Subgroups — #30746508

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558591	C₂	1	2	yes	maximal
2	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863117	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558047	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
8	Q₈⋊(C₆₇⋊C₈)	C₄	1	4	yes	—
9	D₄⋊(C₆₇⋊C₈)	C₄	1	4	yes	—
10	#16863117	—	2	2	no	—
11	#16863117	—	2	1	no	—
12	C₈×C₆₇⋊C₈	—	2	1	no	—
13	C₂×#131×C₆₇	C₄	1	2	yes	—
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	#405×C₆₇	—	2	2	no	—
16	#405×C₆₇	C₄	1	2	yes	—
17	C₄×M₁₆×C₆₇	—	2	1	no	—
18	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	—	2	1	no	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
23	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
28	#131×C₆₇	—	2	4	no	—
29	#131×C₆₇	C₈	1	8	yes	complemented
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	—	2	2	no	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄×D₄×C₆₇	—	2	2	no	—
36	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
37	#117×C₆₇	—	2	1	no	—
38	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	—	2	1	no	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	#97×C₆₇	—	2	2	no	—
42	#97×C₆₇	—	2	1	no	—
43	C₄×C₈×C₆₇	—	2	2	no	—
44	C₄×C₈×C₆₇	—	2	1	no	—
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	D₄	1	2	yes	—
48	C₂×C₆₇⋊C₈	—	2	4	no	—
49	C₂×C₆₇⋊C₈	—	2	4	no	—
50	C₂×C₆₇⋊C₈	—	4	4	no	—
51	#41×C₆₇	—	2	2	no	—
52	#41×C₆₇	C₂×C₈	1	4	yes	—
53	C₂×Q₈×C₆₇	C₄²	1	1	yes	characteristic
54	C₂×D₄×C₆₇	—	2	1	no	—
55	C₂×D₄×C₆₇	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	2	1	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	#41	1	2	yes	—
60	M₁₆×C₆₇	—	2	2	no	—
61	M₁₆×C₆₇	—	2	8	no	—
62	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	2	8	no	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
69	C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₄⋊C₄×C₆₇	C₄²	1	2	yes	—
71	C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	—	2	1	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₆₇	—	2	1	no	—
79	C₂³×C₆₇	#117	1	1	yes	characteristic
80	Q₈×C₆₇	—	2	1	no	—
81	Q₈×C₆₇	C₄×C₈	1	2	yes	—
82	D₄×C₆₇	—	2	4	no	—
83	D₄×C₆₇	—	2	1	no	—
84	D₄×C₆₇	C₄×C₈	1	2	yes	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
96	C₈×C₆₇	—	2	4	no	—
97	C₈×C₆₇	C₂²⋊C₈	1	4	yes	—
98	C₈×C₆₇	C₂²⋊C₈	1	4	yes	—
99	C₆₇⋊C₈	—	4	4	no	—
100	C₆₇⋊C₈	—	4	4	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	—	4	2	no	—
104	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₆₇	#431	1	2	yes	—
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	4	2	no	—
109	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
112	C₄×C₆₇	#444	1	4	yes	—
113	#8185	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	—	2	2	no	—
115	C₂×C₆₇	—	4	2	no	—
116	C₂×C₆₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₆₇	#1765	1	2	yes	—
118	C₄×#131	D₆₇	1	1	yes	characteristic
119	#1476	—	67	2	no	—
120	#1450	—	67	1	no	—
121	C₂×#322	—	67	1	no	—
122	#1278	—	67	2	no	—
123	C₆₇	#8185	1	1	yes	characteristic, complemented, Sylow
124	C₂×#131	Dic₆₇	1	2	yes	—
125	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
126	#431	—	67	2	no	—
127	#405	—	2	2	no	—
128	#405	Dic₆₇	1	2	yes	—
129	C₄×M₁₆	—	2	1	no	—
130	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
131	C₂×C₄×C₈	—	2	1	no	—
132	C₂×C₄×C₈	—	67	2	no	—
133	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
134	Q₈⋊C₈	—	67	4	no	—
135	D₄⋊C₈	—	67	4	no	—
136	#322	—	134	2	no	—
137	#322	—	134	1	no	—
138	C₈²	—	134	1	no	—
139	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
140	#131	—	2	4	no	—
141	#131	C₆₇⋊C₈	1	8	yes	complemented
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	134	2	no	—
146	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
147	C₄×Q₈	C₄×D₆₇	1	2	yes	—
148	C₄×D₄	—	2	2	no	—
149	C₄×D₄	C₄×D₆₇	1	2	yes	—
150	#117	—	2	1	no	—
151	#117	C₂×Dic₆₇	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
154	C₄⋊C₈	—	67	4	no	—
155	C₂²⋊C₈	—	134	2	no	—
156	#97	—	2	2	no	—
157	#97	—	2	1	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	67	4	no	—
161	C₄×C₈	—	134	2	no	—
162	C₄×C₈	—	134	2	no	—
163	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
164	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
166	#41	—	2	2	no	—
167	#41	C₂×C₆₇⋊C₈	1	4	yes	—
168	C₂×Q₈	C₄×Dic₆₇	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₄×Dic₆₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
174	C₂²×C₄	#11761628	1	2	yes	—
175	M₁₆	—	2	2	no	—
176	M₁₆	—	2	8	no	—
177	M₁₆	C₂×C₆₇⋊C₈	1	4	yes	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	268	4	no	—
181	C₂×C₈	—	134	4	no	—
182	C₂×C₈	—	134	4	no	—
183	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
184	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
185	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
186	C₂×C₈	#11761612	1	4	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	C₄×Dic₆₇	1	2	yes	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	C₂×C₄×D₆₇	1	2	yes	—
194	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂³	—	2	1	no	—
197	C₂³	#14524002	1	1	yes	characteristic
198	Q₈	—	2	1	no	—
199	Q₈	C₄×C₆₇⋊C₈	1	2	yes	—
200	D₄	—	2	4	no	—
201	D₄	—	2	1	no	—
202	D₄	C₄×C₆₇⋊C₈	1	2	yes	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
209	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
211	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
212	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
213	C₂×C₄	C₂×#11761612	1	2	yes	—
214	C₈	—	2	4	no	—
215	C₈	—	268	4	no	—
216	C₈	—	268	4	no	—
217	C₈	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
218	C₈	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
223	C₂²	#16863449	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
228	C₄	C₄×#11761612	1	1	yes	characteristic
229	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
230	C₄	#16863516	1	4	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	C₄×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
234	C₂	#22560051	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#8185
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #22558492	C₂	semidirect
3. C₂×#16863117	C₂	semidirect
4. #22558047	C₂	semidirect
29. #131×C₆₇	C₈	semidirect
123. C₆₇	#8185	semidirect
141. #131	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22558591	2
#22558492	2
C₂×#16863117	2
#22558047	2
C₄×#131×C₆₇	2
#8185	67