Subgroups — #30774097

Order 17152.

490 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863212	C₂	1	2	yes	maximal, complemented
2	C₂×#16863209	C₂	1	2	yes	maximal
3	C₄×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523979	—	2	1	no	—
8	C₂×#14523979	C₂²	1	2	yes	—
9	#16863420	—	2	1	no	—
10	#16863212	—	2	4	no	—
11	#16863209	—	2	4	no	—
12	#16863172	—	2	1	no	—
13	C₄×C₆₇⋊C₁₆	—	2	1	no	—
14	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₆₇	—	2	1	no	—
16	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	—	2	1	no	—
18	#495×C₆₇	—	2	1	no	—
19	#495×C₆₇	—	2	1	no	—
20	#495×C₆₇	C₄	1	2	yes	—
21	#495×C₆₇	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	2	no	—
24	#14523983	—	2	2	no	—
25	#14523983	D₄	1	4	yes	—
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	#14523979	—	2	2	no	—
28	#14523979	—	2	2	no	—
29	#14523979	—	4	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
31	C₂²×Q₈×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂×Q₁₆×C₆₇	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	—	2	1	no	—
35	C₂×Q₁₆×C₆₇	—	2	2	no	—
36	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×SD₁₆×C₆₇	—	2	2	no	—
38	C₂×SD₁₆×C₆₇	—	2	2	no	—
39	C₂×SD₁₆×C₆₇	—	4	2	no	—
40	C₂×SD₁₆×C₆₇	—	4	2	no	—
41	C₂×D₈×C₆₇	—	2	2	no	—
42	C₂×D₈×C₆₇	—	2	1	no	—
43	C₂×D₈×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	#124×C₆₇	—	2	2	no	—
48	#124×C₆₇	—	2	2	no	—
49	#124×C₆₇	—	4	2	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
56	#11761612	—	4	2	no	—
57	#11761612	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₆₇⋊C₁₆	—	2	4	no	—
60	C₂⁴×C₆₇	—	4	1	no	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×Q₈×C₆₇	—	4	1	no	—
63	C₂×Q₈×C₆₇	—	4	1	no	—
64	C₂×Q₈×C₆₇	—	4	1	no	—
65	C₂×Q₈×C₆₇	—	4	1	no	—
66	C₂×Q₈×C₆₇	—	4	1	no	—
67	C₂×D₄×C₆₇	—	2	2	no	—
68	C₂×D₄×C₆₇	—	4	1	no	—
69	C₂×D₄×C₆₇	—	4	1	no	—
70	C₂×D₄×C₆₇	—	4	1	no	—
71	C₂×D₄×C₆₇	—	4	1	no	—
72	C₂×D₄×C₆₇	—	4	1	no	—
73	C₂²×C₄×C₆₇	—	4	1	no	—
74	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	Q₁₆×C₆₇	—	2	4	no	—
77	Q₁₆×C₆₇	—	4	2	no	—
78	SD₁₆×C₆₇	—	4	4	no	—
79	SD₁₆×C₆₇	—	4	4	no	—
80	D₈×C₆₇	—	2	4	no	—
81	D₈×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	D₈	1	2	yes	—
85	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	#41	1	2	yes	—
90	C₂²⋊C₄×C₆₇	—	4	4	no	—
91	C₂²⋊C₄×C₆₇	—	4	4	no	—
92	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₆₇	D₈	1	2	yes	—
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	—	8	4	no	—
97	C₂³×C₆₇	—	8	1	no	—
98	C₂³×C₆₇	—	8	1	no	—
99	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	Q₈×C₆₇	—	4	2	no	—
101	Q₈×C₆₇	—	4	2	no	—
102	Q₈×C₆₇	—	8	1	no	—
103	Q₈×C₆₇	—	8	1	no	—
104	D₄×C₆₇	—	4	2	no	—
105	D₄×C₆₇	—	4	2	no	—
106	D₄×C₆₇	—	8	1	no	—
107	D₄×C₆₇	—	8	1	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	8	1	no	—
111	C₂×C₄×C₆₇	—	8	1	no	—
112	C₂×C₄×C₆₇	#102	1	2	yes	—
113	C₂×C₄×C₆₇	#102	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₈×C₆₇	#102	1	2	yes	—
121	C₈×C₆₇	#102	1	2	yes	—
122	C₈×C₆₇	C₄×D₄	1	2	yes	—
123	C₈×C₆₇	#121	1	2	yes	—
124	C₆₇⋊C₈	—	8	2	no	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	—	8	4	no	—
127	C₂²×C₆₇	—	8	2	no	—
128	C₂²×C₆₇	—	16	2	no	—
129	C₂²×C₆₇	—	16	2	no	—
130	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
131	C₂²×C₆₇	#486	1	1	yes	characteristic
132	C₂²×C₆₇	#492	1	1	yes	characteristic
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	8	2	no	—
136	C₄×C₆₇	#390	1	1	yes	characteristic
137	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
138	C₄×C₆₇	#486	1	1	yes	characteristic
139	C₄×C₆₇	#493	1	1	yes	characteristic
140	#13214	—	67	1	no	maximal, Sylow
141	C₂×C₆₇	—	2	1	no	—
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	16	2	no	—
144	C₂×C₆₇	#1957	1	1	yes	characteristic
145	C₂×C₆₇	#2148	1	2	yes	—
146	C₂×#495	D₆₇	1	1	yes	characteristic
147	C₂×#362	—	67	2	no	—
148	C₂×#361	—	67	2	no	—
149	C₄×M₃₂	—	67	1	no	—
