Subgroups — #30774084

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863206	C₂	1	4	yes	maximal, complemented
2	C₄×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560373	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523975	C₂²	1	2	yes	—
8	#16863262	—	2	1	no	—
9	#16863206	—	2	8	no	—
10	#16863172	—	2	1	no	—
11	C₄×C₆₇⋊C₁₆	—	2	1	no	—
12	C₂×#135×C₆₇	—	2	1	no	—
13	C₂×#135×C₆₇	C₂²	1	2	yes	—
14	#532×C₆₇	—	2	1	no	—
15	#495×C₆₇	—	2	1	no	—
16	#495×C₆₇	C₄	1	2	yes	complemented
17	#494×C₆₇	C₄	1	1	yes	characteristic, complemented
18	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
19	#492×C₆₇	—	2	1	no	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	#14523983	—	2	2	no	—
23	#14523983	D₄	1	4	yes	complemented
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	#14523977	—	4	2	no	—
26	#14523975	—	2	4	no	—
27	C₆₇⋊#95	—	4	2	no	—
28	C₂×#41×C₆₇	—	2	2	no	—
29	#135×C₆₇	—	2	4	no	—
30	#135×C₆₇	—	2	4	no	—
31	#135×C₆₇	—	4	2	no	—
32	#135×C₆₇	—	4	2	no	—
33	C₂×Q₁₆×C₆₇	—	2	1	no	—
34	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₆₇	—	2	2	no	—
36	C₂×SD₁₆×C₆₇	—	2	2	no	—
37	C₂×D₈×C₆₇	—	2	1	no	—
38	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₆₇	—	2	1	no	—
42	#127×C₆₇	—	2	1	no	—
43	#124×C₆₇	—	2	2	no	—
44	#121×C₆₇	—	4	2	no	—
45	C₄×D₄×C₆₇	—	4	2	no	—
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×Dic₆₇	—	4	2	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₆₇⋊C₁₆	—	2	4	no	—
55	#41×C₆₇	—	4	2	no	—
56	#41×C₆₇	—	4	2	no	—
57	#41×C₆₇	—	4	2	no	—
58	#41×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	—	4	2	no	—
62	C₂×D₄×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	D₈	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₆₇	—	2	4	no	—
67	Q₁₆×C₆₇	—	4	2	no	—
68	SD₁₆×C₆₇	—	4	4	no	—
69	SD₁₆×C₆₇	—	4	4	no	—
70	D₈×C₆₇	—	2	4	no	—
71	D₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	D₈	1	2	yes	—
75	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	#41	1	2	yes	—
80	C₄⋊C₄×C₆₇	—	4	2	no	—
81	C₂²⋊C₄×C₆₇	—	4	4	no	—
82	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₆₇	SD₁₆	1	2	yes	—
85	C₂³×C₆₇	—	4	2	no	—
86	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₆₇	—	4	2	no	—
88	Q₈×C₆₇	—	4	2	no	—
89	D₄×C₆₇	—	4	2	no	—
90	D₄×C₆₇	—	4	2	no	—
91	D₄×C₆₇	—	8	4	no	—
92	D₄×C₆₇	—	8	2	no	—
93	D₄×C₆₇	—	8	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	8	2	no	—
97	C₂×C₄×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	#102	1	2	yes	—
99	C₂×C₄×C₆₇	#102	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×Dic₆₇	—	8	4	no	—
107	C₂×Dic₆₇	—	8	2	no	—
108	C₈×C₆₇	#102	1	2	yes	—
109	C₈×C₆₇	#102	1	2	yes	—
110	C₈×C₆₇	C₄×D₄	1	2	yes	—
111	C₈×C₆₇	#121	1	2	yes	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	8	4	no	—
114	C₂²×C₆₇	—	8	2	no	—
115	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
116	C₂²×C₆₇	#486	1	1	yes	characteristic
117	C₂²×C₆₇	#493	1	1	yes	characteristic
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	8	2	no	—
121	C₄×C₆₇	#390	1	1	yes	characteristic
122	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
123	C₄×C₆₇	#486	1	1	yes	characteristic
124	C₄×C₆₇	#492	1	1	yes	characteristic
125	Dic₆₇	—	16	2	no	—
126	#13211	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	—	2	1	no	—
128	C₂×C₆₇	—	2	1	no	—
129	C₂×C₆₇	—	16	2	no	—
130	C₂×C₆₇	#1957	1	1	yes	characteristic
131	C₂×C₆₇	#2145	1	2	yes	—
132	#3147	D₆₇	1	1	yes	characteristic
133	C₂×#360	—	67	4	no	—
134	C₄×M₃₂	—	67	1	no	—
135	#1846	—	67	1	no	—
136	C₆₇	#13211	1	1	yes	characteristic, complemented, Sylow
137	C₂×#135	—	2	1	no	—
