Subgroups — #30765920

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558170	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
6	#16863420	—	2	1	no	—
7	C₄×C₆₇⋊C₁₆	—	2	1	no	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#16863149	—	2	1	no	—
10	#16863149	C₂²	1	2	yes	—
11	#532×C₆₇	—	2	1	no	—
12	#500×C₆₇	C₄	1	1	yes	characteristic
13	#493×C₆₇	C₄	1	1	yes	characteristic
14	#486×C₆₇	—	2	1	no	—
15	#466×C₆₇	—	2	1	no	—
16	#466×C₆₇	C₄	1	2	yes	—
17	C₄×D₈×C₆₇	—	2	1	no	—
18	C₄×D₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	4	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	4	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
25	C₂×#41×C₆₇	—	4	1	no	—
26	C₂×D₈×C₆₇	—	2	1	no	—
27	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₆₇	—	2	1	no	—
31	#127×C₆₇	—	2	1	no	—
32	#124×C₆₇	—	4	1	no	—
33	#121×C₆₇	—	2	2	no	—
34	#121×C₆₇	—	4	1	no	—
35	C₄×D₄×C₆₇	—	2	2	no	—
36	C₄×D₄×C₆₇	—	4	1	no	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	#107×C₆₇	—	2	1	no	—
39	#107×C₆₇	C₂×C₄	1	2	yes	—
40	#102×C₆₇	—	2	2	no	—
41	#102×C₆₇	—	2	2	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	#11761612	—	8	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₆₇⋊C₁₆	—	2	4	no	—
49	#41×C₆₇	—	8	2	no	—
50	#41×C₆₇	—	8	2	no	—
51	C₂×Q₈×C₆₇	—	4	1	no	—
52	C₂×D₄×C₆₇	—	2	2	no	—
53	C₂×D₄×C₆₇	—	4	2	no	—
54	C₂×D₄×C₆₇	—	4	1	no	—
55	C₂×D₄×C₆₇	—	4	1	no	—
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
60	D₈×C₆₇	—	2	4	no	—
61	D₈×C₆₇	—	4	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₄⋊C₄×C₆₇	—	4	1	no	—
70	C₂²⋊C₄×C₆₇	—	4	2	no	—
71	C₂²⋊C₄×C₆₇	—	4	2	no	—
72	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₆₇	D₈	1	1	yes	characteristic
74	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₆₇	—	4	2	no	—
77	C₂³×C₆₇	#102	1	1	yes	characteristic
78	Q₈×C₆₇	—	8	2	no	—
79	D₄×C₆₇	—	4	2	no	—
80	D₄×C₆₇	—	4	2	no	—
81	D₄×C₆₇	—	8	4	no	—
82	D₄×C₆₇	—	8	2	no	—
83	D₄×C₆₇	—	8	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	8	2	no	—
89	C₂×C₄×C₆₇	—	8	2	no	—
90	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
94	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	1	no	—
100	C₈×C₆₇	#104	1	2	yes	—
101	C₈×C₆₇	#131	1	2	yes	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	—	2	1	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	—	8	2	no	—
106	C₂²×C₆₇	—	8	2	no	—
107	C₂²×C₆₇	—	8	2	no	—
108	C₂²×C₆₇	#357	1	2	yes	—
109	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	8	2	no	—
113	C₄×C₆₇	#357	1	2	yes	—
114	C₄×C₆₇	#359	1	2	yes	—
115	C₄×C₆₇	#408	1	1	yes	characteristic
116	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
117	C₄×C₆₇	C₂×D₁₆	1	1	yes	characteristic
118	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
119	#11671	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	—	2	2	no	—
121	C₂×C₆₇	—	16	2	no	—
122	C₂×C₆₇	#1492	1	1	yes	characteristic
123	C₂×C₆₇	C₂×#357	1	1	yes	characteristic
124	C₂×C₆₇	C₂×#359	1	1	yes	characteristic
125	#3169	D₆₇	1	1	yes	characteristic
126	C₂×C₄×C₁₆	—	67	1	no	—
127	#1567	—	67	1	no	—
128	#1331	—	67	4	no	—
129	C₆₇	#11671	1	1	yes	characteristic, complemented, Sylow
130	#532	—	2	1	no	—
131	C₂²×C₁₆	—	134	1	no	—
