Subgroups — #30826389

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	#22569139	C₂	1	2	yes	maximal
3	#22559117	C₂	1	4	yes	maximal, complemented
4	#22559027	C₂	1	4	yes	maximal, complemented
5	#22558663	C₂	1	2	yes	maximal
6	C₂×#16863117	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1566	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	#16863523	—	2	2	no	—
14	#16863523	C₂²	1	4	yes	—
15	#16863520	—	2	2	no	—
16	#16863520	C₂²	1	4	yes	—
17	C₄×Dic₂₆₈	C₂²	1	4	yes	—
18	#16863424	—	2	2	no	—
19	#16863354	—	2	2	no	—
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	#16863126	C₂²	1	8	yes	—
23	#16863117	C₂²	1	2	yes	—
24	#16863117	C₂²	1	2	yes	—
25	#423×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#334×C₆₇	C₂²	1	4	yes	—
28	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
31	#14524043	C₂×C₄	1	2	yes	—
32	#14524043	C₂×C₄	1	2	yes	—
33	#14524035	—	2	2	no	—
34	#14524022	—	2	2	no	—
35	#14524022	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₃₄	—	2	2	no	—
37	C₄×Dic₁₃₄	C₂³	1	4	yes	—
38	#14524002	—	2	2	no	—
39	#14523977	C₂×C₄	1	4	yes	—
40	Dic₆₇⋊C₈	—	2	4	no	—
41	Dic₆₇⋊C₈	—	2	4	no	—
42	#14523968	C₂×C₄	1	8	yes	—
43	#14523953	—	2	4	no	—
44	C₄×C₆₇⋊C₈	—	2	1	no	—
45	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₈×Dic₆₇	—	2	4	no	—
50	C₂×M₁₆×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	—	2	2	no	—
54	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
55	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂×Dic₁₃₄	—	2	2	no	—
59	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
60	Dic₂₆₈	—	2	8	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
65	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
66	C₄⋊Dic₆₇	—	2	2	no	—
67	C₄⋊Dic₆₇	—	2	2	no	—
68	C₄⋊Dic₆₇	—	2	2	no	—
69	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
70	Dic₆₇⋊C₄	—	2	4	no	—
71	Dic₆₇⋊C₄	—	2	4	no	—
72	C₂²⋊Dic₆₇	—	2	4	no	—
73	C₄×Dic₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
77	M₁₆×C₆₇	—	4	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂²×C₄	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	—	4	4	no	—
102	C₈×C₆₇	#131	1	4	yes	—
103	C₆₇⋊C₈	—	2	2	no	—
104	C₆₇⋊C₈	—	2	2	no	—
105	C₆₇⋊C₈	—	4	2	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	D₆₇	1	1	yes	characteristic
128	#1493	—	67	2	no	—
129	C₂×#322	—	67	1	no	—
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	C₂×D₆₇	1	2	yes	—
143	#405	—	134	2	no	—
144	C₂×C₄×C₈	—	67	1	no	—
145	C₂×C₄×C₈	—	67	1	no	—
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	#334	C₂×D₆₇	1	4	yes	—
148	Q₈⋊C₈	—	67	8	no	—
149	#322	—	67	2	no	—
150	#322	—	67	2	no	—
151	C₂×Q₁₆	—	67	4	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	—	67	2	no	—
154	C₂²×C₈	—	134	1	no	—
155	C₂²×C₈	C₄×D₆₇	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	—	2	2	no	—
167	C₄⋊C₈	—	134	4	no	—
168	C₄⋊C₈	—	134	4	no	—
169	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
170	#103	—	67	8	no	—
171	#97	—	134	4	no	—
172	C₄×C₈	—	67	2	no	—
173	C₄×C₈	—	67	1	no	—
174	C₄×C₈	—	67	1	no	—
175	C₄×C₈	—	67	2	no	—
176	C₄×C₈	—	134	4	no	—
177	C₄×C₈	—	134	1	no	—
178	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
179	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₆₇	1	2	yes	—
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	D₄×D₆₇	1	1	yes	characteristic
185	C₂²×C₄	#11761629	1	1	yes	characteristic
186	Q₁₆	—	134	8	no	—
187	M₁₆	—	4	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	268	2	no	—
191	C₂×C₈	—	67	2	no	—
192	C₂×C₈	—	67	2	no	—
193	C₂×C₈	—	134	2	no	—
194	C₂×C₈	—	134	4	no	—
195	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
196	C₂×C₈	C₂×C₄×D₆₇	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	D₄×D₆₇	1	1	yes	characteristic
207	C₄²	#11761629	1	1	yes	characteristic
208	C₄²	C₂³×D₆₇	1	1	yes	characteristic
209	C₂³	#14524005	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	#14524005	1	1	yes	characteristic
217	C₂×C₄	#14524005	1	1	yes	characteristic
218	C₂×C₄	#14524005	1	1	yes	characteristic
219	C₂×C₄	#14524072	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
222	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	4	no	—
225	C₈	—	268	2	no	—
226	C₈	—	134	2	no	—
227	C₈	—	134	2	no	—
228	C₈	#14524051	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#16863502	1	2	yes	—
232	C₂²	C₂×#14524005	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	#16863502	1	2	yes	—
238	C₄	#16863517	1	2	yes	—
239	C₄	C₂×#14524051	1	2	yes	—
240	C₄	C₂×#14524072	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#22568920	1	1	yes	characteristic
243	C₂	C₂×#16863502	1	1	yes	characteristic
244	C₂	C₂×#16863517	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#17961
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #22559117	C₂	semidirect
4. #22559027	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
#22569139	2
#22559117	2
#22559027	2
#22558663	2
C₂×#16863117	2
C₆₇×#1566	2
#17961	67