Subgroups — #30864710

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	#22569149	C₂	1	2	yes	maximal
3	#22561186	C₂	1	2	yes	maximal
4	#22560962	C₂	1	2	yes	maximal
5	#22560957	C₂	1	2	yes	maximal
6	#22560384	C₂	1	1	yes	maximal, characteristic
7	#22560093	C₂	1	2	yes	maximal
8	#22560088	C₂	1	2	yes	maximal
9	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#16863953	C₂²	1	2	yes	—
11	#16863805	C₄	1	8	yes	complemented
12	#16863524	—	2	8	no	—
13	C₂×#14523979	C₂²	1	2	yes	—
14	C₂×#14523979	C₂²	1	2	yes	—
15	#16863434	C₂²	1	2	yes	—
16	#16863434	C₂²	1	2	yes	—
17	#16863422	C₂²	1	4	yes	—
18	#16863420	C₂²	1	2	yes	—
19	#16863351	C₂²	1	4	yes	—
20	#16863156	C₂²	1	4	yes	—
21	#16863156	C₂²	1	4	yes	—
22	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₄³×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#11761612	C₂³	1	4	yes	—
30	#14524043	C₂×C₄	1	4	yes	—
31	#14524035	C₂×C₄	1	4	yes	—
32	#14524022	—	2	4	no	—
33	C₄×Dic₁₃₄	—	2	4	no	—
34	#14524002	C₂³	1	4	yes	—
35	#14523979	—	2	4	no	—
36	#14523979	—	2	4	no	—
37	#14523977	C₂×C₄	1	4	yes	—
38	#14523977	C₂×C₄	1	4	yes	—
39	#14523975	C₂×C₄	1	4	yes	—
40	#14523975	C₂×C₄	1	4	yes	—
41	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
42	Dic₆₇⋊C₈	—	2	8	no	—
43	#14523953	—	2	4	no	—
44	C₈×Dic₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	#97×C₆₇	—	2	2	no	—
54	#97×C₆₇	C₂×C₄	1	4	yes	—
55	#97×C₆₇	D₄	1	4	yes	—
56	#97×C₆₇	Q₈	1	4	yes	complemented
57	C₄×C₈×C₆₇	—	2	2	no	—
58	C₄×C₈×C₆₇	Q₈	1	4	yes	complemented
59	C₂×Dic₁₃₄	—	2	4	no	—
60	#11761612	—	2	4	no	—
61	#11761612	—	2	4	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
65	Dic₆₇⋊C₄	—	2	8	no	—
66	C₂²⋊Dic₆₇	—	2	4	no	—
67	C₄×Dic₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	#41	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	2	1	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×Q₈	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	C₄²×C₆₇	—	2	1	no	—
85	C₄²×C₆₇	—	2	1	no	—
86	C₄²×C₆₇	—	4	4	no	—
87	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₆₇	C₂×D₄	1	2	yes	—
89	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
104	C₂×Dic₆₇	—	2	8	no	—
105	Dic₁₃₄	—	8	4	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	C₄×Q₈	1	4	yes	—
108	C₈×C₆₇	C₄×Q₈	1	4	yes	—
109	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
110	C₆₇⋊C₈	—	4	4	no	—
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	—	2	4	no	—
117	C₄×C₆₇	—	4	2	no	—
118	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₄×C₆₇	#532	1	1	yes	characteristic
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	—	67	1	no	—
134	#1773	—	67	2	no	—
135	#1772	—	67	2	no	—
136	C₄×#97	D₆₇	1	1	yes	characteristic
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	—	67	2	no	—
142	C₂×#108	—	67	2	no	—
143	#427	—	67	2	no	—
144	#427	—	67	2	no	—
145	#424	—	67	4	no	—
146	#423	—	67	2	no	—
147	#405	—	67	4	no	—
148	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
149	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
150	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
151	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
154	C₄³	C₂×D₆₇	1	1	yes	characteristic
155	#339	—	67	4	no	—
156	#339	—	67	4	no	—
157	C₂×M₁₆	—	67	4	no	—
158	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
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	—	2	2	no	—
168	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
171	#108	—	134	4	no	—
172	#108	—	134	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	—	134	8	no	—
178	C₄⋊C₈	—	134	4	no	—
179	#97	—	2	2	no	—
180	#97	—	134	4	no	—
181	#97	Dic₁₃₄	1	4	yes	complemented
182	#97	C₄×D₆₇	1	4	yes	—
183	#97	D₂₆₈	1	4	yes	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	134	4	no	—
186	C₄×C₈	Dic₁₃₄	1	4	yes	complemented
187	C₂×Q₈	—	134	4	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	#11761628	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	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	134	4	no	—
197	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
198	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
199	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
200	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
201	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
202	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
203	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
204	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
205	C₂×C₈	#11761628	1	2	yes	—
206	C₂×C₈	#11761628	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	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	4	4	no	—
214	C₄²	—	134	4	no	—
215	C₄²	C₂×C₄×D₆₇	1	2	yes	—
216	C₄²	C₂×D₂₆₈	1	2	yes	—
217	C₄²	C₂×Dic₁₃₄	1	2	yes	—
218	C₄²	C₂×Dic₁₃₄	1	2	yes	—
219	C₂³	C₂×#11761628	1	1	yes	characteristic
220	Q₈	—	536	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	134	8	no	—
227	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
230	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
231	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
233	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	268	4	no	—
236	C₈	C₄×Dic₁₃₄	1	4	yes	—
237	C₈	C₄×Dic₁₃₄	1	4	yes	—
238	C₈	#14524043	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
241	C₂²	#16863953	1	1	yes	characteristic
242	C₂²	#16863953	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	2	no	—
246	C₄	—	268	8	no	—
247	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
248	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
249	C₄	C₂×#14524043	1	1	yes	characteristic
250	C₄	#16863946	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₄×#14524043	1	1	yes	characteristic
254	C₂	#22569120	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#97×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#97	D₆₇
hypocenter	C₆₇	#21736
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
11. #16863805	C₄	semidirect
56. #97×C₆₇	Q₈	semidirect
58. C₄×C₈×C₆₇	Q₈	semidirect
137. C₆₇	#21736	semidirect
181. #97	Dic₁₃₄	semidirect
186. 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
#22570733	2
#22569149	2
#22561186	2
#22560962	2
#22560957	2
#22560384	2
#22560093	2
#22560088	2
C₄×#97×C₆₇	2
#21736	67