Subgroups — #30771207

Order 17152.

296 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×#16863217	C₂	1	1	yes	maximal, characteristic
2	#22561814	C₂	1	2	yes	maximal
3	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
4	#22558295	C₂	1	2	yes	maximal
5	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523983	C₂²	1	1	yes	characteristic
8	C₂×#14523983	C₂²	1	1	yes	characteristic
9	#16863217	C₄	1	2	yes	complemented
10	#16863217	C₄	1	2	yes	complemented
11	#16863174	—	2	4	no	—
12	#16863172	C₄	1	2	yes	complemented
13	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
14	C₂×#108×C₆₇	C₄	1	1	yes	characteristic
15	#427×C₆₇	C₄	1	1	yes	characteristic
16	#405×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#339×C₆₇	C₄	1	2	yes	—
19	#14523983	—	2	2	no	—
20	#14523983	—	2	1	no	—
21	#14523983	—	2	1	no	—
22	#14523983	C₂×C₄	1	4	yes	—
23	#14523983	D₄	1	2	yes	—
24	#14523983	Q₈	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	#117×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	#108×C₆₇	—	2	1	no	—
34	#108×C₆₇	—	2	1	no	—
35	#107×C₆₇	—	2	1	no	—
36	#106×C₆₇	—	2	1	no	—
37	#97×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₆₇⋊C₁₆	—	2	2	no	—
44	C₆₇⋊C₁₆	—	2	4	no	—
45	C₆₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₆₇	—	2	1	no	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₆₇	—	2	2	no	—
49	M₁₆×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
55	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
56	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	#41	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
63	C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂²⋊C₄×C₆₇	—	2	2	no	—
65	C₄²×C₆₇	—	2	1	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	C₄²	1	1	yes	characteristic
68	C₄²×C₆₇	C₄²	1	1	yes	characteristic
69	C₂³×C₆₇	#131	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	2	1	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	—	4	2	no	—
74	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
80	C₈×C₆₇	—	4	2	no	—
81	C₈×C₆₇	#97	1	1	yes	characteristic
82	C₈×C₆₇	#97	1	1	yes	characteristic
83	C₈×C₆₇	#97	1	1	yes	characteristic
84	C₈×C₆₇	#97	1	1	yes	characteristic
85	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₈×C₆₇	#117	1	1	yes	characteristic
87	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
88	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
89	C₂²×C₆₇	—	4	1	no	—
90	C₂²×C₆₇	#405	1	1	yes	characteristic
91	C₂²×C₆₇	#436	1	1	yes	characteristic
92	C₂²×C₆₇	#446	1	1	yes	characteristic
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	—	4	1	no	—
95	C₄×C₆₇	—	8	2	no	—
96	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
98	C₄×C₆₇	#436	1	1	yes	characteristic
99	C₄×C₆₇	#446	1	1	yes	characteristic
100	#12667	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	—	2	1	no	—
103	C₂×C₆₇	—	2	1	no	—
104	C₂×C₆₇	#1770	1	1	yes	characteristic
105	C₂×#364	—	67	1	no	—
106	#2110	—	67	2	no	—
107	C₄×M₃₂	—	67	1	no	—
108	#1772	D₆₇	1	1	yes	characteristic
109	#1378	—	67	2	no	—
110	C₆₇	#12667	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₃₂	—	67	1	no	—
112	C₂×M₃₂	—	67	1	no	—
113	C₂×M₃₂	—	67	2	no	—
114	C₂×#108	Dic₆₇	1	1	yes	characteristic
115	#427	Dic₆₇	1	1	yes	characteristic
116	#405	C₂×D₆₇	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	#364	—	67	2	no	—
119	#364	—	67	2	no	—
120	#347	—	134	4	no	—
121	#346	—	67	2	no	—
122	C₄×C₁₆	—	67	2	no	—
123	#339	Dic₆₇	1	2	yes	—
124	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
125	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	#117	C₂×Dic₆₇	1	2	yes	—
128	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
129	M₃₂	—	67	2	no	—
130	M₃₂	—	67	2	no	—
131	M₃₂	—	67	4	no	—
132	M₃₂	—	134	2	no	—
133	M₃₂	—	134	1	no	—
134	M₃₂	—	134	1	no	—
135	C₂×C₁₆	—	67	2	no	—
136	C₂×C₁₆	—	67	2	no	—
137	C₂×C₁₆	—	134	2	no	—
138	#108	—	2	1	no	—
139	#108	—	2	1	no	—
140	#107	—	2	1	no	—
141	#106	—	2	1	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
145	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
146	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
147	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
155	C₂×C₈	#11761612	1	1	yes	characteristic
156	C₂×C₈	#11761612	1	1	yes	characteristic
157	C₂×C₈	#11761612	1	1	yes	characteristic
158	C₂×C₈	#11761612	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
161	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
163	C₂×C₈	#11761627	1	1	yes	characteristic
164	C₂×C₈	#11761629	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
170	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
171	C₁₆	—	67	4	no	—
172	C₁₆	—	134	2	no	—
173	C₁₆	—	134	4	no	—
174	C₂³	#14524050	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
180	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
182	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
183	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
184	C₂×C₄	#14524050	1	1	yes	characteristic
185	C₈	—	4	2	no	—
186	C₈	#14523954	1	1	yes	characteristic
187	C₈	#14523954	1	1	yes	characteristic
188	C₈	#14523954	1	1	yes	characteristic
189	C₈	#14523954	1	1	yes	characteristic
190	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
191	C₈	#14524001	1	1	yes	characteristic
192	C₈	#14524005	1	1	yes	characteristic
193	C₈	#14524008	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#16863352	1	1	yes	characteristic
196	C₂²	#16863474	1	1	yes	characteristic
197	C₂²	#16863527	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
202	C₄	C₂×#14523954	1	1	yes	characteristic
203	C₄	#16863474	1	1	yes	characteristic
204	C₄	#16863527	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#22560078	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14524001
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1772	C₂
socle	C₂×C₆₇	#1770
prime radical	C₂³×C₆₇	#131
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1772	D₆₇
hypocenter	C₆₇	#12667
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
9. #16863217	C₄	semidirect
10. #16863217	C₄	semidirect
12. #16863172	C₄	semidirect
13. C₄×C₆₇⋊C₁₆	C₄	semidirect
110. C₆₇	#12667	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22561814	2
C₄×#14523983	2
#22558295	2
C₆₇×#1772	2
#12667	67