138	C₂×#135	C₂×D₆₇	1	2	yes	—
139	#532	—	2	1	no	—
140	C₂×M₃₂	—	67	2	no	—
141	#495	—	2	1	no	—
142	#495	Dic₆₇	1	2	yes	complemented
143	#494	Dic₆₇	1	1	yes	characteristic, complemented
144	#493	Dic₆₇	1	1	yes	characteristic, complemented
145	#492	—	2	1	no	—
146	C₂×#107	—	134	1	no	—
147	C₂×#106	—	67	2	no	—
148	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
149	#382	—	134	1	no	—
150	#360	—	134	8	no	—
151	#346	—	134	1	no	—
152	C₄×C₁₆	—	134	1	no	—
153	C₂×#41	—	2	2	no	—
154	#135	—	2	4	no	—
155	#135	—	2	4	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	1	no	—
167	#127	—	2	1	no	—
168	#124	—	2	2	no	—
169	#121	—	4	2	no	—
170	C₄×D₄	—	4	2	no	—
171	C₂×C₄⋊C₄	—	134	2	no	—
172	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
173	M₃₂	—	67	4	no	—
174	M₃₂	—	134	2	no	—
175	C₂×C₁₆	—	134	2	no	—
176	#107	—	268	2	no	—
177	#106	—	134	4	no	—
178	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
179	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	#95	—	268	2	no	—
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	268	2	no	—
193	C₂²×C₄	#11761613	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	Q₁₆	—	2	4	no	—
196	Q₁₆	—	4	2	no	—
197	SD₁₆	—	4	4	no	—
198	SD₁₆	—	4	4	no	—
199	D₈	—	2	4	no	—
200	D₈	—	4	2	no	—
201	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
203	C₂×C₈	#11761613	1	2	yes	—
204	C₂×C₈	#11761616	1	2	yes	—
205	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
206	C₂×C₈	D₄×D₆₇	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
208	C₂×C₈	#11761627	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	268	2	no	—
211	C₄⋊C₄	—	268	2	no	—
212	C₂²⋊C₄	—	4	4	no	—
213	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
214	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
215	C₄²	#11761616	1	2	yes	—
216	C₁₆	—	134	4	no	—
217	C₂³	—	4	2	no	—
218	C₂³	C₂×#11761613	1	1	yes	characteristic
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	536	4	no	—
231	C₂×C₄	—	536	2	no	—
232	C₂×C₄	#14523964	1	2	yes	—
233	C₂×C₄	#14523964	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
235	C₂×C₄	#14524030	1	1	yes	characteristic
236	C₂×C₄	#14524039	1	1	yes	characteristic
237	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
238	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
239	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
240	C₈	#14523964	1	2	yes	—
241	C₈	#14523964	1	2	yes	—
242	C₈	D₄×Dic₆₇	1	2	yes	—
243	C₈	#14524014	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	8	2	no	—
247	C₂²	C₂×#14523964	1	1	yes	characteristic
248	C₂²	#16863742	1	1	yes	characteristic
249	C₂²	#16863772	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	1072	2	no	—
254	C₄	#16863299	1	1	yes	characteristic
255	C₄	C₂×#14523964	1	1	yes	characteristic
256	C₄	#16863742	1	1	yes	characteristic
257	C₄	#16863768	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	2	no	—
261	C₂	#22561016	1	1	yes	characteristic
262	C₂	#22561945	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#3147	C₂
socle	C₂²×C₆₇	C₂×#102
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3147	D₆₇
hypocenter	C₆₇	#13211
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16863206	C₂	semidirect
2. C₄×#14523983	C₂	semidirect
3. #22560373	C₂	semidirect
16. #495×C₆₇	C₄	semidirect
17. #494×C₆₇	C₄	semidirect
18. #493×C₆₇	C₄	semidirect
23. #14523983	D₄	semidirect
136. C₆₇	#13211	semidirect
142. #495	Dic₆₇	semidirect
143. #494	Dic₆₇	semidirect
144. #493	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863206	2
C₄×#14523983	2
#22560373	2
C₆₇×#3147	2
#13211	67