150	#1848	—	67	1	no	—
151	C₆₇	#13214	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
153	C₂²×SD₁₆	—	2	1	no	—
154	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
155	C₂×#124	—	2	1	no	—
156	C₂×M₃₂	—	67	2	no	—
157	#495	—	2	1	no	—
158	#495	—	2	1	no	—
159	#495	Dic₆₇	1	2	yes	—
160	#495	Dic₆₇	1	2	yes	—
161	C₂×#108	—	67	2	no	—
162	C₂×#108	—	134	1	no	—
163	#423	—	134	1	no	—
164	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
165	#362	—	134	4	no	—
166	#361	—	134	4	no	—
167	#346	—	134	1	no	—
168	C₄×C₁₆	—	134	1	no	—
169	C₂²×Q₈	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂×Q₁₆	—	2	2	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	—	2	2	no	—
174	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×SD₁₆	—	4	2	no	—
178	C₂×SD₁₆	—	4	2	no	—
179	C₂×D₈	—	2	2	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
186	#124	—	2	2	no	—
187	#124	—	2	2	no	—
188	#124	—	4	2	no	—
189	C₂×C₂²⋊C₄	—	4	2	no	—
190	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
191	M₃₂	—	67	4	no	—
192	M₃₂	—	134	2	no	—
193	C₂×C₁₆	—	134	2	no	—
194	#108	—	268	2	no	—
195	#108	—	134	2	no	—
196	#108	—	134	2	no	—
197	C₄⋊C₈	—	268	2	no	—
198	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
199	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
201	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
202	C₂⁴	—	4	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	#11761616	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	4	2	no	—
220	SD₁₆	—	4	4	no	—
221	SD₁₆	—	4	4	no	—
222	D₈	—	2	4	no	—
223	D₈	—	4	2	no	—
224	M₁₆	—	268	2	no	—
225	M₁₆	—	268	2	no	—
226	C₂×C₈	—	268	2	no	—
227	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₂×C₈	#11761613	1	2	yes	—
230	C₂×C₈	#11761616	1	2	yes	—
231	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
232	C₂×C₈	D₄×D₆₇	1	2	yes	—
233	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
234	C₂×C₈	#11761627	1	2	yes	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
238	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
239	C₄²	#11761613	1	2	yes	—
240	C₁₆	—	134	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	8	4	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	C₂×#11761616	1	1	yes	characteristic
246	Q₈	—	4	2	no	—
247	Q₈	—	4	2	no	—
248	Q₈	—	8	1	no	—
249	Q₈	—	8	1	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	1	no	—
253	D₄	—	8	1	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	#14523964	1	2	yes	—
259	C₂×C₄	#14523964	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
261	C₂×C₄	#14524030	1	1	yes	characteristic
262	C₂×C₄	#14524039	1	1	yes	characteristic
263	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
264	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
265	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
266	C₈	—	536	2	no	—
267	C₈	#14523964	1	2	yes	—
268	C₈	#14523964	1	2	yes	—
269	C₈	D₄×Dic₆₇	1	2	yes	—
270	C₈	#14524014	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₂×#14523964	1	1	yes	characteristic
277	C₂²	#16863742	1	1	yes	characteristic
278	C₂²	#16863768	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	8	2	no	—
282	C₄	#16863299	1	1	yes	characteristic
283	C₄	C₂×#14523964	1	1	yes	characteristic
284	C₄	#16863742	1	1	yes	characteristic
285	C₄	#16863772	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	16	2	no	—
289	C₂	#22561016	1	1	yes	characteristic
290	C₂	#22561958	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#495×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂²×D₈×C₆₇	C₂²
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#495	D₆₇
hypocenter	C₆₇	#13214
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. C₂×#16863212	C₂	semidirect
3. C₄×#14523983	C₂	semidirect
4. #22560381	C₂	semidirect
151. C₆₇	#13214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863212	2
C₂×#16863209	2
C₄×#14523983	2
#22560381	2
C₂×#495×C₆₇	2
#13214	67