Subgroups — #30760737

Order 17152.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560967	C₂	1	2	yes	maximal
2	#22560369	C₂	1	1	yes	maximal, characteristic
3	#22558093	C₂	1	2	yes	maximal
4	#22558044	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523977	—	2	1	no	—
8	C₂×#14523976	—	2	2	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863266	C₂²	1	2	yes	—
11	#16863158	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
16	#336×C₆₇	—	2	1	no	—
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	#14523977	—	4	2	no	—
22	#14523977	—	4	2	no	—
23	#14523976	—	4	4	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	—
26	C₂²×C₈×C₆₇	—	2	1	no	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	—	2	1	no	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	2	no	—
34	C₄⋊C₈×C₆₇	—	4	2	no	—
35	C₄×C₈×C₆₇	—	2	2	no	—
36	#95×C₆₇	—	2	1	no	—
37	C₂²×Dic₆₇	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	2	4	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₄⋊Dic₆₇	—	4	2	no	—
42	C₄⋊Dic₆₇	—	4	2	no	—
43	C₄⋊Dic₆₇	—	4	2	no	—
44	C₄⋊Dic₆₇	—	4	2	no	—
45	C₄⋊Dic₆₇	—	8	2	no	—
46	C₂²×C₄×C₆₇	—	2	1	no	—
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	1	no	—
52	C₂×C₈×C₆₇	—	2	1	no	—
53	C₂×C₈×C₆₇	—	2	1	no	—
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	4	1	no	—
56	C₂×C₈×C₆₇	—	4	1	no	—
57	C₂×C₈×C₆₇	—	4	1	no	—
58	C₂×C₈×C₆₇	D₈	1	2	yes	—
59	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
60	C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
63	C₄²×C₆₇	—	4	1	no	—
64	C₄²×C₆₇	—	4	1	no	—
65	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
66	C₄²×C₆₇	#41	1	2	yes	—
67	C₂³×C₆₇	#121	1	1	yes	characteristic
68	C₂×C₄×C₆₇	—	2	1	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	4	1	no	—
72	C₂×C₄×C₆₇	—	4	1	no	—
73	C₂×C₄×C₆₇	—	4	1	no	—
74	C₂×C₄×C₆₇	—	4	1	no	—
75	C₂×C₄×C₆₇	—	4	1	no	—
76	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#125	1	2	yes	—
79	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
84	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#137	1	2	yes	—
87	C₂×Dic₆₇	—	8	2	no	—
88	C₂×Dic₆₇	—	8	2	no	—
89	C₂×Dic₆₇	—	8	2	no	—
90	C₈×C₆₇	—	2	2	no	—
91	C₈×C₆₇	—	2	2	no	—
92	C₈×C₆₇	—	8	1	no	—
93	C₆₇⋊C₈	—	4	4	no	—
94	C₂²×C₆₇	#398	1	1	yes	characteristic
95	C₂²×C₆₇	#459	1	1	yes	characteristic
96	C₂²×C₆₇	#462	1	1	yes	characteristic
97	C₂²×C₆₇	#466	1	1	yes	characteristic
98	C₂²×C₆₇	#470	1	1	yes	characteristic
99	C₂²×C₆₇	#472	1	1	yes	characteristic
100	C₂²×C₆₇	#473	1	1	yes	characteristic
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	4	2	no	—
103	C₄×C₆₇	—	8	1	no	—
104	C₄×C₆₇	#474	1	2	yes	—
105	C₄×C₆₇	#479	1	2	yes	—
106	C₄×C₆₇	#480	1	1	yes	characteristic
107	C₄×C₆₇	#484	1	1	yes	characteristic
108	C₄×C₆₇	#500	1	1	yes	characteristic
109	C₄×C₆₇	#501	1	1	yes	characteristic
110	Dic₆₇	—	16	2	no	—
111	#10649	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	#1667	1	1	yes	characteristic
113	C₂×C₆₇	#1674	1	1	yes	characteristic
114	C₂×C₆₇	#1689	1	1	yes	characteristic
115	C₂×C₆₇	#1696	1	1	yes	characteristic
116	C₂×C₆₇	#2083	1	1	yes	characteristic
117	C₂×C₆₇	#2086	1	1	yes	characteristic
118	C₂×C₆₇	#2089	1	1	yes	characteristic
119	#1947	—	67	2	no	—
120	#1939	D₆₇	1	1	yes	characteristic
121	#1845	—	67	1	no	—
122	#1296	—	67	2	no	—
123	#1277	—	67	1	no	—
124	C₆₇	#10649	1	1	yes	characteristic, complemented, Sylow
