Subgroups — #30864703

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	#22570733	C₂	1	1	yes	maximal, characteristic
2	#22569146	C₂	1	2	yes	maximal
3	#22561185	C₂	1	2	yes	maximal
4	#22560960	C₂	1	2	yes	maximal
5	#22560955	C₂	1	2	yes	maximal
6	#22560096	C₂	1	2	yes	maximal
7	#22560091	C₂	1	2	yes	maximal
8	#97×Dic₆₇	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic
10	#16863953	C₂²	1	2	yes	—
11	#16863805	C₄	1	8	yes	complemented
12	#16863525	—	2	8	no	—
13	#16863434	C₂²	1	2	yes	—
14	#16863434	C₂²	1	2	yes	—
15	#16863424	C₂²	1	4	yes	—
16	#16863354	C₂²	1	4	yes	—
17	C₂×#14523954	C₂²	1	1	yes	characteristic
18	C₂×#14523953	C₂²	1	1	yes	characteristic
19	C₂×#14523953	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
22	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
23	#16863157	C₂²	1	4	yes	—
24	#16863157	C₂²	1	4	yes	—
25	C₂×#108×C₆₇	C₂²	1	2	yes	—
26	C₂×#108×C₆₇	C₂²	1	2	yes	—
27	#423×C₆₇	C₂²	1	2	yes	—
28	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
31	#14524043	C₂×C₄	1	4	yes	—
32	#14524035	C₂×C₄	1	4	yes	—
33	#14524022	—	2	4	no	—
34	C₄×Dic₁₃₄	—	2	4	no	—
35	#14524002	C₂³	1	4	yes	—
36	C₂×C₄×Dic₆₇	—	2	2	no	—
37	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
39	#14523977	C₂×C₄	1	4	yes	—
40	#14523977	C₂×C₄	1	4	yes	—
41	#14523975	C₂×C₄	1	4	yes	—
42	#14523975	C₂×C₄	1	4	yes	—
43	Dic₆₇⋊C₈	—	2	8	no	—
44	#14523954	D₄	1	4	yes	—
45	#14523953	—	2	4	no	—
46	#14523953	—	2	2	no	—
47	#14523953	Q₈	1	4	yes	complemented
48	C₈×Dic₆₇	—	2	4	no	—
49	C₈×Dic₆₇	—	2	2	no	—
50	C₈×Dic₆₇	Q₈	1	4	yes	complemented
51	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#108×C₆₇	—	2	4	no	—
56	#108×C₆₇	—	2	4	no	—
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	#97×C₆₇	C₂×C₄	1	4	yes	—
59	C₂×Dic₁₃₄	—	2	4	no	—
60	C₂²×Dic₆₇	—	2	1	no	—
61	C₂²×Dic₆₇	—	2	1	no	—
62	C₂×C₆₇⋊C₈	—	2	1	no	—
63	C₂×C₆₇⋊C₈	—	2	1	no	—
64	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
65	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
66	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
67	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
70	Dic₆₇⋊C₄	—	2	8	no	—
71	C₂²⋊Dic₆₇	—	2	4	no	—
72	C₄×Dic₆₇	—	2	4	no	—
73	C₄×Dic₆₇	—	2	1	no	—
74	C₄×Dic₆₇	—	2	1	no	—
75	C₄×Dic₆₇	—	4	4	no	—
76	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
77	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₆₇	—	2	4	no	—
81	M₁₆×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₆₇	#41	1	2	yes	—
88	C₂×C₈×C₆₇	#41	1	2	yes	—
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	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
100	C₂×Dic₆₇	—	2	4	no	—
101	C₂×Dic₆₇	—	2	2	no	—
102	C₂×Dic₆₇	—	2	8	no	—
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂×Dic₆₇	—	4	2	no	—
105	Dic₁₃₄	—	8	4	no	—
106	C₈×C₆₇	—	4	4	no	—
107	C₈×C₆₇	C₄×Q₈	1	4	yes	—
108	C₈×C₆₇	C₄×Q₈	1	4	yes	—
109	C₆₇⋊C₈	—	2	2	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	C₂×C₄×Q₈	1	1	yes	characteristic
117	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
118	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₄×C₆₇	#532	1	1	yes	characteristic
120	Dic₆₇	—	2	4	no	—
121	Dic₆₇	—	4	2	no	—
122	Dic₆₇	—	4	8	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	—	67	1	no	—
129	#2961	—	67	2	no	—
130	#1994	—	67	2	no	—
