Subgroups — #30869722

Order 17152.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568927	C₂	1	2	yes	maximal, complemented
2	#22561042	C₂	1	2	yes	maximal, complemented
3	#22560538	C₂	1	8	yes	maximal, complemented
4	#22560347	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559960	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524050	C₂²	1	4	yes	—
8	#16863468	—	2	8	no	—
9	#16863420	C₂²	1	2	yes	—
10	C₂×#14523970	C₂²	1	8	yes	—
11	#16863363	C₂²	1	4	yes	—
12	C₂×#14523954	C₂²	1	1	yes	characteristic
13	#16863242	C₂²	1	1	yes	characteristic
14	#16863161	C₂²	1	4	yes	—
15	#16863150	C₂²	1	4	yes	—
16	C₂×#135×C₆₇	C₂²	1	4	yes	—
17	#532×C₆₇	C₂²	1	2	yes	—
18	#495×C₆₇	C₄	1	4	yes	complemented
19	#494×C₆₇	C₄	1	1	yes	characteristic, complemented
20	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
21	#492×C₆₇	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#14524050	—	4	8	no	—
24	C₂×#11761612	—	2	4	no	—
25	C₂×#11761612	C₂³	1	4	yes	—
26	C₂²×C₆₇⋊C₈	—	2	4	no	—
27	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
29	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
30	#14523970	—	4	8	no	—
31	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
32	#14523954	D₄	1	4	yes	—
33	C₆₇⋊#95	—	2	2	no	—
34	C₂×#41×C₆₇	C₂³	1	4	yes	—
35	#135×C₆₇	—	4	8	no	—
36	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
38	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
41	#127×C₆₇	C₂×C₄	1	2	yes	—
42	#124×C₆₇	C₂×C₄	1	4	yes	—
43	#121×C₆₇	—	2	4	no	—
44	C₄×D₄×C₆₇	—	2	4	no	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂²×Dic₆₇	—	2	2	no	—
50	#11761612	—	2	8	no	—
51	#11761612	—	4	8	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₂×C₆₇⋊C₈	—	4	8	no	—
55	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
56	C₄×Dic₆₇	—	2	4	no	—
57	#41×C₆₇	—	2	8	no	—
58	#41×C₆₇	—	2	8	no	—
59	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂×D₄×C₆₇	—	2	4	no	—
61	C₂×D₄×C₆₇	—	2	4	no	—
62	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	#41	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	Q₁₆×C₆₇	—	4	4	no	—
67	SD₁₆×C₆₇	—	4	8	no	—
68	D₈×C₆₇	—	4	4	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
72	C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₂²⋊C₄×C₆₇	—	2	8	no	—
74	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	2	yes	—
77	C₂³×C₆₇	—	2	4	no	—
78	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
79	Q₈×C₆₇	—	2	8	no	—
80	D₄×C₆₇	—	2	8	no	—
81	D₄×C₆₇	—	4	8	no	—
82	D₄×C₆₇	—	8	4	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	4	2	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	4	4	no	—
87	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
92	C₂×Dic₆₇	—	4	4	no	—
93	C₂×Dic₆₇	—	4	2	no	—
94	C₈×C₆₇	—	2	2	no	—
95	C₈×C₆₇	—	2	2	no	—
96	C₆₇⋊C₈	—	2	4	no	—
97	C₆₇⋊C₈	—	4	4	no	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	4	4	no	—
100	C₂²×C₆₇	—	4	8	no	—
101	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂²×C₆₇	#540	1	2	yes	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	2	no	—
105	C₄×C₆₇	—	4	4	no	—
106	C₄×C₆₇	#518	1	2	yes	—
107	C₄×C₆₇	#535	1	2	yes	—
108	Dic₆₇	—	8	2	no	—
109	#22282	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	—	2	2	no	—
111	C₂×C₆₇	—	8	4	no	—
112	C₂×C₆₇	#2328	1	1	yes	characteristic
113	C₂×C₆₇	#2957	1	2	yes	—
114	#3147	D₆₇	1	1	yes	characteristic
115	#2932	—	67	2	no	—
116	#1963	—	67	2	no	—
117	#1876	—	67	8	no	—
118	#1837	—	67	1	no	—
119	#1746	—	67	1	no	—
120	C₆₇	#22282	1	1	yes	characteristic, complemented, Sylow
