Subgroups — #30826691

Order 17152.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#22569306	C₂	1	2	yes	maximal
3	#22559079	C₂	1	4	yes	maximal, complemented
4	#22559058	C₂	1	4	yes	maximal, complemented
5	#22558681	C₂	1	2	yes	maximal
6	#22558501	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1568	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	2	yes	—
13	#16863637	C₄	1	2	yes	—
14	#16863524	—	2	4	no	—
15	#16863523	C₂²	1	4	yes	—
16	#16863520	C₂²	1	4	yes	—
17	C₄×Dic₂₆₈	C₂²	1	2	yes	—
18	C₄×Dic₂₆₈	C₂²	1	2	yes	—
19	#16863454	—	2	4	no	—
20	#16863449	C₂²	1	1	yes	characteristic
21	#16863449	C₂²	1	1	yes	characteristic
22	#16863126	C₂²	1	8	yes	—
23	#16863118	C₂²	1	2	yes	—
24	#16863118	C₂²	1	2	yes	—
25	#432×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#335×C₆₇	C₂²	1	2	yes	—
28	#334×C₆₇	C₂²	1	2	yes	—
29	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₆₈	C₂×C₄	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	2	yes	—
40	#14523977	C₂×C₄	1	2	yes	—
41	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
43	Dic₆₇⋊C₈	—	2	4	no	—
44	Dic₆₇⋊C₈	—	2	4	no	—
45	#14523968	C₂×C₄	1	4	yes	—
46	#14523968	C₂×C₄	1	4	yes	—
47	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
48	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
49	#14523953	—	2	4	no	—
50	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
51	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
52	C₈×Dic₆₇	—	2	4	no	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
57	C₂²⋊C₈×C₆₇	—	2	2	no	—
58	#97×C₆₇	—	2	2	no	—
59	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₃₄	—	2	2	no	—
64	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
65	Dic₂₆₈	—	4	2	no	—
66	Dic₂₆₈	—	4	2	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
69	C₄⋊Dic₆₇	—	2	2	no	—
70	C₄⋊Dic₆₇	—	2	2	no	—
71	C₄⋊Dic₆₇	—	2	2	no	—
72	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
73	Dic₆₇⋊C₄	—	2	4	no	—
74	Dic₆₇⋊C₄	—	2	4	no	—
75	C₂²⋊Dic₆₇	—	2	4	no	—
76	C₄×Dic₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	4	1	no	—
82	C₂×C₈×C₆₇	—	4	1	no	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	#41	1	1	yes	characteristic
90	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	4	1	no	—
96	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₆₇	—	2	4	no	—
103	C₂×Dic₆₇	—	2	4	no	—
104	Dic₁₃₄	—	2	4	no	—
105	Dic₁₃₄	—	2	4	no	—
106	Dic₁₃₄	—	8	2	no	—
107	C₈×C₆₇	—	2	1	no	—
108	C₈×C₆₇	—	2	1	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	—	2	1	no	—
111	C₈×C₆₇	—	4	4	no	—
112	C₆₇⋊C₈	—	2	4	no	—
113	C₆₇⋊C₈	—	4	2	no	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	1	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	#568	1	1	yes	characteristic
123	C₄×C₆₇	#568	1	1	yes	characteristic
124	C₄×C₆₇	#578	1	2	yes	—
125	Dic₆₇	—	4	4	no	—
126	Dic₆₇	—	4	4	no	—
127	#17991	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	—	4	1	no	—
129	C₂×C₆₇	#2932	1	1	yes	characteristic
130	C₂×C₆₇	#2950	1	1	yes	characteristic
131	C₂×C₆₇	#2957	1	1	yes	characteristic
132	#3170	—	67	1	no	—
133	#2985	—	67	2	no	—
134	#1587	—	67	4	no	—
135	#1584	—	67	4	no	—
136	#1568	D₆₇	1	1	yes	characteristic
137	#1497	—	67	2	no	—
138	#1452	—	67	1	no	—
139	C₆₇	#17991	1	1	yes	characteristic, complemented, Sylow
140	#533	—	67	2	no	—
141	#500	—	67	1	no	—
142	#494	—	67	1	no	—
143	#487	—	67	2	no	—
144	#467	—	67	2	no	—
