Subgroups — #30826391

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 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	#22559119	C₂	1	4	yes	maximal, complemented
4	#22559029	C₂	1	4	yes	maximal, complemented
5	#22558965	C₂	1	1	yes	maximal, characteristic
6	#22558665	C₂	1	2	yes	maximal
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₂²	1	2	yes	—
9	#16863801	C₄	1	1	yes	characteristic
10	#16863776	C₄	1	1	yes	characteristic
11	#16863747	C₄	1	2	yes	—
12	#16863637	C₄	1	4	yes	—
13	#16863526	—	2	2	no	—
14	#16863526	C₂²	1	4	yes	—
15	C₈×Dic₁₃₄	—	2	2	no	—
16	C₈×Dic₁₃₄	C₂²	1	4	yes	—
17	C₄×Dic₂₆₈	C₂²	1	4	yes	—
18	#16863422	—	2	2	no	—
19	#16863420	C₂²	1	2	yes	—
20	#16863351	—	2	2	no	—
21	#16863147	C₂²	1	4	yes	—
22	#16863128	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#322×C₆₇	C₂²	1	2	yes	—
27	#322×C₆₇	C₂²	1	2	yes	—
28	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
29	C₂×#11761612	—	2	2	no	—
30	#14524043	C₂×C₄	1	2	yes	—
31	#14524043	C₂×C₄	1	2	yes	—
32	#14524035	—	2	2	no	—
33	#14524022	—	2	2	no	—
34	#14524022	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₃₄	—	2	2	no	—
36	C₄×Dic₁₃₄	C₂³	1	4	yes	—
37	#14524002	—	2	2	no	—
38	#14523977	C₂×C₄	1	4	yes	—
39	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
41	Dic₆₇⋊C₈	—	2	4	no	—
42	Dic₆₇⋊C₈	—	2	4	no	—
43	#14523968	C₂×C₄	1	8	yes	—
44	#14523953	—	2	4	no	—
45	C₈×Dic₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₆₇	—	2	1	no	—
51	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	2	yes	—
54	C₄×C₈×C₆₇	C₂³	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₃₄	—	2	2	no	—
59	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
60	Dic₂₆₈	—	2	8	no	—
61	#11761612	—	4	4	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₄⋊Dic₆₇	—	2	2	no	—
64	C₄⋊Dic₆₇	—	2	2	no	—
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
67	Dic₆₇⋊C₄	—	2	4	no	—
68	Dic₆₇⋊C₄	—	2	4	no	—
69	C₂²⋊Dic₆₇	—	2	4	no	—
70	C₄×Dic₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	4	2	no	—
78	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#137	1	4	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₆₇	—	2	4	no	—
97	C₂×Dic₆₇	—	2	4	no	—
98	Dic₁₃₄	—	2	8	no	—
99	Dic₁₃₄	—	4	4	no	—
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	—	2	2	no	—
103	C₈×C₆₇	—	4	2	no	—
104	C₈×C₆₇	#131	1	4	yes	—
105	C₆₇⋊C₈	—	4	4	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	#441	1	2	yes	—
109	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	#441	1	2	yes	—
113	C₄×C₆₇	#444	1	2	yes	—
114	C₄×C₆₇	C₂×#131	1	2	yes	—
115	C₄×C₆₇	C₂×#137	1	2	yes	—
116	Dic₆₇	—	4	4	no	—
117	Dic₆₇	—	4	4	no	—
118	#17961	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	—	2	2	no	—
120	C₂×C₆₇	#2931	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
123	#3170	—	67	1	no	—
124	#2960	—	67	2	no	—
125	#1593	—	67	4	no	—
126	#1579	—	67	4	no	—
127	#1566	—	67	1	no	—
128	#1493	—	67	2	no	—
129	C₂×#322	D₆₇	1	1	yes	characteristic
130	C₆₇	#17961	1	1	yes	characteristic, complemented, Sylow
131	#533	—	67	2	no	—
132	#500	—	67	1	no	—
133	#494	—	67	1	no	—
134	#487	—	67	2	no	—
135	#467	—	67	4	no	—
136	#446	—	67	4	no	—
137	#446	—	134	2	no	—
138	C₈×Q₈	—	67	4	no	—
139	C₈×Q₈	—	134	2	no	—
