Subgroups — #30747015

Order 17152.

470 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	C₄⋊C₈×Dic₆₇	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863148	C₂	1	2	yes	maximal
4	#22558044	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₇	—	2	2	no	—
9	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
10	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
11	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
12	#16863150	—	2	2	no	—
13	#16863148	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
25	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₈×Dic₆₇	—	2	4	no	—
27	C₈×Dic₆₇	—	4	4	no	—
28	C₈×Dic₆₇	C₈	1	8	yes	complemented
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₆₇	—	2	2	no	—
33	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
36	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₆₇⋊C₈	—	2	2	no	—
38	C₂×C₆₇⋊C₈	—	2	4	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
42	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
43	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
44	C₄×Dic₆₇	—	2	2	no	—
45	C₄×Dic₆₇	—	2	4	no	—
46	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
47	C₄×Dic₆₇	M₁₆	1	4	yes	—
48	C₄×Dic₆₇	D₈	1	2	yes	—
49	C₄×Dic₆₇	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	—	4	2	no	—
57	C₂×C₈×C₆₇	—	4	2	no	—
58	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
59	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
66	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#135	1	2	yes	—
72	C₂×Dic₆₇	—	2	2	no	—
73	C₂×Dic₆₇	—	2	2	no	—
74	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
75	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
76	C₂×Dic₆₇	#107	1	4	yes	—
77	C₂×Dic₆₇	#108	1	4	yes	—
78	C₈×C₆₇	—	2	4	no	—
79	C₈×C₆₇	—	4	4	no	—
80	C₆₇⋊C₈	—	2	2	no	—
81	C₆₇⋊C₈	—	4	4	no	—
82	C₆₇⋊C₈	C₄×C₈	1	4	yes	complemented
83	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
85	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
86	C₂²×C₆₇	#427	1	1	yes	characteristic
87	C₂²×C₆₇	C₂×#108	1	2	yes	—
88	C₄×C₆₇	—	2	2	no	—
89	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
91	C₄×C₆₇	#431	1	2	yes	—
92	C₄×C₆₇	C₄×D₈	1	2	yes	—
93	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
94	Dic₆₇	—	2	4	no	—
95	Dic₆₇	#335	1	8	yes	complemented
96	C₄×#335	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	C₂×#335	1	2	yes	—
98	C₂×C₆₇	#1564	1	2	yes	—
99	C₂×C₆₇	C₄×C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₆₇	C₄×#107	1	1	yes	characteristic
101	C₂×C₆₇	C₄×#108	1	1	yes	characteristic
102	C₄×C₄⋊C₈	—	67	2	no	—
103	C₄²×C₈	—	67	1	no	—
104	C₂×#335	—	67	2	no	—
105	#1277	—	67	1	no	—
106	#1277	D₆₇	1	1	yes	characteristic
107	C₆₇	C₄×#335	1	1	yes	characteristic, complemented, Sylow
108	C₂×C₄⋊C₈	—	67	2	no	—
109	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
110	C₂×C₄×C₈	—	67	2	no	—
111	C₂×C₄×C₈	—	67	1	no	—
112	C₂×C₄×C₈	—	67	2	no	—
113	C₂×C₄×C₈	—	134	2	no	—
114	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
115	C₄³	—	67	1	no	—
116	#336	—	134	2	no	—
117	#335	—	67	8	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	67	1	no	—
120	C₂²×C₈	—	67	1	no	—
121	C₂²×C₈	—	134	2	no	—
122	C₂²×C₈	C₄×D₆₇	1	2	yes	—
123	C₂×C₄²	—	67	2	no	—
124	C₂×C₄²	—	67	1	no	—
125	C₂×C₄²	—	67	1	no	—
126	C₂×C₄²	—	67	2	no	—
127	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	67	4	no	—
130	C₄⋊C₈	—	134	2	no	—
131	C₄⋊C₈	C₄×D₆₇	1	4	yes	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	268	4	no	—
134	C₄×C₈	—	67	8	no	—
135	C₄×C₈	—	67	4	no	—
136	C₄×C₈	—	134	4	no	—
137	C₂²×C₄	—	67	2	no	—
138	C₂²×C₄	—	67	2	no	—
139	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
141	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	268	2	no	—
148	C₂×C₈	—	268	2	no	—
149	C₂×C₈	—	67	2	no	—
150	C₂×C₈	—	67	4	no	—
151	C₂×C₈	—	67	2	no	—
152	C₂×C₈	—	134	4	no	—
153	C₂×C₈	—	134	2	no	—
154	C₂×C₈	C₈×D₆₇	1	4	yes	—
155	C₄²	—	67	4	no	—
156	C₄²	—	67	4	no	—
157	C₄²	—	67	2	no	—
158	C₄²	—	67	2	no	—
159	C₄²	—	134	4	no	—
160	C₄²	—	134	2	no	—
161	C₄²	C₂×C₄×D₆₇	1	2	yes	—
162	C₄²	#11761627	1	2	yes	—
163	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	67	4	no	—
166	C₂×C₄	—	67	4	no	—
167	C₂×C₄	—	67	4	no	—
168	C₂×C₄	—	67	4	no	—
169	C₂×C₄	—	134	2	no	—
170	C₂×C₄	—	134	2	no	—
171	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
172	C₂×C₄	#14523999	1	1	yes	characteristic
173	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
174	C₂×C₄	#14524008	1	2	yes	—
175	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
176	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
177	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
178	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
179	C₂×C₄	#14524064	1	2	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	4	4	no	—
182	C₈	—	268	4	no	—
183	C₈	—	67	4	no	—
184	C₈	—	134	2	no	—
185	C₂²	#16863247	1	1	yes	characteristic
186	C₂²	C₄⋊C₈×D₆₇	1	2	yes	—
187	C₂²	#107×D₆₇	1	1	yes	characteristic
188	C₂²	#16863431	1	1	yes	characteristic
189	C₂²	#108×D₆₇	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	67	8	no	—
192	C₄	—	134	4	no	—
193	C₄	C₄×C₈×D₆₇	1	1	yes	characteristic
194	C₄	#16863348	1	1	yes	characteristic
195	C₄	#16863446	1	2	yes	—
196	C₄	#16863476	1	2	yes	—
197	C₄	#16863488	1	2	yes	—
198	C₂	#335×D₆₇	1	2	yes	—
199	C₂	#22558954	1	2	yes	—
200	C₂	#22560057	1	1	yes	characteristic
201	C₂	#22560101	1	1	yes	characteristic
202	C₂	#22560110	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1277	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1277	D₆₇
hypocenter	C₆₇	C₄×#335
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
13. #16863148	C₄	semidirect
28. C₈×Dic₆₇	C₈	semidirect
82. C₆₇⋊C₈	C₄×C₈	semidirect
95. Dic₆₇	#335	semidirect
107. C₆₇	C₄×#335	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₆₇	2
C₄×C₈×Dic₆₇	2
C₂×#16863148	2
#22558044	2
C₆₇×#1277	2
C₄×#335	67