Subgroups — #30754394

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#107×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×#107×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	#107×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	1	no	—
40	C₂×C₆₇⋊C₈	—	2	1	no	—
41	C₂×C₆₇⋊C₈	—	2	1	no	—
42	C₂×C₆₇⋊C₈	—	2	1	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
49	C₂×C₆₇⋊C₈	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	—	2	2	no	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
63	C₄²×C₆₇	—	4	2	no	—
64	C₄²×C₆₇	—	4	2	no	—
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₆₇	#41	1	1	yes	characteristic
69	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	2	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	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
89	C₈×C₆₇	—	2	2	no	—
90	C₈×C₆₇	—	2	2	no	—
91	C₆₇⋊C₈	—	2	2	no	—
92	C₆₇⋊C₈	—	2	2	no	—
93	C₆₇⋊C₈	—	4	2	no	—
94	C₆₇⋊C₈	—	8	2	no	—
95	C₂²×C₆₇	#389	1	1	yes	characteristic
96	C₂²×C₆₇	#435	1	2	yes	—
97	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
99	C₂²×C₆₇	#444	1	2	yes	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	—	4	4	no	—
103	C₄×C₆₇	—	8	2	no	—
104	C₄×C₆₇	#423	1	1	yes	characteristic
105	C₄×C₆₇	#424	1	1	yes	characteristic
106	C₄×C₆₇	#446	1	2	yes	—
107	C₄×C₆₇	#466	1	1	yes	characteristic
108	C₄×C₆₇	#467	1	1	yes	characteristic
109	C₄×C₆₇	#499	1	1	yes	characteristic
110	C₄×C₆₇	#500	1	1	yes	characteristic
111	#9598	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	#1581	1	2	yes	—
113	C₂×C₆₇	#1584	1	2	yes	—
114	C₂×C₆₇	#1940	1	1	yes	characteristic
115	C₂×C₆₇	#1942	1	1	yes	characteristic
116	C₂×C₆₇	#1946	1	1	yes	characteristic
117	#1939	—	67	2	no	—
118	C₄×#107	D₆₇	1	1	yes	characteristic
119	C₂×#335	—	67	1	no	—
120	C₂×#322	—	67	1	no	—
121	#1279	—	67	2	no	—
122	C₆₇	#9598	1	1	yes	characteristic, complemented, Sylow
123	C₂×#107	Dic₆₇	1	2	yes	—
124	C₂×C₄⋊C₈	—	67	2	no	—
125	C₂×C₄⋊C₈	—	134	2	no	—
126	C₂×C₄×C₈	—	67	1	no	—
127	C₂×C₄×C₈	—	67	1	no	—
128	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
130	#340	Dic₆₇	1	1	yes	characteristic
131	#340	Dic₆₇	1	1	yes	characteristic
132	#336	—	134	2	no	—
133	#335	—	134	2	no	—
134	#322	—	134	2	no	—
135	C₂²×C₈	—	67	1	no	—
136	C₂²×C₈	—	67	1	no	—
137	C₂²×C₈	—	134	2	no	—
138	C₂²×C₈	—	134	1	no	—
139	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
140	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
145	#107	—	2	4	no	—
146	C₄⋊C₈	—	268	4	no	—
147	C₄⋊C₈	—	134	2	no	—
148	C₄⋊C₈	—	134	2	no	—
149	C₄×C₈	—	67	2	no	—
150	C₄×C₈	—	134	1	no	—
151	C₄×C₈	—	134	1	no	—
152	C₄×C₈	—	134	1	no	—
153	C₄×C₈	Dic₁₃₄	1	2	yes	—
154	C₄×C₈	D₂₆₈	1	2	yes	—
155	#95	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
159	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
160	C₂²×C₄	#11761627	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	268	2	no	—
164	C₂×C₈	—	268	2	no	—
165	C₂×C₈	—	268	2	no	—
166	C₂×C₈	—	268	2	no	—
167	C₂×C₈	—	67	2	no	—
168	C₂×C₈	—	67	2	no	—
169	C₂×C₈	—	134	2	no	—
170	C₂×C₈	—	134	1	no	—
171	C₂×C₈	—	134	1	no	—
172	C₂×C₈	—	134	1	no	—
173	C₂×C₈	—	134	1	no	—
174	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
175	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	#11761612	1	4	yes	—
180	C₄²	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
183	C₄²	Q₈×D₆₇	1	1	yes	characteristic
184	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
185	C₄²	#11761629	1	1	yes	characteristic
186	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
195	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
196	C₂×C₄	#14524018	1	1	yes	characteristic
197	C₂×C₄	#14524023	1	1	yes	characteristic
198	C₂×C₄	#14524034	1	1	yes	characteristic
199	C₂×C₄	#14524042	1	1	yes	characteristic
200	C₂×C₄	C₂×#11761612	1	2	yes	—
201	C₂×C₄	#14524050	1	2	yes	—
202	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
204	C₂×C₄	#14524062	1	1	yes	characteristic
205	C₂×C₄	#14524066	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	268	2	no	—
209	C₈	—	536	2	no	—
210	C₈	—	134	2	no	—
211	C₈	—	134	2	no	—
212	C₂²	#16863295	1	1	yes	characteristic
213	C₂²	#16863468	1	2	yes	—
214	C₂²	D₈×Dic₆₇	1	1	yes	characteristic
215	C₂²	Q₁₆×Dic₆₇	1	1	yes	characteristic
216	C₂²	#16863518	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	#16863420	1	1	yes	characteristic
222	C₄	#16863423	1	1	yes	characteristic
223	C₄	#16863527	1	2	yes	—
224	C₄	#16863633	1	1	yes	characteristic
225	C₄	#16863638	1	1	yes	characteristic
226	C₄	#16863797	1	1	yes	characteristic
227	C₄	#16863800	1	1	yes	characteristic
228	C₂	#22559039	1	2	yes	—
229	C₂	#22559060	1	2	yes	—
230	C₂	#22560933	1	1	yes	characteristic
231	C₂	#22560943	1	1	yes	characteristic
232	C₂	#22560963	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×#107×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#107	D₆₇
hypocenter	C₆₇	#9598
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
122. C₆₇	#9598	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×#107×C₆₇	2
#9598	67