132	#500	Dic₆₇	1	1	yes	characteristic
133	#493	Dic₆₇	1	1	yes	characteristic
134	#486	—	2	1	no	—
135	#466	—	2	1	no	—
136	#466	Dic₆₇	1	2	yes	—
137	C₄×D₈	—	2	1	no	—
138	C₄×D₈	C₂×D₆₇	1	2	yes	—
139	#423	—	134	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	C₄×C₁₆	—	67	2	no	—
142	C₄×C₁₆	—	134	1	no	—
143	#335	—	67	2	no	—
144	#335	—	134	1	no	—
145	C₂×#41	—	4	1	no	—
146	C₂×D₈	—	2	1	no	—
147	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
148	C₂×M₁₆	—	268	1	no	—
149	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
150	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
151	C₄⋊Q₈	—	2	1	no	—
152	#127	—	2	1	no	—
153	#124	—	4	1	no	—
154	#121	—	2	2	no	—
155	#121	—	4	1	no	—
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	—	4	1	no	—
158	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂×C₁₆	—	134	4	no	—
160	C₂×C₁₆	—	134	2	no	—
161	#107	—	2	1	no	—
162	#107	C₂×Dic₆₇	1	2	yes	—
163	C₄⋊C₈	—	268	1	no	—
164	C₄⋊C₈	—	134	2	no	—
165	#102	—	2	2	no	—
166	#102	—	2	2	no	—
167	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
171	#41	—	8	2	no	—
172	#41	—	8	2	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
180	C₂²×C₄	#11761613	1	1	yes	characteristic
181	C₂²×C₄	#11761616	1	1	yes	characteristic
182	D₈	—	2	4	no	—
183	D₈	—	4	2	no	—
184	M₁₆	—	536	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	268	2	no	—
188	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
197	C₄²	#11761613	1	1	yes	characteristic
198	C₄²	#11761616	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₁₆	—	134	4	no	—
201	C₂³	—	4	2	no	—
202	C₂³	#14523964	1	1	yes	characteristic
203	Q₈	—	8	2	no	—
204	D₄	—	4	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	8	4	no	—
207	D₄	—	8	2	no	—
208	D₄	—	8	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	#14523964	1	1	yes	characteristic
216	C₂×C₄	#14523964	1	1	yes	characteristic
217	C₂×C₄	#14523964	1	1	yes	characteristic
218	C₂×C₄	#14523984	1	2	yes	—
219	C₂×C₄	#14523987	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
221	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
222	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	536	2	no	—
226	C₈	#14523970	1	2	yes	—
227	C₈	#14524050	1	2	yes	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	#16863197	1	2	yes	—
234	C₂²	C₂×#14523964	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	8	2	no	—
238	C₄	#16863197	1	2	yes	—
239	C₄	#16863203	1	2	yes	—
240	C₄	#16863363	1	1	yes	characteristic
241	C₄	C₂×#14523970	1	1	yes	characteristic
242	C₄	C₂×#14523984	1	1	yes	characteristic
243	C₄	C₂×#14523987	1	1	yes	characteristic
244	C₂	—	2	2	no	—
245	C₂	—	16	2	no	—
246	C₂	#22558660	1	1	yes	characteristic
247	C₂	C₂×#16863197	1	1	yes	characteristic
248	C₂	C₂×#16863203	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#3169	C₂
socle	C₂²×C₆₇	C₂×#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#3169	D₆₇
hypocenter	C₆₇	#11671
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
2. #22558968	C₂	semidirect
3. #22558170	C₂	semidirect
129. C₆₇	#11671	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₆₇⋊C₁₆	2
#22558968	2
#22558170	2
C₆₇×#3169	2
#11671	67