131	#1946	—	67	2	no	—
132	#1945	—	67	2	no	—
133	#1849	D₆₇	1	1	yes	characteristic
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	—	67	2	no	—
139	#501	—	67	8	no	—
140	#446	—	134	8	no	—
141	C₂×#108	C₂×D₆₇	1	2	yes	—
142	C₂×#108	C₂×D₆₇	1	2	yes	—
143	#427	—	67	2	no	—
144	#427	—	67	2	no	—
145	#424	—	67	4	no	—
146	#423	C₂×D₆₇	1	2	yes	—
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	C₂²×D₆₇	1	4	yes	—
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	—	67	4	no	—
163	#125	—	67	4	no	—
164	C₂²⋊Q₈	—	134	4	no	—
165	C₄×Q₈	—	134	4	no	—
166	#117	—	67	4	no	—
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	—	2	4	no	—
172	#108	—	2	4	no	—
173	#107	—	67	4	no	—
174	#107	—	67	4	no	—
175	#106	—	67	4	no	—
176	#106	—	67	4	no	—
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	—	134	8	no	—
179	#97	—	67	4	no	—
180	#97	—	67	4	no	—
181	#97	—	134	4	no	—
182	#97	—	134	2	no	—
183	#97	C₄×D₆₇	1	4	yes	—
184	C₄×C₈	—	67	4	no	—
185	C₄×C₈	—	134	4	no	—
186	C₄×C₈	—	134	2	no	—
187	C₂×Q₈	—	134	4	no	—
188	C₂²×C₄	—	134	1	no	—
189	C₂²×C₄	—	134	1	no	—
190	C₂²×C₄	#11761629	1	2	yes	—
191	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	67	2	no	—
196	C₂×C₈	—	67	2	no	—
197	C₂×C₈	—	67	2	no	—
198	C₂×C₈	—	67	2	no	—
199	C₂×C₈	—	134	1	no	—
200	C₂×C₈	—	134	1	no	—
201	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
203	C₂×C₈	Q₈×D₆₇	1	2	yes	—
204	C₂×C₈	Q₈×D₆₇	1	2	yes	—
205	C₂×C₈	#11761627	1	2	yes	—
206	C₂×C₈	#11761627	1	2	yes	—
207	C₄⋊C₄	—	67	8	no	—
208	C₄⋊C₄	—	134	8	no	—
209	C₄⋊C₄	—	134	4	no	—
210	C₂²⋊C₄	—	134	4	no	—
211	C₄²	—	268	4	no	—
212	C₄²	—	67	2	no	—
213	C₄²	—	67	2	no	—
214	C₄²	—	134	4	no	—
215	C₄²	—	134	1	no	—
216	C₄²	—	134	1	no	—
217	C₄²	C₂×C₄×D₆₇	1	2	yes	—
218	C₄²	D₄×D₆₇	1	2	yes	—
219	C₂³	C₂×#11761629	1	1	yes	characteristic
220	Q₈	—	536	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	268	2	no	—
223	C₂×C₄	—	134	8	no	—
224	C₂×C₄	—	134	2	no	—
225	C₂×C₄	—	134	4	no	—
226	C₂×C₄	—	134	2	no	—
227	C₂×C₄	#14524005	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₆₇	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×#11761627	1	1	yes	characteristic
233	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
234	C₈	—	4	4	no	—
235	C₈	—	67	4	no	—
236	C₈	—	134	2	no	—
237	C₈	#14524008	1	4	yes	—
238	C₈	#14524008	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×#14524005	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	—	268	2	no	—
245	C₄	—	268	8	no	—
246	C₄	—	134	4	no	—
247	C₄	C₂×#14524008	1	1	yes	characteristic
248	C₄	C₂×#14524008	1	1	yes	characteristic
249	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
250	C₄	#16863944	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	#22562776	1	1	yes	characteristic
254	C₂	#22569118	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×#1849	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1849	D₆₇
hypocenter	C₆₇	#21736
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
11. #16863805	C₄	semidirect
47. #14523953	Q₈	semidirect
50. C₈×Dic₆₇	Q₈	semidirect
137. C₆₇	#21736	semidirect

Maximal subgroups

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

subgroup	index
#22570733	2
#22569146	2
#22561185	2
#22560960	2
#22560955	2
#22560096	2
#22560091	2
#97×Dic₆₇	2
C₆₇×#1849	2
#21736	67