121	C₂×#135	C₂×D₆₇	1	4	yes	—
122	C₂×#131	—	67	4	no	—
123	#532	C₂×D₆₇	1	2	yes	—
124	#495	Dic₆₇	1	4	yes	complemented
125	#494	Dic₆₇	1	1	yes	characteristic, complemented
126	#493	Dic₆₇	1	1	yes	characteristic, complemented
127	#492	Dic₆₇	1	2	yes	complemented
128	#435	—	134	8	no	—
129	#423	—	67	2	no	—
130	C₂×#104	—	67	8	no	—
131	#408	—	67	4	no	—
132	C₂×#97	—	67	1	no	—
133	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
134	#376	—	67	1	no	—
135	#341	—	67	4	no	—
136	#336	—	67	4	no	—
137	C₂×#41	C₂²×D₆₇	1	4	yes	—
138	#135	—	4	8	no	—
139	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
140	C₂×SD₁₆	C₂×Dic₆₇	1	8	yes	—
141	C₂×D₈	C₂×Dic₆₇	1	4	yes	—
142	#131	—	268	8	no	—
143	C₂×M₁₆	—	67	4	no	—
144	C₂×M₁₆	—	134	4	no	—
145	C₂²×C₈	—	67	2	no	—
146	C₂²×C₈	—	134	4	no	—
147	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
149	#127	C₂×Dic₆₇	1	2	yes	—
150	#124	C₂×Dic₆₇	1	4	yes	—
151	#121	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄²	—	67	2	no	—
154	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
155	C₄⋊C₈	—	134	4	no	—
156	#104	—	268	8	no	—
157	C₂²⋊C₈	—	134	8	no	—
158	#97	—	67	4	no	—
159	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
160	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
161	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
162	#95	—	134	2	no	—
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×Dic₆₇	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	134	2	no	—
171	C₂²×C₄	#11761627	1	2	yes	—
172	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	268	8	no	—
177	M₁₆	—	134	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	268	8	no	—
180	C₂×C₈	—	67	4	no	—
181	C₂×C₈	—	134	4	no	—
182	C₂×C₈	—	134	4	no	—
183	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
184	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	134	4	no	—
188	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
189	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
190	C₄²	D₄×D₆₇	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#11761627	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	268	4	no	—
202	C₂×C₄	—	268	2	no	—
203	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
204	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
205	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
207	C₂×C₄	#14524088	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	268	4	no	—
211	C₈	—	134	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	8	no	—
215	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
216	C₂²	#16864000	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	536	2	no	—
221	C₄	#16863871	1	2	yes	—
222	C₄	#16863959	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	#22562935	1	1	yes	characteristic
226	C₂	#22569126	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3147	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3147	D₆₇
hypocenter	C₆₇	#22282
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #22568927	C₂	semidirect
2. #22561042	C₂	semidirect
3. #22560538	C₂	semidirect
4. #22560347	C₂	semidirect
5. #22559960	C₂	semidirect
18. #495×C₆₇	C₄	semidirect
19. #494×C₆₇	C₄	semidirect
20. #493×C₆₇	C₄	semidirect
21. #492×C₆₇	C₄	semidirect
120. C₆₇	#22282	semidirect
124. #495	Dic₆₇	semidirect
125. #494	Dic₆₇	semidirect
126. #493	Dic₆₇	semidirect
127. #492	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568927	2
#22561042	2
#22560538	2
#22560347	2
#22559960	2
C₆₇×#3147	2
#22282	67