Subgroups — #30864711

Order 17152.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569150	C₂	1	2	yes	maximal
2	#22561187	C₂	1	2	yes	maximal
3	#22560963	C₂	1	2	yes	maximal
4	#22560958	C₂	1	2	yes	maximal
5	#22560384	C₂	1	1	yes	maximal, characteristic
6	#22560095	C₂	1	2	yes	maximal
7	#22560090	C₂	1	2	yes	maximal
8	#97×Dic₆₇	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#3162	C₂	1	1	yes	maximal, characteristic
10	#16863527	—	2	8	no	—
11	C₂×#14523979	C₂²	1	2	yes	—
12	C₂×#14523979	C₂²	1	2	yes	—
13	#16863423	C₂²	1	4	yes	—
14	#16863420	C₂²	1	2	yes	—
15	#16863352	C₂²	1	4	yes	—
16	C₂×#14523954	C₂²	1	1	yes	characteristic
17	C₂×#14523953	C₂²	1	1	yes	characteristic
18	C₂×#14523953	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
21	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
22	#16863155	C₂²	1	4	yes	—
23	#16863155	C₂²	1	4	yes	—
24	#533×C₆₇	C₂²	1	2	yes	—
25	#501×C₆₇	C₄	1	8	yes	complemented
26	#427×C₆₇	C₂²	1	2	yes	—
27	#427×C₆₇	C₂²	1	2	yes	—
28	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#11761612	C₂³	1	4	yes	—
30	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
35	#14523979	—	2	4	no	—
36	#14523979	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
39	#14523954	—	2	4	no	—
40	#14523954	D₄	1	4	yes	—
41	#14523953	—	2	2	no	—
42	#14523953	Q₈	1	4	yes	complemented
43	C₄×C₆₇⋊C₈	—	2	4	no	—
44	C₈×Dic₆₇	—	2	2	no	—
45	C₈×Dic₆₇	Q₈	1	4	yes	complemented
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
49	#125×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²⋊Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	—	2	4	no	—
52	#117×C₆₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#107×C₆₇	C₂×C₄	1	4	yes	—
55	#107×C₆₇	C₂×C₄	1	4	yes	—
56	#106×C₆₇	C₂×C₄	1	4	yes	—
57	#106×C₆₇	C₂×C₄	1	4	yes	—
58	#97×C₆₇	C₂×C₄	1	4	yes	—
59	C₂²×Dic₆₇	—	2	1	no	—
60	C₂²×Dic₆₇	—	2	1	no	—
61	#11761612	—	2	4	no	—
62	#11761612	—	2	4	no	—
63	C₂×C₆₇⋊C₈	—	2	1	no	—
64	C₂×C₆₇⋊C₈	—	2	1	no	—
65	C₂×C₆₇⋊C₈	—	2	4	no	—
66	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
67	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
68	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
69	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
70	C₄×Dic₆₇	—	2	1	no	—
71	C₄×Dic₆₇	—	2	1	no	—
72	C₄×Dic₆₇	—	4	4	no	—
73	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
74	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
75	C₂×Q₈×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₆₇	#41	1	2	yes	—
83	C₂×C₈×C₆₇	#41	1	2	yes	—
84	C₄⋊C₄×C₆₇	—	2	8	no	—
85	C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
87	C₂²⋊C₄×C₆₇	—	2	4	no	—
88	C₄²×C₆₇	—	2	4	no	—
89	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₆₇	C₂×D₄	1	2	yes	—
91	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
92	Q₈×C₆₇	—	8	4	no	—
93	C₂×C₄×C₆₇	—	2	8	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₆₇	—	2	4	no	—
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂×Dic₆₇	—	2	2	no	—
105	C₂×Dic₆₇	—	4	2	no	—
106	C₈×C₆₇	C₄×Q₈	1	4	yes	—
107	C₈×C₆₇	C₄×Q₈	1	4	yes	—
108	C₆₇⋊C₈	—	2	2	no	—
109	C₆₇⋊C₈	—	4	4	no	—
110	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂²×C₆₇	#533	1	1	yes	characteristic
114	C₂²×C₆₇	#533	1	1	yes	characteristic
115	C₄×C₆₇	—	2	4	no	—
116	C₄×C₆₇	—	4	8	no	—
117	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
118	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₄×C₆₇	#532	1	1	yes	characteristic
121	Dic₆₇	—	2	4	no	—
122	Dic₆₇	—	4	2	no	—
123	#21736	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	2	1	no	—
126	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₆₇	#2956	1	2	yes	—
128	#3162	D₆₇	1	1	yes	characteristic
129	#2961	—	67	2	no	—
130	#1994	—	67	2	no	—
131	#1946	—	67	2	no	—
132	#1945	—	67	2	no	—
133	#1849	—	67	1	no	—
134	#1773	—	67	2	no	—
135	#1772	—	67	2	no	—
136	C₄×#97	—	67	1	no	—