125	C₂×#107	—	134	1	no	—
126	C₂×#107	—	134	1	no	—
127	C₂×#107	—	134	2	no	—
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	—	67	2	no	—
130	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	#384	—	67	2	no	—
133	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
134	#340	—	134	2	no	—
135	#336	—	2	1	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	134	2	no	—
138	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	134	2	no	—
141	C₂×C₄⋊C₄	—	134	2	no	—
142	C₂×C₄⋊C₄	—	134	2	no	—
143	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
144	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
147	#107	—	268	2	no	—
148	#107	—	268	2	no	—
149	#107	—	268	4	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	4	2	no	—
152	C₄⋊C₈	—	67	4	no	—
153	C₄⋊C₈	—	134	2	no	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	268	2	no	—
159	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
161	C₂²×C₄	#11761628	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	268	2	no	—
170	C₂×C₈	—	268	2	no	—
171	C₂×C₈	—	134	4	no	—
172	C₂×C₈	#11761613	1	2	yes	—
173	C₂×C₈	#11761618	1	2	yes	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	268	2	no	—
177	C₄⋊C₄	—	268	2	no	—
178	C₄⋊C₄	—	268	2	no	—
179	C₄⋊C₄	—	268	2	no	—
180	C₄⋊C₄	—	536	2	no	—
181	C₄⋊C₄	D₅₃₆	1	2	yes	—
182	C₄²	—	4	1	no	—
183	C₄²	—	4	1	no	—
184	C₄²	Q₈×D₆₇	1	2	yes	—
185	C₄²	#11761627	1	2	yes	—
186	C₂³	#14524017	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	536	2	no	—
196	C₂×C₄	—	536	2	no	—
197	C₂×C₄	—	536	2	no	—
198	C₂×C₄	#14524023	1	2	yes	—
199	C₂×C₄	#14524024	1	1	yes	characteristic
200	C₂×C₄	#14524033	1	2	yes	—
201	C₂×C₄	#14524041	1	1	yes	characteristic
202	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
203	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
204	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
205	C₂×C₄	#14524065	1	1	yes	characteristic
206	C₂×C₄	#14524068	1	1	yes	characteristic
207	C₂×C₄	#14524073	1	1	yes	characteristic
208	C₂×C₄	#14524076	1	2	yes	—
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	8	1	no	—
212	C₈	—	268	4	no	—
213	C₂²	#16863326	1	1	yes	characteristic
214	C₂²	#16863591	1	1	yes	characteristic
215	C₂²	#16863606	1	1	yes	characteristic
216	C₂²	#16863632	1	1	yes	characteristic
217	C₂²	#16863654	1	1	yes	characteristic
218	C₂²	#16863666	1	1	yes	characteristic
219	C₂²	#16863671	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	1072	2	no	—
224	C₄	#16863677	1	2	yes	—
225	C₄	#16863704	1	2	yes	—
226	C₄	#16863707	1	1	yes	characteristic
227	C₄	#16863729	1	1	yes	characteristic
228	C₄	#16863799	1	1	yes	characteristic
229	C₄	#16863806	1	1	yes	characteristic
230	C₂	#22559564	1	1	yes	characteristic
231	C₂	#22559603	1	1	yes	characteristic
232	C₂	#22559698	1	1	yes	characteristic
233	C₂	#22559737	1	1	yes	characteristic
234	C₂	#22561680	1	1	yes	characteristic
235	C₂	#22561697	1	1	yes	characteristic
236	C₂	#22561714	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1939	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	#1939	D₆₇
hypocenter	C₆₇	#10649
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
124. C₆₇	#10649	semidirect

Maximal subgroups

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

subgroup	index
#22560967	2
#22560369	2
#22558093	2
#22558044	2
C₆₇×#1939	2
#10649	67