Subgroups — #30754404

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560928	C₂	1	2	yes	maximal
2	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
4	#22558050	C₂	1	2	yes	maximal
5	C₄×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#16863151	—	2	2	no	—
11	#16863149	—	2	2	no	—
12	#16863118	—	2	2	no	—
13	C₂×#106×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	#340×C₆₇	C₄	1	1	yes	characteristic
17	#340×C₆₇	C₄	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	C₂²×C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
25	C₄×C₆₇⋊C₈	—	2	1	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₄×C₆₇⋊C₈	—	2	1	no	—
28	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	—	2	2	no	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	#106×C₆₇	—	2	4	no	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₄×C₈×C₆₇	Q₈	1	2	yes	—
38	#95×C₆₇	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	2	no	—
40	C₂×C₆₇⋊C₈	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	1	no	—
43	C₂×C₆₇⋊C₈	—	2	1	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
62	C₄²×C₆₇	—	4	2	no	—
63	C₄²×C₆₇	—	4	2	no	—
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₆₇	#41	1	1	yes	characteristic
68	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	1	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	—	4	2	no	—
74	C₂×C₄×C₆₇	—	4	2	no	—
75	C₂×C₄×C₆₇	—	4	2	no	—
76	C₂×C₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₆₇	#131	1	2	yes	—
85	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×C₄×C₆₇	#137	1	2	yes	—
87	C₈×C₆₇	—	2	2	no	—
88	C₈×C₆₇	—	2	2	no	—
89	C₆₇⋊C₈	—	2	4	no	—
90	C₆₇⋊C₈	—	4	1	no	—
91	C₆₇⋊C₈	—	4	1	no	—
92	C₆₇⋊C₈	—	8	2	no	—
93	C₂²×C₆₇	#389	1	1	yes	characteristic
94	C₂²×C₆₇	#435	1	2	yes	—
95	C₂²×C₆₇	#440	1	2	yes	—
96	C₂²×C₆₇	#443	1	1	yes	characteristic
97	C₂²×C₆₇	#443	1	1	yes	characteristic
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	4	4	no	—
101	C₄×C₆₇	—	8	2	no	—
102	C₄×C₆₇	#423	1	1	yes	characteristic
103	C₄×C₆₇	#424	1	1	yes	characteristic
104	C₄×C₆₇	#446	1	2	yes	—
105	C₄×C₆₇	#469	1	1	yes	characteristic
106	C₄×C₆₇	#470	1	1	yes	characteristic
107	C₄×C₆₇	#501	1	2	yes	—
108	#9600	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	#1585	1	2	yes	—
110	C₂×C₆₇	#1588	1	2	yes	—
111	C₂×C₆₇	#1940	1	1	yes	characteristic
112	C₂×C₆₇	#1944	1	1	yes	characteristic
113	C₂×C₆₇	#1945	1	1	yes	characteristic
114	#1939	—	67	2	no	—
115	C₄×#106	D₆₇	1	1	yes	characteristic
116	C₂×#335	—	67	1	no	—
117	C₂×#322	—	67	1	no	—
118	#1279	—	67	2	no	—
119	C₆₇	#9600	1	1	yes	characteristic, complemented, Sylow
120	C₂×#106	Dic₆₇	1	2	yes	—
121	C₂×C₄⋊C₈	—	67	2	no	—
122	C₂×C₄⋊C₈	—	134	2	no	—
123	C₂×C₄×C₈	—	67	1	no	—
124	C₂×C₄×C₈	—	67	1	no	—
125	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
126	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
127	#340	Dic₆₇	1	1	yes	characteristic
128	#340	Dic₆₇	1	1	yes	characteristic
129	#336	—	134	2	no	—
130	#335	—	134	2	no	—
131	#322	—	134	2	no	—
132	C₂²×C₈	—	67	1	no	—
133	C₂²×C₈	—	67	1	no	—
134	C₂²×C₈	—	134	2	no	—
135	C₂²×C₈	—	134	1	no	—
136	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	268	4	no	—
144	C₄⋊C₈	—	134	2	no	—
145	C₄⋊C₈	—	134	2	no	—
146	C₄×C₈	—	67	2	no	—
147	C₄×C₈	—	134	1	no	—
148	C₄×C₈	—	134	1	no	—
149	C₄×C₈	—	134	1	no	—
150	C₄×C₈	Dic₁₃₄	1	2	yes	—
151	C₄×C₈	D₂₆₈	1	2	yes	—
152	#95	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
156	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
157	C₂²×C₄	#11761627	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	268	2	no	—
161	C₂×C₈	—	268	2	no	—
162	C₂×C₈	—	268	2	no	—
163	C₂×C₈	—	134	1	no	—
164	C₂×C₈	—	134	1	no	—
165	C₂×C₈	—	134	2	no	—
166	C₂×C₈	—	134	2	no	—
167	C₂×C₈	—	134	2	no	—
168	C₂×C₈	—	134	2	no	—
169	C₂×C₈	—	134	2	no	—
170	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
171	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	#11761612	1	4	yes	—
176	C₄²	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
179	C₄²	Q₈×D₆₇	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
181	C₄²	#11761629	1	1	yes	characteristic
182	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
192	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
193	C₂×C₄	#14524018	1	1	yes	characteristic
194	C₂×C₄	#14524023	1	1	yes	characteristic
195	C₂×C₄	#14524034	1	1	yes	characteristic
196	C₂×C₄	#14524042	1	1	yes	characteristic
197	C₂×C₄	C₂×#11761612	1	2	yes	—
198	C₂×C₄	#14524050	1	2	yes	—
199	C₂×C₄	#14524070	1	2	yes	—
200	C₂×C₄	#14524072	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	268	1	no	—
204	C₈	—	268	1	no	—
205	C₈	—	536	2	no	—
206	C₈	—	134	4	no	—
207	C₂²	#16863295	1	1	yes	characteristic
208	C₂²	#16863468	1	2	yes	—
209	C₂²	#16863497	1	2	yes	—
210	C₂²	#16863513	1	1	yes	characteristic
211	C₂²	#16863513	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	8	2	no	—
216	C₄	#16863420	1	1	yes	characteristic
217	C₄	#16863423	1	1	yes	characteristic
218	C₄	#16863527	1	2	yes	—
219	C₄	#16863650	1	1	yes	characteristic
220	C₄	#16863655	1	1	yes	characteristic
221	C₄	#16863803	1	2	yes	—
222	C₂	#22559067	1	2	yes	—
223	C₂	#22559088	1	2	yes	—
224	C₂	#22560933	1	1	yes	characteristic
225	C₂	#22560953	1	1	yes	characteristic
226	C₂	#22560958	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#106×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#106	D₆₇
hypocenter	C₆₇	#9600
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
119. C₆₇	#9600	semidirect

Maximal subgroups

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

subgroup	index
#22560928	2
C₂×#16863149	2
C₂×#16863118	2
#22558050	2
C₄×#106×C₆₇	2
#9600	67