137	C₆₇	#21736	1	1	yes	characteristic, complemented, Sylow
138	#533	C₂×D₆₇	1	2	yes	—
139	#501	Dic₆₇	1	8	yes	complemented
140	#446	—	134	8	no	—
141	C₂×#108	—	67	2	no	—
142	C₂×#108	—	67	2	no	—
143	#427	C₂×D₆₇	1	2	yes	—
144	#427	C₂×D₆₇	1	2	yes	—
145	#424	—	67	4	no	—
146	#423	—	67	2	no	—
147	#405	—	67	4	no	—
148	C₂×#97	—	67	1	no	—
149	C₂×#97	—	67	1	no	—
150	C₂×#97	—	67	1	no	—
151	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
152	C₂×C₄×C₈	—	67	1	no	—
153	C₂×C₄×C₈	—	67	1	no	—
154	C₄³	—	67	1	no	—
155	#339	—	67	4	no	—
156	#339	—	67	4	no	—
157	C₂×M₁₆	—	67	4	no	—
158	C₂²×C₈	—	67	1	no	—
159	C₂²×C₈	—	67	1	no	—
160	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
162	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
163	#125	C₂×Dic₆₇	1	4	yes	—
164	C₂²⋊Q₈	—	2	4	no	—
165	C₄×Q₈	—	2	4	no	—
166	#117	C₂²×D₆₇	1	4	yes	—
167	C₂×C₄²	—	67	1	no	—
168	C₂×C₄²	—	67	1	no	—
169	C₂×C₄²	—	134	2	no	—
170	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
171	#108	—	134	4	no	—
172	#108	—	134	4	no	—
173	#107	C₂×Dic₆₇	1	4	yes	—
174	#107	C₂×Dic₆₇	1	4	yes	—
175	#106	C₂×Dic₆₇	1	4	yes	—
176	#106	C₂×Dic₆₇	1	4	yes	—
177	C₄⋊C₈	—	134	8	no	—
178	C₄⋊C₈	—	134	4	no	—
179	#97	—	67	4	no	—
180	#97	—	67	4	no	—
181	#97	—	134	4	no	—
182	#97	—	134	2	no	—
183	#97	C₂×Dic₆₇	1	4	yes	—
184	C₄×C₈	—	67	4	no	—
185	C₄×C₈	—	134	4	no	—
186	C₄×C₈	—	134	2	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂²×C₄	—	134	1	no	—
189	C₂²×C₄	—	134	1	no	—
190	C₂²×C₄	#11761627	1	2	yes	—
191	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
192	M₁₆	—	134	4	no	—
193	M₁₆	—	134	4	no	—
194	C₂×C₈	—	67	2	no	—
195	C₂×C₈	—	67	2	no	—
196	C₂×C₈	—	67	2	no	—
197	C₂×C₈	—	67	2	no	—
198	C₂×C₈	—	134	4	no	—
199	C₂×C₈	—	134	1	no	—
200	C₂×C₈	—	134	1	no	—
201	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
202	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
203	C₂×C₈	Q₈×D₆₇	1	2	yes	—
204	C₂×C₈	Q₈×D₆₇	1	2	yes	—
205	C₂×C₈	#11761629	1	2	yes	—
206	C₂×C₈	#11761629	1	2	yes	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	268	4	no	—
213	C₄²	—	67	2	no	—
214	C₄²	—	67	2	no	—
215	C₄²	—	134	1	no	—
216	C₄²	—	134	1	no	—
217	C₄²	C₂²×Dic₆₇	1	2	yes	—
218	C₄²	D₄×D₆₇	1	2	yes	—
219	C₂³	C₂×#11761627	1	1	yes	characteristic
220	Q₈	—	8	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	268	2	no	—
224	C₂×C₄	—	134	2	no	—
225	C₂×C₄	—	134	4	no	—
226	C₂×C₄	—	134	2	no	—
227	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
228	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
230	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
231	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
233	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
234	C₈	—	268	4	no	—
235	C₈	—	67	4	no	—
236	C₈	—	134	2	no	—
237	C₈	Q₈×Dic₆₇	1	4	yes	—
238	C₈	Q₈×Dic₆₇	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
241	C₂²	#16863951	1	1	yes	characteristic
242	C₂²	#16863951	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	4	8	no	—
245	C₄	—	268	2	no	—
246	C₄	—	134	4	no	—
247	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
248	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
249	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
250	C₄	#16863947	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
254	C₂	#22569119	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3162	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3162	D₆₇
hypocenter	C₆₇	#21736
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

Complements

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

N	H ≅ G/N	type
25. #501×C₆₇	C₄	semidirect
42. #14523953	Q₈	semidirect
45. C₈×Dic₆₇	Q₈	semidirect
137. C₆₇	#21736	semidirect
139. #501	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569150	2
#22561187	2
#22560963	2
#22560958	2
#22560384	2
#22560095	2
#22560090	2
#97×Dic₆₇	2
C₆₇×#3162	2
#21736	67