140	C₄×Q₁₆	—	67	4	no	—
141	#424	—	134	2	no	—
142	#423	—	67	2	no	—
143	#405	—	134	2	no	—
144	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	#334	—	67	4	no	—
148	Q₈⋊C₈	—	67	8	no	—
149	#322	C₂×D₆₇	1	2	yes	—
150	#322	C₂×D₆₇	1	2	yes	—
151	C₂×Q₁₆	—	67	4	no	—
152	C₂×M₁₆	—	134	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₄×D₆₇	1	2	yes	—
155	C₂²×C₈	D₂₆₈	1	2	yes	—
156	C₄⋊Q₈	—	67	2	no	—
157	C₄⋊Q₈	—	67	2	no	—
158	#125	—	134	2	no	—
159	C₂²⋊Q₈	—	67	4	no	—
160	C₂²⋊Q₈	—	134	2	no	—
161	C₄×Q₈	—	67	4	no	—
162	C₄×Q₈	—	134	2	no	—
163	#117	—	134	2	no	—
164	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
165	#107	—	67	4	no	—
166	C₄⋊C₈	—	67	4	no	—
167	C₄⋊C₈	—	134	4	no	—
168	C₄⋊C₈	—	134	4	no	—
169	C₄⋊C₈	—	134	2	no	—
170	#103	—	67	8	no	—
171	#97	—	134	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	134	4	no	—
174	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
175	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
176	C₄×C₈	D₂₆₈	1	2	yes	—
177	C₄×C₈	C₂²×D₆₇	1	2	yes	—
178	C₄×C₈	C₂²×D₆₇	1	2	yes	—
179	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
181	C₂×Q₈	—	67	4	no	—
182	C₂×Q₈	—	134	2	no	—
183	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
185	C₂²×C₄	#11761628	1	1	yes	characteristic
186	Q₁₆	—	134	8	no	—
187	M₁₆	—	268	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	134	4	no	—
193	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
195	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
196	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
197	C₄⋊C₄	—	67	4	no	—
198	C₄⋊C₄	—	134	2	no	—
199	C₄⋊C₄	—	134	2	no	—
200	C₄⋊C₄	—	134	4	no	—
201	C₄⋊C₄	—	134	2	no	—
202	C₄⋊C₄	—	134	4	no	—
203	C₂²⋊C₄	—	134	4	no	—
204	C₄²	—	134	4	no	—
205	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
206	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
207	C₄²	#11761628	1	1	yes	characteristic
208	C₄²	C₂³×D₆₇	1	1	yes	characteristic
209	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
210	Q₈	—	268	4	no	—
211	Q₈	—	134	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	134	4	no	—
215	C₂×C₄	—	134	4	no	—
216	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
217	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
218	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
219	C₂×C₄	#14524076	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
222	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	268	4	no	—
228	C₈	#14524049	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#16863504	1	2	yes	—
232	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	268	4	no	—
236	C₄	—	268	4	no	—
237	C₄	#16863504	1	2	yes	—
238	C₄	#16863519	1	2	yes	—
239	C₄	C₂×#14524049	1	2	yes	—
240	C₄	C₂×#14524076	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#22568922	1	1	yes	characteristic
243	C₂	C₂×#16863504	1	1	yes	characteristic
244	C₂	C₂×#16863519	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	#17961
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. #22559119	C₂	semidirect
4. #22559029	C₂	semidirect
130. C₆₇	#17961	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
#22559119	2
#22559029	2
#22558965	2
#22558665	2
C₂×#322×C₆₇	2
#17961	67