Subgroups — #30771997

Order 17152.

438 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	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863221	C₂	1	2	yes	maximal, complemented
3	#22560238	C₂	1	1	yes	maximal, characteristic
4	#22558300	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	—	2	4	no	—
7	C₂²×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#14523977	C₂²	1	2	yes	—
11	#16863256	—	2	1	no	—
12	#16863221	—	2	4	no	—
13	#16863159	—	2	1	no	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	#380×C₆₇	—	2	1	no	—
17	#340×C₆₇	—	2	1	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
20	C₂×C₆₇⋊C₁₆	—	2	8	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
25	#14523977	—	2	2	no	—
26	#14523977	—	2	2	no	—
27	C₆₇⋊#95	—	4	1	no	—
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
37	#107×C₆₇	—	2	2	no	—
38	#107×C₆₇	—	2	2	no	—
39	#95×C₆₇	—	4	1	no	—
40	C₂²×Dic₆₇	—	4	2	no	—
41	C₄⋊Dic₆₇	—	4	2	no	—
42	C₄⋊Dic₆₇	—	4	2	no	—
43	C₂²⋊Dic₆₇	—	8	2	no	—
44	C₆₇⋊C₁₆	—	2	2	no	—
45	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
46	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	—	2	2	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₂²⋊C₄×C₆₇	—	8	2	no	—
66	C₂³×C₆₇	—	2	2	no	—
67	C₂³×C₆₇	—	2	2	no	—
68	C₂³×C₆₇	#107	1	2	yes	—
69	C₂³×C₆₇	D₁₆	1	2	yes	—
70	C₂³×C₆₇	Q₃₂	1	2	yes	—
71	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	8	2	no	—
77	C₂×C₄×C₆₇	—	8	2	no	—
78	C₂×C₄×C₆₇	—	8	2	no	—
79	C₂×C₄×C₆₇	#107	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#121	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#135	1	2	yes	—
84	C₂×Dic₆₇	—	8	2	no	—
85	C₂×Dic₆₇	—	8	2	no	—
86	C₂×Dic₆₇	—	8	2	no	—
87	C₈×C₆₇	—	2	2	no	—
88	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₆₇	#117	1	1	yes	characteristic
90	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
91	C₂²×C₆₇	—	2	2	no	—
92	C₂²×C₆₇	—	2	2	no	—
93	C₂²×C₆₇	—	2	4	no	—
94	C₂²×C₆₇	—	2	4	no	—
95	C₂²×C₆₇	#366	1	4	yes	complemented
96	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
97	C₂²×C₆₇	#466	1	2	yes	—
98	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
99	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
100	C₂²×C₆₇	#508	1	2	yes	—
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	16	2	no	—
103	C₄×C₆₇	#380	1	1	yes	characteristic
104	C₄×C₆₇	#427	1	1	yes	characteristic
105	C₄×C₆₇	#467	1	2	yes	—
106	Dic₆₇	—	16	2	no	—
107	#12834	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	—	2	4	no	—
109	C₂×C₆₇	#1808	1	1	yes	characteristic
110	C₂×C₆₇	C₂×#366	1	1	yes	characteristic
111	C₂×C₆₇	#2158	1	1	yes	characteristic
112	C₂×C₆₇	#2215	1	2	yes	—
113	C₂×C₆₇	#2216	1	2	yes	—
114	C₂³×C₁₆	—	67	1	no	—
115	C₂×#366	—	67	2	no	—
116	#1808	—	67	1	no	—
117	#1808	D₆₇	1	1	yes	characteristic
118	#1380	—	67	2	no	—
119	C₆₇	#12834	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
121	C₂²×C₁₆	—	67	1	no	—
122	C₂²×C₁₆	—	67	4	no	—
123	C₂²×C₁₆	—	67	1	no	—
124	C₂²×C₁₆	—	134	4	no	—
125	C₂×#107	—	67	2	no	—
126	C₂×#107	C₂×D₆₇	1	2	yes	—
127	#380	—	2	1	no	—
128	#380	—	134	1	no	—
129	#366	—	134	4	no	—
130	#340	—	2	1	no	—
131	#340	—	134	1	no	—
132	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	Dic₁₃₄	1	2	yes	—
136	C₂²×C₈	C₄×D₆₇	1	2	yes	—
137	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	134	2	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₂²⋊C₄	—	268	1	no	—
143	C₂×C₁₆	—	67	2	no	—
144	C₂×C₁₆	—	67	4	no	—
145	C₂×C₁₆	—	67	2	no	—
146	C₂×C₁₆	—	134	8	no	—
147	C₂×C₁₆	—	134	2	no	—
148	#107	—	2	2	no	—
149	#107	—	2	2	no	—
150	#107	—	134	2	no	—
151	#107	—	134	2	no	—
152	#95	—	4	1	no	—
153	#95	—	268	1	no	—
154	C₂⁴	#11761613	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	268	2	no	—
159	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
160	C₂²×C₄	#11761613	1	1	yes	characteristic
161	C₂²×C₄	#11761618	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
168	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
169	C₂×C₈	D₄×D₆₇	1	2	yes	—
170	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
171	C₂×C₈	#11761627	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	268	2	no	—
175	C₄⋊C₄	—	268	2	no	—
176	C₂²⋊C₄	—	8	2	no	—
177	C₂²⋊C₄	—	536	2	no	—
178	C₁₆	—	67	4	no	—
179	C₁₆	—	134	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	#14523976	1	2	yes	—
183	C₂³	#14523984	1	2	yes	—
184	C₂³	#14523989	1	2	yes	—
185	C₂³	C₂×#11761613	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	536	2	no	—
194	C₂×C₄	—	536	2	no	—
195	C₂×C₄	—	536	2	no	—
196	C₂×C₄	#14523976	1	2	yes	—
197	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#14524015	1	2	yes	—
199	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
200	C₂×C₄	#14524063	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
203	C₈	#14523999	1	1	yes	characteristic
204	C₈	#14524020	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	#16863221	1	4	yes	complemented
210	C₂²	C₂×#14523976	1	1	yes	characteristic
211	C₂²	#16863630	1	2	yes	—
212	C₂²	C₂×#14523984	1	1	yes	characteristic
213	C₂²	C₂×#14523989	1	1	yes	characteristic
214	C₂²	#16863826	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	16	2	no	—
217	C₄	—	1072	2	no	—
218	C₄	#16863254	1	1	yes	characteristic
219	C₄	#16863432	1	1	yes	characteristic
220	C₄	#16863635	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	#22560236	1	1	yes	characteristic
223	C₂	C₂×#16863221	1	1	yes	characteristic
224	C₂	#22561991	1	1	yes	characteristic
225	C₂	#22562293	1	2	yes	—
226	C₂	#22562298	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1808	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1808	D₆₇
hypocenter	C₆₇	#12834
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
2. C₂×#16863221	C₂	semidirect
4. #22558300	C₂	semidirect
7. C₂²×C₆₇⋊C₁₆	C₄	semidirect
45. C₆₇⋊C₁₆	C₂²⋊C₄	semidirect
95. C₂²×C₆₇	#366	semidirect
119. C₆₇	#12834	semidirect
209. C₂²	#16863221	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₆₇⋊C₁₆	2
C₂×#16863221	2
#22560238	2
#22558300	2
C₆₇×#1808	2
#12834	67