Subgroups — #30755978

Order 17152.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568607	C₂	1	1	yes	maximal, characteristic
2	#22560326	C₂	1	1	yes	maximal, characteristic
3	#22558093	C₂	1	4	yes	maximal
4	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524043	—	2	1	no	—
6	C₂×#14524043	—	2	1	no	—
7	C₂×#14524043	C₄	1	2	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	#16863266	—	2	1	no	—
10	#16863266	—	2	1	no	—
11	#16863266	C₄	1	2	yes	—
12	#16863266	C₂²	1	2	yes	—
13	#16863151	—	2	1	no	—
14	#16863151	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#97×C₆₇	—	2	1	no	—
17	C₄³×C₆₇	C₂²	1	1	yes	characteristic
18	#336×C₆₇	—	2	1	no	—
19	C₂²×Dic₁₃₄	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	#14524043	—	4	4	no	—
22	#14524043	—	4	4	no	—
23	#14524043	—	4	4	no	—
24	C₂×C₄⋊Dic₆₇	—	2	2	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
30	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	C₄⋊C₈×C₆₇	Q₈	1	4	yes	complemented
41	#97×C₆₇	—	4	2	no	—
42	C₂×Dic₁₃₄	—	4	4	no	—
43	C₂×Dic₁₃₄	—	4	4	no	—
44	C₂×Dic₁₃₄	—	4	4	no	—
45	C₂²×Dic₆₇	—	2	4	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₄⋊Dic₆₇	—	4	4	no	—
50	C₄⋊Dic₆₇	—	4	4	no	—
51	C₄⋊Dic₆₇	—	4	4	no	—
52	C₄⋊Dic₆₇	—	4	4	no	—
53	C₄⋊Dic₆₇	—	4	4	no	—
54	C₄⋊Dic₆₇	—	4	4	no	—
55	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	4	2	no	—
64	C₂×C₈×C₆₇	—	4	2	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	—	2	2	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	SD₁₆	1	2	yes	—
71	C₄²×C₆₇	Q₁₆	1	2	yes	—
72	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
73	C₄²×C₆₇	#41	1	2	yes	—
74	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
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	#100	1	2	yes	—
86	C₂×C₄×C₆₇	#101	1	2	yes	—
87	C₂×C₄×C₆₇	#103	1	2	yes	—
88	C₂×C₄×C₆₇	#103	1	2	yes	—
89	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
90	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₆₇	#137	1	2	yes	—
98	C₂×Dic₆₇	—	4	4	no	—
99	C₂×Dic₆₇	—	4	4	no	—
100	C₂×Dic₆₇	—	4	4	no	—
101	Dic₁₃₄	—	8	8	no	—
102	C₈×C₆₇	—	4	4	no	—
103	C₆₇⋊C₈	—	8	2	no	—
104	C₂²×C₆₇	#391	1	1	yes	characteristic
105	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
106	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
107	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
108	C₂²×C₆₇	#418	1	1	yes	characteristic
109	C₂²×C₆₇	#418	1	1	yes	characteristic
110	C₂²×C₆₇	#419	1	1	yes	characteristic
111	C₄×C₆₇	—	2	2	no	—
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	—	2	2	no	—
114	C₄×C₆₇	—	2	2	no	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	#476	1	2	yes	—
117	C₄×C₆₇	#477	1	2	yes	—
118	C₄×C₆₇	#489	1	2	yes	—
119	C₄×C₆₇	#501	1	2	yes	—
120	Dic₆₇	—	8	4	no	—
121	#9903	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	#1548	1	1	yes	characteristic
123	C₂×C₆₇	#1549	1	1	yes	characteristic
124	C₂×C₆₇	#1555	1	1	yes	characteristic
125	C₂×C₆₇	#1556	1	1	yes	characteristic
126	C₂×C₆₇	#1987	1	1	yes	characteristic
127	C₂×C₆₇	#1993	1	1	yes	characteristic
128	C₂×C₆₇	#1997	1	1	yes	characteristic
129	#2868	—	67	1	no	—
130	#1831	—	67	1	no	—
131	#1767	D₆₇	1	1	yes	characteristic
132	#1296	—	67	4	no	—
133	C₆₇	#9903	1	1	yes	characteristic, complemented, Sylow
134	C₂×C₄⋊Q₈	—	67	2	no	—
