Subgroups — #30826579

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570795	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	2	yes	maximal
3	#22559082	C₂	1	8	yes	maximal, complemented
4	#22558965	C₂	1	1	yes	maximal, characteristic
5	#22558665	C₂	1	2	yes	maximal
6	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863953	C₂²	1	2	yes	—
8	#16863801	C₄	1	1	yes	characteristic
9	#16863776	C₄	1	1	yes	characteristic
10	#16863747	C₄	1	2	yes	—
11	#16863637	C₄	1	4	yes	—
12	#16863526	—	2	2	no	—
13	#16863526	C₂²	1	4	yes	—
14	C₈×Dic₁₃₄	—	2	2	no	—
15	C₈×Dic₁₃₄	C₂²	1	4	yes	—
16	C₄×Dic₂₆₈	C₂²	1	4	yes	—
17	#16863422	—	2	2	no	—
18	#16863420	C₂²	1	2	yes	—
19	#16863351	—	2	2	no	—
20	#16863147	C₂²	1	4	yes	—
21	#16863128	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#322×C₆₇	C₂²	1	4	yes	—
26	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
27	C₂×#11761612	—	2	2	no	—
28	#14524043	C₂×C₄	1	2	yes	—
29	#14524043	C₂×C₄	1	2	yes	—
30	#14524035	—	2	2	no	—
31	#14524022	—	2	2	no	—
32	#14524022	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₃₄	—	2	2	no	—
34	C₄×Dic₁₃₄	C₂³	1	4	yes	—
35	#14524002	—	2	2	no	—
36	#14523977	C₂×C₄	1	4	yes	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
39	Dic₆₇⋊C₈	—	2	4	no	—
40	Dic₆₇⋊C₈	—	2	4	no	—
41	#14523968	C₂×C₄	1	8	yes	—
42	#14523953	—	2	4	no	—
43	C₈×Dic₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₆₇	—	2	1	no	—
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂×Dic₁₃₄	—	2	2	no	—
57	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
58	Dic₂₆₈	—	4	4	no	—
59	#11761612	—	4	4	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₄⋊Dic₆₇	—	2	2	no	—
62	C₄⋊Dic₆₇	—	2	2	no	—
63	C₄⋊Dic₆₇	—	2	2	no	—
64	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
65	Dic₆₇⋊C₄	—	2	4	no	—
66	Dic₆₇⋊C₄	—	2	4	no	—
67	C₂²⋊Dic₆₇	—	2	4	no	—
68	C₄×Dic₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
80	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	#41	1	1	yes	characteristic
83	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#137	1	4	yes	—
91	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
94	C₂×Dic₆₇	—	2	4	no	—
95	C₂×Dic₆₇	—	2	4	no	—
96	Dic₁₃₄	—	2	8	no	—
97	Dic₁₃₄	—	4	4	no	—
98	C₈×C₆₇	—	2	4	no	—
99	C₈×C₆₇	—	2	2	no	—
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	—	2	2	no	—
103	C₆₇⋊C₈	—	4	4	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	#441	1	2	yes	—
107	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	#441	1	2	yes	—
111	C₄×C₆₇	#443	1	2	yes	—
112	C₄×C₆₇	#568	1	2	yes	—
113	C₄×C₆₇	C₂×#137	1	2	yes	—
114	Dic₆₇	—	4	4	no	—
115	Dic₆₇	—	4	4	no	—
116	#17980	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	—	2	2	no	—
118	C₂×C₆₇	#2932	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
121	#3170	—	67	1	no	—
122	#2960	—	67	2	no	—
123	#1587	—	67	8	no	—
124	#1566	—	67	1	no	—
125	#1493	—	67	2	no	—
126	C₂×#322	D₆₇	1	1	yes	characteristic
127	C₆₇	#17980	1	1	yes	characteristic, complemented, Sylow
128	#533	—	67	2	no	—
129	#500	—	67	1	no	—
130	#494	—	67	1	no	—
131	#487	—	67	2	no	—
132	#467	—	67	4	no	—
133	#446	—	67	4	no	—
134	#446	—	134	2	no	—
135	C₈×Q₈	—	67	4	no	—
136	C₈×Q₈	—	134	2	no	—
137	C₄×Q₁₆	—	67	4	no	—
