Subgroups — #30744293

Order 17152.

448 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568587	C₂	1	1	yes	maximal, characteristic
2	#22559955	C₂	1	1	yes	maximal, characteristic
3	#22558034	C₂	1	4	yes	maximal
4	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524043	C₄	1	1	yes	characteristic
6	C₂×#14524043	C₄	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
8	#16863302	—	2	1	no	—
9	#16863302	—	2	1	no	—
10	#16863302	C₄	1	2	yes	—
11	#16863302	C₂²	1	2	yes	—
12	#16863262	—	2	1	no	—
13	#16863262	—	2	1	no	—
14	#16863151	—	2	1	no	—
15	#16863151	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
17	C₂×#97×C₆₇	—	2	1	no	—
18	C₄³×C₆₇	C₂²	1	1	yes	characteristic
19	#336×C₆₇	C₂²	1	2	yes	—
20	C₂²×Dic₁₃₄	—	2	1	no	—
21	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	#14524043	—	4	2	no	—
24	#14524043	—	4	2	no	—
25	C₂×C₄⋊Dic₆₇	—	2	1	no	—
26	C₂×C₄⋊Dic₆₇	—	2	1	no	—
27	C₂×C₄⋊Dic₆₇	—	2	1	no	—
28	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
29	C₄×C₆₇⋊C₈	—	2	4	no	—
30	C₆₇⋊#95	—	2	2	no	—
31	C₆₇⋊#95	—	2	2	no	—
32	C₆₇⋊#95	—	2	2	no	—
33	C₆₇⋊#95	—	2	2	no	—
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	4	no	—
43	#97×C₆₇	—	4	2	no	—
44	C₂×Dic₁₃₄	—	2	4	no	—
45	C₂×Dic₁₃₄	—	4	2	no	—
46	C₂×Dic₁₃₄	—	4	2	no	—
47	C₂×Dic₁₃₄	—	4	2	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂²×Dic₆₇	—	2	2	no	—
51	C₂²×Dic₆₇	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₄⋊Dic₆₇	—	4	2	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	—	4	2	no	—
68	C₂×C₈×C₆₇	—	4	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	—	4	1	no	—
75	C₄²×C₆₇	SD₁₆	1	2	yes	—
76	C₄²×C₆₇	Q₁₆	1	2	yes	—
77	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	#100	1	2	yes	—
92	C₂×C₄×C₆₇	#103	1	2	yes	—
93	C₂×C₄×C₆₇	#103	1	2	yes	—
94	C₂×C₄×C₆₇	#104	1	2	yes	—
95	C₂×C₄×C₆₇	#104	1	2	yes	—
96	C₂×C₄×C₆₇	#131	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#137	1	2	yes	—
100	C₂×Dic₆₇	—	4	2	no	—
101	C₂×Dic₆₇	—	4	2	no	—
102	C₂×Dic₆₇	—	4	2	no	—
103	C₂×Dic₆₇	—	4	2	no	—
104	C₂×Dic₆₇	—	4	2	no	—
105	C₂×Dic₆₇	—	4	2	no	—
106	Dic₁₃₄	—	8	4	no	—
107	Dic₁₃₄	—	8	4	no	—
108	C₈×C₆₇	—	4	4	no	—
109	C₆₇⋊C₈	—	4	4	no	—
110	C₂²×C₆₇	#408	1	1	yes	characteristic
111	C₂²×C₆₇	#410	1	1	yes	characteristic
112	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
114	C₂²×C₆₇	#419	1	1	yes	characteristic
115	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
116	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	4	1	no	—
124	C₄×C₆₇	#329	1	4	yes	—
125	Dic₆₇	—	8	2	no	—
126	Dic₆₇	—	8	2	no	—
127	#7682	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	#1469	1	1	yes	characteristic
129	C₂×C₆₇	#1485	1	1	yes	characteristic
130	C₂×C₆₇	#1493	1	1	yes	characteristic
131	C₂×C₆₇	#1503	1	1	yes	characteristic
132	C₂×C₆₇	#1506	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#329	1	1	yes	characteristic
134	C₂×C₆₇	#1548	1	1	yes	characteristic
135	#2862	—	67	1	no	—
136	#1767	D₆₇	1	1	yes	characteristic
137	#1744	—	67	1	no	—
138	#1272	—	67	4	no	—
139	C₆₇	#7682	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄⋊Q₈	—	67	1	no	—
141	C₂×C₄⋊Q₈	—	67	1	no	—
