Subgroups — #30772274

Order 17152.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863223	C₂	1	2	yes	maximal
2	C₂×#16863221	C₂	1	2	yes	maximal
3	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
4	#22560373	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14523977	C₂²	1	1	yes	characteristic
8	C₂×#14523977	C₂²	1	1	yes	characteristic
9	C₂×#14523975	—	2	1	no	—
10	#16863262	—	2	1	no	—
11	#16863223	—	2	4	no	—
12	#16863221	—	2	4	no	—
13	#16863172	C₄	1	4	yes	complemented
14	C₂×#107×C₆₇	—	2	1	no	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#384×C₆₇	—	2	1	no	—
18	C₂×C₄⋊Dic₆₇	—	2	1	no	—
19	C₂×C₄⋊Dic₆₇	—	2	1	no	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
21	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
22	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	—
23	#14523977	—	2	2	no	—
24	#14523977	—	2	2	no	—
25	#14523975	—	4	2	no	—
26	C₆₇⋊#95	—	4	2	no	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	#107×C₆₇	—	2	2	no	—
33	#107×C₆₇	—	2	2	no	—
34	#107×C₆₇	—	4	2	no	—
35	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₇	Q₈	1	2	yes	—
37	C₂²×Dic₆₇	—	4	1	no	—
38	C₂²×Dic₆₇	—	4	1	no	—
39	C₄⋊Dic₆₇	—	4	1	no	—
40	C₄⋊Dic₆₇	—	4	1	no	—
41	C₄⋊Dic₆₇	—	4	1	no	—
42	C₄⋊Dic₆₇	—	4	1	no	—
43	C₆₇⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
44	C₂²×C₄×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	D₈	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₇	D₈	1	2	yes	—
49	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	4	2	no	—
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₄⋊C₄×C₆₇	—	8	4	no	—
57	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₄²×C₆₇	Q₁₆	1	2	yes	—
59	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	8	2	no	—
63	C₂×C₄×C₆₇	—	8	4	no	—
64	C₂×C₄×C₆₇	#107	1	4	yes	—
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₆₇	—	8	1	no	—
72	C₂×Dic₆₇	—	8	1	no	—
73	C₂×Dic₆₇	—	8	2	no	—
74	C₂×Dic₆₇	—	8	2	no	—
75	C₈×C₆₇	#107	1	4	yes	—
76	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
78	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
79	C₂²×C₆₇	#493	1	1	yes	characteristic
80	C₂²×C₆₇	#500	1	1	yes	characteristic
81	C₂²×C₆₇	#509	1	2	yes	—
82	C₂²×C₆₇	#510	1	2	yes	—
83	C₄×C₆₇	—	2	4	no	—
84	C₄×C₆₇	—	16	2	no	—
85	C₄×C₆₇	#384	1	1	yes	characteristic
86	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
87	C₄×C₆₇	#494	1	1	yes	characteristic
88	C₄×C₆₇	#500	1	1	yes	characteristic
89	Dic₆₇	—	16	1	no	—
90	Dic₆₇	—	16	1	no	—
91	#12908	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	#1845	1	1	yes	characteristic
93	C₂×C₆₇	#2159	1	2	yes	—
94	C₂×C₆₇	#2250	1	1	yes	characteristic
95	C₂×C₆₇	#2251	1	1	yes	characteristic
96	C₂×C₆₇	#2255	1	2	yes	—
97	C₂×#367	—	67	2	no	—
98	C₂×#366	—	67	2	no	—
99	C₂×#346	—	67	1	no	—
100	#1846	—	67	1	no	—
101	#1845	D₆₇	1	1	yes	characteristic
102	C₆₇	#12908	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₁₆	—	67	2	no	—
104	C₂×#107	—	2	1	no	—
105	C₂×#107	—	67	1	no	—
106	C₂×#107	—	67	1	no	—
107	C₂×#107	C₂×D₆₇	1	2	yes	—
108	C₂×#106	—	134	1	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	#384	—	2	1	no	—
111	#382	—	134	1	no	—
112	#367	—	134	4	no	—
113	#366	—	134	4	no	—
114	#346	—	67	4	no	—
115	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	4	2	no	—
119	C₂×C₄⋊C₄	—	134	1	no	—
120	C₂×C₄⋊C₄	—	134	1	no	—
121	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂×C₁₆	—	67	4	no	—
123	C₂×C₁₆	—	67	4	no	—
124	C₂×C₁₆	—	67	4	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#107	—	4	2	no	—
128	#107	—	134	2	no	—
129	#107	—	134	2	no	—
130	#106	—	268	2	no	—
131	C₄×C₈	Dic₁₃₄	1	2	yes	—
132	C₄×C₈	C₄×D₆₇	1	2	yes	—
133	#95	—	268	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	268	1	no	—
136	C₂²×C₄	—	268	1	no	—
137	C₂²×C₄	#11761613	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
139	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
140	C₂×C₈	#11761613	1	2	yes	—
141	C₂×C₈	#11761618	1	2	yes	—
142	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
143	C₂×C₈	D₄×D₆₇	1	2	yes	—
144	C₂×C₈	Q₈×D₆₇	1	2	yes	—
145	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	8	4	no	—
149	C₄⋊C₄	—	268	1	no	—
150	C₄⋊C₄	—	268	1	no	—
151	C₄⋊C₄	—	268	1	no	—
152	C₄⋊C₄	—	268	1	no	—
153	C₄²	Dic₆₇⋊C₄	1	2	yes	—
154	C₄²	#11761618	1	2	yes	—
155	C₁₆	—	67	8	no	—
156	C₂³	C₂×#11761613	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	536	1	no	—
162	C₂×C₄	—	536	1	no	—
163	C₂×C₄	—	536	2	no	—
164	C₂×C₄	—	536	2	no	—
165	C₂×C₄	#14523976	1	4	yes	—
166	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#14524039	1	1	yes	characteristic
168	C₂×C₄	#14524041	1	1	yes	characteristic
169	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
170	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
171	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
172	C₈	#14523976	1	4	yes	—
173	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
174	C₈	#14524042	1	2	yes	—
175	C₂²	C₂×#14523976	1	1	yes	characteristic
176	C₂²	#16863772	1	1	yes	characteristic
177	C₂²	#16863799	1	1	yes	characteristic
178	C₂²	#16863832	1	2	yes	—
179	C₂²	#16863837	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	16	2	no	—
182	C₄	—	1072	1	no	—
183	C₄	—	1072	1	no	—
184	C₄	#16863265	1	1	yes	characteristic
185	C₄	C₂×#14523976	1	1	yes	characteristic
186	C₄	#16863775	1	1	yes	characteristic
187	C₄	#16863799	1	1	yes	characteristic
188	C₂	#22560368	1	1	yes	characteristic
189	C₂	#22561995	1	2	yes	—
190	C₂	#22562473	1	1	yes	characteristic
191	C₂	#22562478	1	1	yes	characteristic
192	C₂	#22562492	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#12908
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
13. #16863172	C₄	semidirect
43. C₆₇⋊C₁₆	C₄⋊C₄	semidirect
102. C₆₇	#12908	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863223	2
C₂×#16863221	2
C₂×#16863172	2
#22560373	2
C₆₇×#1845	2
#12908	67