Subgroups — #30856154

Order 17152.

778 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560354	C₂	1	2	yes	maximal, complemented
2	#22560329	C₂	1	2	yes	maximal, complemented
3	#22560280	C₂	1	2	yes	maximal
4	#22560193	C₂	1	2	yes	maximal
5	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
7	#22559955	C₂	1	2	yes	maximal, complemented
8	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
12	#16863151	C₂²	1	4	yes	—
13	#16863151	C₂²	1	4	yes	—
14	#16863151	C₂²	1	4	yes	—
15	#16863151	C₂²	1	4	yes	—
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	—
20	C₄×C₂²⋊C₄×C₆₇	C₄	1	4	yes	—
21	#376×C₆₇	C₂²	1	1	yes	characteristic
22	#376×C₆₇	C₂²	1	1	yes	characteristic
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	C₄³×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
26	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
28	C₄×C₆₇⋊C₈	—	2	8	no	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	#117×C₆₇	—	2	4	no	—
32	#117×C₆₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	—	2	4	no	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	#95×C₆₇	C₂×C₄	1	4	yes	—
48	#95×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₆₇⋊C₈	—	2	8	no	—
50	C₂×C₆₇⋊C₈	—	2	8	no	—
51	C₂×C₆₇⋊C₈	—	2	8	no	—
52	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	2	8	no	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	#41	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₂²⋊C₄×C₆₇	—	2	4	no	—
69	C₂²⋊C₄×C₆₇	—	2	4	no	—
70	C₄²×C₆₇	—	2	4	no	—
71	C₄²×C₆₇	—	2	8	no	—
72	C₄²×C₆₇	—	2	8	no	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	—	2	2	no	—
77	C₄²×C₆₇	#41	1	4	yes	—
78	C₄²×C₆₇	#41	1	4	yes	—
79	C₂³×C₆₇	—	2	1	no	—
80	C₂³×C₆₇	—	2	1	no	—
81	C₂³×C₆₇	—	2	4	no	—
82	C₂³×C₆₇	#117	1	2	yes	—
83	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	8	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	—	2	4	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	4	8	no	—
97	C₂×C₄×C₆₇	#117	1	4	yes	—
98	C₂×C₄×C₆₇	#117	1	2	yes	—
99	C₂×C₄×C₆₇	#117	1	4	yes	—
100	C₂×C₄×C₆₇	#131	1	8	yes	—
101	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
104	C₆₇⋊C₈	—	4	8	no	—
105	C₂²×C₆₇	—	2	4	no	—
106	C₂²×C₆₇	—	2	2	no	—
107	C₂²×C₆₇	—	4	2	no	—
108	C₂²×C₆₇	—	4	2	no	—
109	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
110	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
111	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
112	C₂²×C₆₇	C₂×#131	1	4	yes	—
113	C₄×C₆₇	—	2	4	no	—
114	C₄×C₆₇	—	2	2	no	—
115	C₄×C₆₇	—	4	2	no	—
116	C₄×C₆₇	—	4	2	no	—
117	C₄×C₆₇	—	4	2	no	—
118	C₄×C₆₇	—	4	2	no	—
119	C₄×C₆₇	#433	1	8	yes	—
120	C₄×C₆₇	#529	1	4	yes	—
121	#20782	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	—	4	2	no	—
123	C₂×C₆₇	#2283	1	1	yes	characteristic
124	C₂×C₆₇	C₂×#433	1	2	yes	—
125	C₂×C₆₇	#2922	1	2	yes	—
126	C₂×C₆₇	#2922	1	2	yes	—
127	C₄×#117	D₆₇	1	1	yes	characteristic
128	#1840	—	67	2	no	—
129	#1832	—	67	2	no	—
130	#1818	—	67	2	no	—
131	#1798	—	67	2	no	—
132	C₄×C₄⋊C₈	—	67	2	no	—
133	C₄×C₂²⋊C₈	—	67	2	no	—
134	#1744	—	67	2	no	—
135	C₆₇	#20782	1	1	yes	characteristic, complemented, Sylow
136	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
137	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
138	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	67	4	no	—
140	C₂×C₂²⋊C₈	—	67	4	no	—
141	C₂×C₄×C₈	—	67	4	no	—
142	C₄×C₄⋊C₄	Dic₆₇	1	4	yes	—
143	C₄×C₂²⋊C₄	Dic₆₇	1	4	yes	—
144	#376	C₂×D₆₇	1	1	yes	characteristic
145	#376	C₂×D₆₇	1	1	yes	characteristic
146	C₄³	C₂×D₆₇	1	1	yes	characteristic
147	C₄³	C₂×D₆₇	1	1	yes	characteristic
148	#336	—	67	4	no	—
149	#336	—	67	4	no	—
150	#336	—	67	4	no	—
151	#336	—	67	4	no	—
152	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
153	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
154	C₂²×C₈	—	67	8	no	—
155	#117	—	2	4	no	—
156	#117	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
163	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
164	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
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	C₄⋊C₈	—	134	8	no	—
172	C₂²⋊C₈	—	134	8	no	—
173	C₄×C₈	—	134	8	no	—
174	#95	C₂×Dic₆₇	1	4	yes	—
175	#95	C₂×Dic₆₇	1	4	yes	—
176	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
185	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
187	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
188	C₂²×C₄	#11761628	1	4	yes	—
189	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
190	C₂×C₈	—	134	8	no	—
191	C₂×C₈	—	134	8	no	—
192	C₂×C₈	—	134	8	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	8	no	—
199	C₄²	—	2	8	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	#11761628	1	4	yes	—
205	C₄²	#11761628	1	4	yes	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	4	no	—
209	C₂³	#14524002	1	2	yes	—
210	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	#14524002	1	4	yes	—
225	C₂×C₄	#14524002	1	2	yes	—
226	C₂×C₄	#14524002	1	4	yes	—
227	C₂×C₄	#14524050	1	8	yes	—
228	C₂×C₄	C₂×#11761628	1	2	yes	—
229	C₂×C₄	C₂×#11761628	1	2	yes	—
230	C₂×C₄	C₂×#11761628	1	2	yes	—
231	C₈	—	268	8	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	C₂×#14524002	1	1	yes	characteristic
237	C₂²	C₂×#14524002	1	1	yes	characteristic
238	C₂²	C₂×#14524002	1	1	yes	characteristic
239	C₂²	C₂×#14524050	1	4	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	#16863457	1	8	yes	—
247	C₄	#16863930	1	4	yes	—
248	C₂	—	4	2	no	—
249	C₂	#22562607	1	1	yes	characteristic
250	C₂	C₂×#16863457	1	2	yes	—
251	C₂	#22568853	1	2	yes	—
252	C₂	#22568853	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#117×C₆₇	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#117	D₆₇
hypocenter	C₆₇	#20782
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
1. #22560354	C₂	semidirect
2. #22560329	C₂	semidirect
5. C₄×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
7. #22559955	C₂	semidirect
135. C₆₇	#20782	semidirect

Maximal subgroups

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

subgroup	index
#22560354	2
#22560329	2
#22560280	2
#22560193	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₄×C₂²⋊(C₆₇⋊C₈)	2
#22559955	2
C₄×#117×C₆₇	2
#20782	67