138	#424	—	134	2	no	—
139	#423	—	67	2	no	—
140	#405	—	134	2	no	—
141	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
144	#334	—	67	4	no	—
145	Q₈⋊C₈	—	67	8	no	—
146	#322	C₂×D₆₇	1	4	yes	—
147	C₂×Q₁₆	—	67	4	no	—
148	C₂×M₁₆	—	134	2	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	C₄×D₆₇	1	2	yes	—
151	C₂²×C₈	D₂₆₈	1	2	yes	—
152	C₄⋊Q₈	—	67	2	no	—
153	C₄⋊Q₈	—	67	2	no	—
154	#125	—	134	2	no	—
155	C₂²⋊Q₈	—	67	4	no	—
156	C₂²⋊Q₈	—	134	2	no	—
157	C₄×Q₈	—	67	4	no	—
158	C₄×Q₈	—	134	2	no	—
159	#117	—	134	2	no	—
160	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
161	#107	—	67	4	no	—
162	C₄⋊C₈	—	67	4	no	—
163	C₄⋊C₈	—	134	4	no	—
164	C₄⋊C₈	—	134	4	no	—
165	C₄⋊C₈	—	134	2	no	—
166	#103	—	67	8	no	—
167	#97	—	134	4	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	134	4	no	—
170	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
171	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
172	C₄×C₈	D₂₆₈	1	1	yes	characteristic
173	C₄×C₈	D₂₆₈	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₆₇	1	2	yes	—
175	C₄×C₈	C₂²×D₆₇	1	2	yes	—
176	C₄×C₈	C₂²×D₆₇	1	2	yes	—
177	C₂×Q₈	—	67	4	no	—
178	C₂×Q₈	—	134	2	no	—
179	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
181	C₂²×C₄	#11761628	1	1	yes	characteristic
182	Q₁₆	—	268	4	no	—
183	M₁₆	—	268	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	134	4	no	—
189	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
191	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
192	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
193	C₄⋊C₄	—	67	4	no	—
194	C₄⋊C₄	—	134	2	no	—
195	C₄⋊C₄	—	134	2	no	—
196	C₄⋊C₄	—	134	4	no	—
197	C₄⋊C₄	—	134	2	no	—
198	C₄⋊C₄	—	134	4	no	—
199	C₂²⋊C₄	—	134	4	no	—
200	C₄²	—	134	4	no	—
201	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
202	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
203	C₄²	#11761628	1	1	yes	characteristic
204	C₄²	C₂³×D₆₇	1	1	yes	characteristic
205	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
206	Q₈	—	268	4	no	—
207	Q₈	—	134	8	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	134	4	no	—
211	C₂×C₄	—	134	4	no	—
212	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
213	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
214	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
215	C₂×C₄	#14524076	1	4	yes	—
216	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
218	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	268	4	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#16863504	1	2	yes	—
228	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	268	4	no	—
232	C₄	—	268	4	no	—
233	C₄	#16863504	1	2	yes	—
234	C₄	#16863514	1	2	yes	—
235	C₄	#16864162	1	2	yes	—
236	C₄	C₂×#14524076	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	#22568928	1	1	yes	characteristic
239	C₂	C₂×#16863504	1	1	yes	characteristic
240	C₂	C₂×#16863514	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#17980
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #22559082	C₂	semidirect
127. C₆₇	#17980	semidirect

Maximal subgroups

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

subgroup	index
#22570795	2
#22569143	2
#22559082	2
#22558965	2
#22558665	2
C₂×#322×C₆₇	2
#17980	67