Subgroups — #30745256

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	2	yes	maximal
2	#22558040	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863453	—	2	2	no	—
5	#16863453	—	2	2	no	—
6	#16863449	C₄	1	4	yes	—
7	#16863449	C₂²	1	4	yes	—
8	#16863420	—	2	2	no	—
9	#16863360	—	2	2	no	—
10	C₄×#11761612	C₄	1	4	yes	—
11	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₆₇	—	2	2	no	—
14	C₄×M₁₆×C₆₇	—	2	2	no	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
17	C₂×#11761612	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
22	#14523954	—	2	4	no	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
27	C₂×M₁₆×C₆₇	—	2	4	no	—
28	C₂×M₁₆×C₆₇	—	2	8	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#97×C₆₇	—	2	4	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	—	2	2	no	—
41	#11761612	—	4	8	no	—
42	C₂×C₆₇⋊C₈	—	2	8	no	—
43	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	M₁₆×C₆₇	—	2	16	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	2	4	no	—
62	C₂×C₈×C₆₇	M₁₆	1	8	yes	complemented
63	C₄²×C₆₇	C₄²	1	1	yes	characteristic
64	C₄²×C₆₇	C₄²	1	1	yes	characteristic
65	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₆₇	C₂×D₄	1	2	yes	—
73	C₂³×C₆₇	—	2	2	no	—
74	C₂³×C₆₇	—	4	1	no	—
75	C₂³×C₆₇	—	4	1	no	—
76	C₂³×C₆₇	#95	1	2	yes	—
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	4	2	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#95	1	2	yes	—
92	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
99	C₈×C₆₇	—	2	8	no	—
100	C₆₇⋊C₈	—	4	8	no	—
101	C₂²×C₆₇	—	2	1	no	—
102	C₂²×C₆₇	—	2	1	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	—	2	2	no	—
105	C₂²×C₆₇	—	2	2	no	—
106	C₂²×C₆₇	—	4	4	no	—
107	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
108	C₂²×C₆₇	#423	1	2	yes	—
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	2	2	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	2	1	no	—
113	C₄×C₆₇	—	2	2	no	—
114	C₄×C₆₇	H(3,ℤ/4)	1	2	yes	—
115	C₄×C₆₇	#338	1	2	yes	—
116	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
117	C₄×C₆₇	#407	1	2	yes	—
118	#7862	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	—	2	2	no	—
120	C₂×C₆₇	—	4	2	no	—
121	C₂×C₆₇	#1728	1	1	yes	characteristic
122	C₂×C₆₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
123	C₂×C₆₇	C₂×#338	1	1	yes	characteristic
124	#2920	—	67	2	no	—
125	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
126	#1275	—	67	4	no	—
127	C₆₇	#7862	1	1	yes	characteristic, complemented, Sylow
128	C₂²×M₁₆	Dic₆₇	1	2	yes	—
129	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
130	#432	—	134	2	no	—
131	#432	—	134	2	no	—
132	#431	—	67	4	no	—
133	#431	—	67	4	no	—
134	#423	—	134	2	no	—
135	#407	—	134	2	no	—
136	C₄×M₁₆	—	2	2	no	—
137	C₄×M₁₆	—	2	2	no	—
138	C₄×M₁₆	—	67	4	no	—
139	C₂×#97	C₂×D₆₇	1	2	yes	—
140	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
141	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
142	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
143	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	—	134	4	no	—
147	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
148	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₆₇	1	2	yes	—
150	C₂×C₄²	C₄×D₆₇	1	2	yes	—
151	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
154	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
155	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
156	C₄⋊C₈	—	134	4	no	—
157	C₄⋊C₈	—	134	4	no	—
158	C₂²⋊C₈	—	134	4	no	—
159	C₂²⋊C₈	—	134	4	no	—
160	#97	—	2	4	no	—
161	#97	—	134	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	67	8	no	—
165	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
170	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
171	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
172	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
173	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
174	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
175	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
181	M₁₆	—	2	16	no	—
182	M₁₆	—	268	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	134	8	no	—
186	C₂×C₈	#11761612	1	8	yes	complemented
187	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
188	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
189	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
190	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
191	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
192	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
193	C₄²	Dic₆₇⋊C₄	1	2	yes	—
194	C₄²	Dic₆₇⋊C₄	1	2	yes	—
195	C₄²	C₂×C₄×D₆₇	1	2	yes	—
196	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₆₇⋊#95	1	2	yes	—
201	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
214	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
215	C₂×C₄	C₆₇⋊#95	1	2	yes	—
216	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
217	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
219	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
221	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
222	C₂×C₄	C₂×#11761612	1	4	yes	—
223	C₈	—	2	8	no	—
224	C₈	—	268	8	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
232	C₂²	#16863420	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	#16863153	1	2	yes	—
239	C₄	C₆₇⋊#338	1	2	yes	—
240	C₄	C₄×#11761612	1	2	yes	—
241	C₄	#16863360	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	#22559903	1	1	yes	characteristic
245	C₂	C₂×#16863153	1	1	yes	characteristic
246	C₂	C₂×C₆₇⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#7862
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22558040	C₂	semidirect
62. C₂×C₈×C₆₇	M₁₆	semidirect
127. C₆₇	#7862	semidirect
186. C₂×C₈	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
#22558040	2
C₂×C₄×M₁₆×C₆₇	2
#7862	67