Subgroups — #30825142

Order 17152.

612 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569341	C₂	1	2	yes	maximal
2	#22558998	C₂	1	6	yes	maximal, complemented
3	#22558968	C₂	1	3	yes	maximal, complemented
4	#22558957	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₆₇⋊C₈	—	2	2	no	—
7	#16863474	—	2	6	no	—
8	#16863449	C₂²	1	6	yes	—
9	#16863420	C₂²	1	6	yes	—
10	#16863149	C₂²	1	6	yes	—
11	#16863149	C₂²	1	1	yes	characteristic
12	#16863149	C₂²	1	3	yes	—
13	#16863147	C₂²	1	6	yes	—
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
17	C₈×D₄×C₆₇	C₄	1	3	yes	—
18	C₈×D₄×C₆₇	C₄	1	3	yes	—
19	#431×C₆₇	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
21	C₂×#11761612	—	2	6	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
23	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
24	C₄⋊(C₆₇⋊C₈)	C₂³	1	6	yes	—
25	C₂²⋊(C₆₇⋊C₈)	—	2	6	no	—
26	C₄×C₆₇⋊C₈	—	2	6	no	—
27	C₂×#41×C₆₇	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
30	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×D₄×C₆₇	C₂×C₄	1	3	yes	—
33	C₄×D₄×C₆₇	D₄	1	3	yes	—
34	#117×C₆₇	Q₈	1	3	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
36	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
37	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
38	C₂²⋊C₈×C₆₇	C₂×C₄	1	6	yes	—
39	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₆₇	C₂³	1	3	yes	—
42	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#11761612	—	8	6	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	6	no	—
46	#41×C₆₇	—	2	4	no	—
47	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₆₇	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
52	C₂×C₈×C₆₇	—	2	6	no	—
53	C₂×C₈×C₆₇	—	4	3	no	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₆₇	C₄⋊C₄	1	3	yes	—
59	C₄⋊C₄×C₆₇	C₄⋊C₄	1	3	yes	—
60	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	6	yes	—
61	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
62	C₄²×C₆₇	C₂×D₄	1	3	yes	—
63	C₄²×C₆₇	C₂×Q₈	1	3	yes	—
64	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂³×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₆₇	#108	1	4	yes	—
67	D₄×C₆₇	—	2	6	no	—
68	C₂×C₄×C₆₇	—	2	3	no	—
69	C₂×C₄×C₆₇	—	2	3	no	—
70	C₂×C₄×C₆₇	—	2	3	no	—
71	C₂×C₄×C₆₇	—	2	3	no	—
72	C₂×C₄×C₆₇	—	2	6	no	—
73	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₆₇	—	2	3	no	—
80	C₈×C₆₇	—	2	1	no	—
81	C₈×C₆₇	—	2	3	no	—
82	C₈×C₆₇	#131	1	2	yes	—
83	C₆₇⋊C₈	—	4	6	no	—
84	C₆₇⋊C₈	—	4	2	no	—
85	C₂²×C₆₇	—	2	3	no	—
86	C₂²×C₆₇	—	2	3	no	—
87	C₂²×C₆₇	—	2	3	no	—
88	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₆₇	—	2	3	no	—
90	C₄×C₆₇	—	2	3	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	C₂×#108	1	6	yes	—
93	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
94	C₄×C₆₇	#568	1	1	yes	characteristic
95	C₄×C₆₇	#576	1	1	yes	characteristic
96	C₄×C₆₇	#578	1	1	yes	characteristic
97	#17837	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	—	4	3	no	—
99	C₂×C₆₇	#2906	1	1	yes	characteristic
100	C₂×C₆₇	#2913	1	1	yes	characteristic
101	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
102	#2991	—	67	2	no	—
103	C₈×#41	D₆₇	1	1	yes	characteristic
104	#1574	—	67	6	no	—
105	#1567	—	67	3	no	—
106	#1564	—	67	3	no	—
107	C₆₇	#17837	1	1	yes	characteristic, complemented, Sylow
108	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
109	C₈×Q₈	—	134	2	no	—
110	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
111	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
