Subgroups — #30767265

Order 17152.

330 conjugacy classes of subgroups.

The subgroup lattice has 208 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	#22558267	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	4	no	—
47	C₂²×C₄×C₆₇	—	4	2	no	—
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	4	2	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₆₇	D₈	1	1	yes	characteristic
63	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₆₇	#102	1	1	yes	characteristic
66	Q₈×C₆₇	—	8	4	no	—
67	Q₈×C₆₇	—	8	4	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	—	8	4	no	—
73	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
77	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
81	C₈×C₆₇	—	2	1	no	—
82	C₈×C₆₇	—	2	1	no	—
83	C₈×C₆₇	#101	1	2	yes	—
84	C₈×C₆₇	#137	1	2	yes	—
85	C₆₇⋊C₈	—	8	2	no	—
86	C₂²×C₆₇	—	2	1	no	—
87	C₂²×C₆₇	—	2	1	no	—
88	C₂²×C₆₇	#358	1	2	yes	—
89	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
90	C₄×C₆₇	—	2	1	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	8	4	no	—
93	C₄×C₆₇	#358	1	2	yes	—
94	C₄×C₆₇	#361	1	2	yes	—
95	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
96	C₄×C₆₇	#419	1	1	yes	characteristic
97	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
98	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
99	#11972	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	2	no	—
101	C₂×C₆₇	#1546	1	1	yes	characteristic
102	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
103	C₂×C₆₇	C₂×#361	1	1	yes	characteristic
104	C₂×#500	D₆₇	1	1	yes	characteristic
105	#2162	—	67	1	no	—
106	#1567	—	67	1	no	—
107	#1365	—	67	4	no	—
108	C₆₇	#11972	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄⋊Q₈	—	2	1	no	—
110	#500	—	2	1	no	—
111	#500	—	2	1	no	—
112	#500	Dic₆₇	1	2	yes	—
113	#500	C₂×D₆₇	1	2	yes	—
114	C₂×#107	—	2	1	no	—
115	C₂×#107	Dic₆₇	1	2	yes	—
116	#423	—	134	1	no	—
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₄⋊C₁₆	—	67	2	no	—
119	C₂²⋊C₁₆	—	134	1	no	—
120	C₄×C₁₆	—	134	1	no	—
121	#335	—	67	2	no	—
122	#335	—	134	1	no	—
123	C₂²×Q₈	—	4	1	no	—
124	C₂×M₁₆	—	268	1	no	—
125	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	4	1	no	—
130	C₄⋊Q₈	—	4	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
134	C₂×C₁₆	—	134	2	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	C₂×Dic₆₇	1	4	yes	—
139	C₄⋊C₈	—	268	1	no	—
140	C₄⋊C₈	—	134	2	no	—
141	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
144	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	8	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
150	C₂²×C₄	#11761613	1	1	yes	characteristic
151	C₂²×C₄	#11761616	1	1	yes	characteristic
152	M₁₆	—	536	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	268	2	no	—
156	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
158	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
165	C₄²	#11761613	1	1	yes	characteristic
166	C₄²	#11761616	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
168	C₁₆	—	268	2	no	—
169	C₂³	#14523964	1	1	yes	characteristic
170	Q₈	—	8	4	no	—
171	Q₈	—	8	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	#14523964	1	1	yes	characteristic
178	C₂×C₄	#14523964	1	1	yes	characteristic
179	C₂×C₄	#14523964	1	1	yes	characteristic
180	C₂×C₄	#14523986	1	2	yes	—
181	C₂×C₄	#14523989	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
183	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	536	2	no	—
188	C₈	#14523962	1	2	yes	—
189	C₈	#14524074	1	2	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#16863200	1	2	yes	—
193	C₂²	C₂×#14523964	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	4	no	—
197	C₄	#16863200	1	2	yes	—
198	C₄	#16863209	1	2	yes	—
199	C₄	C₂×#14523962	1	1	yes	characteristic
200	C₄	#16863405	1	1	yes	characteristic
201	C₄	C₂×#14523986	1	1	yes	characteristic
202	C₄	C₂×#14523989	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#22558881	1	1	yes	characteristic
205	C₂	C₂×#16863200	1	1	yes	characteristic
206	C₂	C₂×#16863209	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	#11972
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. #22558267	C₂	semidirect
108. C₆₇	#11972	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
#22558267	2
C₂×#500×C₆₇	2
#11972	67