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×#97	—	2	1	no	—
144	C₂×C₄×C₈	—	134	1	no	—
145	#391	—	67	2	no	—
146	#391	—	67	2	no	—
147	#391	—	134	1	no	—
148	#391	—	134	1	no	—
149	#382	—	134	1	no	—
150	#382	—	134	1	no	—
151	C₄³	C₂×D₆₇	1	1	yes	characteristic
152	#336	—	67	2	no	—
153	#336	—	134	1	no	—
154	#336	C₂×D₆₇	1	2	yes	—
155	C₂²×Q₈	—	67	2	no	—
156	C₂²×Q₈	—	134	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	134	2	no	—
159	C₄⋊Q₈	—	268	2	no	—
160	C₄⋊Q₈	—	268	2	no	—
161	C₂×C₄⋊C₄	—	67	2	no	—
162	C₂×C₄⋊C₄	—	134	1	no	—
163	C₂×C₄⋊C₄	—	134	1	no	—
164	C₂×C₄⋊C₄	—	134	1	no	—
165	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
167	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
168	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
172	C₄⋊C₈	—	2	4	no	—
173	#97	—	4	2	no	—
174	C₄×C₈	—	134	4	no	—
175	#95	—	134	2	no	—
176	#95	—	134	2	no	—
177	#95	—	134	2	no	—
178	#95	—	134	2	no	—
179	C₂×Q₈	—	268	2	no	—
180	C₂×Q₈	—	268	2	no	—
181	C₂×Q₈	—	268	2	no	—
182	C₂×Q₈	—	268	2	no	—
183	C₂×Q₈	—	268	2	no	—
184	C₂×Q₈	—	134	4	no	—
185	C₂²×C₄	—	134	2	no	—
186	C₂²×C₄	—	134	2	no	—
187	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	268	2	no	—
198	C₂×C₈	—	268	2	no	—
199	C₂×C₈	—	134	4	no	—
200	C₄⋊C₄	—	268	2	no	—
201	C₄⋊C₄	—	268	2	no	—
202	C₄⋊C₄	—	268	2	no	—
203	C₄⋊C₄	—	268	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	4	1	no	—
210	C₄²	#11761616	1	2	yes	—
211	C₄²	#11761618	1	2	yes	—
212	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
213	Q₈	—	536	4	no	—
214	Q₈	—	536	4	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	268	2	no	—
229	C₂×C₄	—	268	2	no	—
230	C₂×C₄	—	268	2	no	—
231	C₂×C₄	—	268	2	no	—
232	C₂×C₄	—	268	2	no	—
233	C₂×C₄	—	268	2	no	—
234	C₂×C₄	#14523959	1	2	yes	—
235	C₂×C₄	#14523966	1	2	yes	—
236	C₂×C₄	#14523966	1	2	yes	—
237	C₂×C₄	#14523969	1	2	yes	—
238	C₂×C₄	#14523971	1	2	yes	—
239	C₂×C₄	#14524051	1	2	yes	—
240	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
241	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
242	C₂×C₄	#14524076	1	2	yes	—
243	C₈	—	4	4	no	—
244	C₈	—	268	4	no	—
245	C₂²	#16863364	1	1	yes	characteristic
246	C₂²	#16863369	1	1	yes	characteristic
247	C₂²	C₂×#14523959	1	1	yes	characteristic
248	C₂²	C₂×#14523966	1	1	yes	characteristic
249	C₂²	#16863406	1	1	yes	characteristic
250	C₂²	C₂×#14523969	1	1	yes	characteristic
251	C₂²	C₂×#14523971	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	536	2	no	—
260	C₄	—	536	2	no	—
261	C₄	#16863135	1	4	yes	—
262	C₂	#22558562	1	1	yes	characteristic
263	C₂	#22558631	1	1	yes	characteristic
264	C₂	#22558663	1	1	yes	characteristic
265	C₂	#22558707	1	1	yes	characteristic
266	C₂	#22558722	1	1	yes	characteristic
267	C₂	C₂×#16863135	1	1	yes	characteristic
268	C₂	#22558891	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1767	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1767	D₆₇
hypocenter	C₆₇	#7682
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
139. C₆₇	#7682	semidirect

Maximal subgroups

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

subgroup	index
#22568587	2
#22559955	2
#22558034	2
C₆₇×#1767	2
#7682	67