135	C₂×C₄⋊Q₈	—	134	1	no	—
136	C₂×C₄⋊Q₈	—	134	1	no	—
137	C₂×C₄⋊C₈	—	67	2	no	—
138	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
139	C₂×#97	—	2	1	no	—
140	#384	—	67	2	no	—
141	#384	—	67	2	no	—
142	#384	—	134	1	no	—
143	#384	—	134	1	no	—
144	C₄³	C₂×D₆₇	1	1	yes	characteristic
145	#336	—	2	1	no	—
146	#336	—	134	1	no	—
147	#336	—	134	1	no	—
148	C₂²×Q₈	—	134	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	134	2	no	—
151	C₄⋊Q₈	—	268	4	no	—
152	C₄⋊Q₈	—	268	4	no	—
153	C₄⋊Q₈	—	268	4	no	—
154	C₂×C₄⋊C₄	—	67	4	no	—
155	C₂×C₄⋊C₄	—	134	2	no	—
156	C₂×C₄⋊C₄	—	134	2	no	—
157	C₂×C₄⋊C₄	—	134	2	no	—
158	C₂×C₄⋊C₄	—	134	2	no	—
159	C₂×C₄⋊C₄	—	134	2	no	—
160	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
161	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
162	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
163	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	134	4	no	—
169	C₄⋊C₈	Dic₁₃₄	1	4	yes	complemented
170	#97	—	4	2	no	—
171	C₂×Q₈	—	268	4	no	—
172	C₂×Q₈	—	268	4	no	—
173	C₂×Q₈	—	268	4	no	—
174	C₂²×C₄	—	134	4	no	—
175	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	268	2	no	—
186	C₂×C₈	—	268	2	no	—
187	C₂×C₈	—	268	2	no	—
188	C₄⋊C₄	—	268	4	no	—
189	C₄⋊C₄	—	268	4	no	—
190	C₄⋊C₄	—	268	4	no	—
191	C₄⋊C₄	—	268	4	no	—
192	C₄⋊C₄	—	268	4	no	—
193	C₄⋊C₄	—	268	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	#11761616	1	2	yes	—
200	C₄²	#11761618	1	2	yes	—
201	C₄²	C₂×Dic₁₃₄	1	2	yes	—
202	C₄²	#11761628	1	2	yes	—
203	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
204	Q₈	—	536	8	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	268	4	no	—
216	C₂×C₄	—	268	4	no	—
217	C₂×C₄	—	268	4	no	—
218	C₂×C₄	#14523959	1	2	yes	—
219	C₂×C₄	#14523961	1	2	yes	—
220	C₂×C₄	#14523966	1	2	yes	—
221	C₂×C₄	#14523966	1	2	yes	—
222	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
223	C₂×C₄	#14524022	1	2	yes	—
224	C₂×C₄	#14524031	1	1	yes	characteristic
225	C₂×C₄	#14524043	1	1	yes	characteristic
226	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
227	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
228	C₂×C₄	#14524068	1	2	yes	—
229	C₂×C₄	#14524071	1	2	yes	—
230	C₂×C₄	#14524076	1	2	yes	—
231	C₈	—	4	4	no	—
232	C₈	—	536	2	no	—
233	C₂²	#16863302	1	1	yes	characteristic
234	C₂²	C₂×#14523959	1	1	yes	characteristic
235	C₂²	C₂×#14523961	1	1	yes	characteristic
236	C₂²	C₂×#14523966	1	1	yes	characteristic
237	C₂²	#16863400	1	1	yes	characteristic
238	C₂²	#16863402	1	1	yes	characteristic
239	C₂²	#16863406	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	536	4	no	—
246	C₄	#16863686	1	2	yes	—
247	C₄	#16863691	1	2	yes	—
248	C₄	#16863756	1	2	yes	—
249	C₄	#16863805	1	2	yes	—
250	C₂	#22558891	1	1	yes	characteristic
251	C₂	#22558895	1	1	yes	characteristic
252	C₂	#22558922	1	1	yes	characteristic
253	C₂	#22558927	1	1	yes	characteristic
254	C₂	#22561157	1	1	yes	characteristic
255	C₂	#22561182	1	1	yes	characteristic
256	C₂	#22561198	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	#9903
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
40. C₄⋊C₈×C₆₇	Q₈	semidirect
133. C₆₇	#9903	semidirect
169. C₄⋊C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22568607	2
#22560326	2
#22558093	2
C₆₇×#1767	2
#9903	67