	—	2	2	no	—
208	C₂	#22558881	1	1	yes	characteristic
209	C₂	C₂×#16863197	1	1	yes	characteristic
210	C₂	C₂×#16863212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#11971
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #22558264	C₂	semidirect
110. C₆₇	#11971	semidirect

Maximal subgroups

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

subgroup	index
#22562008	2
#22558968	2
#22558264	2
C₂×#500×C₆₇	2
#11971	67