112	#436	—	134	6	no	—
113	C₈×D₄	Dic₆₇	1	3	yes	—
114	C₈×D₄	Dic₆₇	1	3	yes	—
115	#431	—	67	6	no	—
116	#431	C₂×D₆₇	1	3	yes	—
117	#423	—	67	6	no	—
118	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
119	#335	—	67	1	no	—
120	#335	—	67	3	no	—
121	#335	—	67	6	no	—
122	#334	—	67	6	no	—
123	C₂×#41	Dic₁₃₄	1	1	yes	characteristic
124	C₂×M₁₆	—	134	6	no	—
125	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
126	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
127	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
128	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
129	C₄×D₄	D₂₆₈	1	3	yes	—
130	C₄×D₄	C₂×Dic₆₇	1	3	yes	—
131	#117	Dic₁₃₄	1	3	yes	—
132	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
133	C₄⋊C₈	—	67	2	no	—
134	C₄⋊C₈	—	67	6	no	—
135	C₄⋊C₈	—	134	6	no	—
136	C₄⋊C₈	C₂×Dic₆₇	1	3	yes	—
137	C₄⋊C₈	C₂×Dic₆₇	1	3	yes	—
138	C₂²⋊C₈	—	134	6	no	—
139	C₂²⋊C₈	C₂×Dic₆₇	1	6	yes	—
140	C₄×C₈	—	134	6	no	—
141	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
142	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
143	C₄×C₈	C₂²×D₆₇	1	3	yes	—
144	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
145	#41	—	2	4	no	—
146	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
147	C₂×D₄	C₄⋊Dic₆₇	1	3	yes	—
148	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
149	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
150	C₂²×C₄	C₂×Dic₁₃₄	1	3	yes	—
151	M₁₆	—	536	6	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	—	4	3	no	—
154	C₂×C₈	—	134	2	no	—
155	C₂×C₈	—	134	6	no	—
156	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
158	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
159	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
160	C₄⋊C₄	C₄⋊Dic₆₇	1	3	yes	—
161	C₄⋊C₄	C₄⋊Dic₆₇	1	3	yes	—
162	C₂²⋊C₄	C₄⋊Dic₆₇	1	6	yes	—
163	C₄²	C₂²×Dic₆₇	1	3	yes	—
164	C₄²	C₂×D₂₆₈	1	3	yes	—
165	C₄²	C₂×Dic₁₃₄	1	3	yes	—
166	C₄²	C₂³×D₆₇	1	1	yes	characteristic
167	C₂³	C₂×C₄⋊Dic₆₇	1	3	yes	—
168	Q₈	#14523979	1	4	yes	—
169	D₄	—	2	6	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	3	no	—
172	C₂×C₄	—	2	3	no	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	2	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
178	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
179	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
180	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
181	C₈	—	2	3	no	—
182	C₈	—	2	1	no	—
183	C₈	—	2	3	no	—
184	C₈	—	268	6	no	—
185	C₈	—	268	2	no	—
186	C₈	#14524050	1	2	yes	—
187	C₂²	—	2	3	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
191	C₄	—	2	3	no	—
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#14523979	1	6	yes	—
195	C₄	C₂×#14524050	1	1	yes	characteristic
196	C₄	#16864163	1	1	yes	characteristic
197	C₄	#16864201	1	1	yes	characteristic
198	C₄	#16864213	1	1	yes	characteristic
199	C₂	—	4	3	no	—
200	C₂	#22568777	1	1	yes	characteristic
201	C₂	#22568809	1	1	yes	characteristic
202	C₂	C₂²×#14523979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₇	Q₈
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#17837
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #22558998	C₂	semidirect
3. #22558968	C₂	semidirect
4. #22558957	C₂	semidirect
107. C₆₇	#17837	semidirect

Maximal subgroups

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

subgroup	index
#22569341	2
#22558998	2
#22558968	2
#22558957	2
C₈×#41×C₆₇	2
#17837	67