145	#467	—	67	2	no	—
146	#446	—	67	4	no	—
147	#446	—	134	4	no	—
148	C₈×Q₈	—	67	4	no	—
149	C₄×Q₁₆	—	67	2	no	—
150	C₄×Q₁₆	—	67	2	no	—
151	#433	—	134	4	no	—
152	#432	C₂×D₆₇	1	2	yes	—
153	#431	—	67	1	no	—
154	#431	—	67	1	no	—
155	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
156	#335	C₂×D₆₇	1	2	yes	—
157	#334	C₂×D₆₇	1	2	yes	—
158	Q₈⋊C₈	—	67	8	no	—
159	#322	—	67	2	no	—
160	#322	—	67	2	no	—
161	C₂×Q₁₆	—	67	2	no	—
162	C₂×Q₁₆	—	67	2	no	—
163	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
165	C₄⋊Q₈	—	67	2	no	—
166	C₄⋊Q₈	—	67	2	no	—
167	#125	—	134	2	no	—
168	C₂²⋊Q₈	—	67	4	no	—
169	C₂²⋊Q₈	—	134	2	no	—
170	C₄×Q₈	—	67	4	no	—
171	C₄×Q₈	—	134	2	no	—
172	#117	—	134	2	no	—
173	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
174	#107	—	67	2	no	—
175	#107	—	67	2	no	—
176	C₄⋊C₈	—	67	2	no	—
177	C₄⋊C₈	—	134	4	no	—
178	C₄⋊C₈	—	134	4	no	—
179	C₄⋊C₈	—	134	1	no	—
180	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
181	#103	—	67	4	no	—
182	#103	—	67	4	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₂²⋊C₈	—	67	2	no	—
185	C₂²⋊C₈	—	134	1	no	—
186	#97	—	2	2	no	—
187	#97	—	134	4	no	—
188	C₄×C₈	—	67	2	no	—
189	C₄×C₈	—	67	2	no	—
190	C₄×C₈	—	134	4	no	—
191	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
192	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	C₂×Q₈	—	67	4	no	—
196	C₂×Q₈	—	134	2	no	—
197	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
198	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
199	C₂²×C₄	#11761629	1	1	yes	characteristic
200	Q₁₆	—	268	2	no	—
201	Q₁₆	—	268	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	67	4	no	—
206	C₂×C₈	—	134	2	no	—
207	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
211	C₄⋊C₄	—	67	4	no	—
212	C₄⋊C₄	—	134	2	no	—
213	C₄⋊C₄	—	134	2	no	—
214	C₄⋊C₄	—	134	4	no	—
215	C₄⋊C₄	—	134	2	no	—
216	C₄⋊C₄	—	134	4	no	—
217	C₂²⋊C₄	—	134	4	no	—
218	C₄²	—	134	4	no	—
219	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
220	C₄²	D₄×D₆₇	1	1	yes	characteristic
221	C₄²	#11761629	1	1	yes	characteristic
222	C₄²	C₂³×D₆₇	1	1	yes	characteristic
223	C₂³	#14524005	1	1	yes	characteristic
224	Q₈	—	536	2	no	—
225	Q₈	—	134	4	no	—
226	Q₈	—	134	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	134	4	no	—
232	C₂×C₄	—	134	4	no	—
233	C₂×C₄	#14524005	1	1	yes	characteristic
234	C₂×C₄	#14524005	1	1	yes	characteristic
235	C₂×C₄	#14524005	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
238	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	4	4	no	—
244	C₈	—	268	2	no	—
245	C₈	—	134	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×#14524005	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	268	4	no	—
255	C₄	—	268	4	no	—
256	C₄	#16864162	1	1	yes	characteristic
257	C₄	#16864162	1	1	yes	characteristic
258	C₄	#16864210	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#22568926	1	1	yes	characteristic
261	C₂	#22569076	1	1	yes	characteristic
262	C₂	#22569125	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×#1568	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#17991
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #22559079	C₂	semidirect
4. #22559058	C₂	semidirect
139. C₆₇	#17991	semidirect

Maximal subgroups

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

subgroup	index
#22570795	2
#22569306	2
#22559079	2
#22559058	2
#22558681	2
#22558501	2
C₆₇×#1568	2
#17991	67