Subgroups — #30761521

Order 17152.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863166	C₂	1	2	yes	maximal
2	#22559935	C₂	1	2	yes	maximal
3	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂²×#11761612	C₂²	1	2	yes	—
8	#16863166	—	2	4	no	—
9	#16863160	C₄	1	8	yes	complemented
10	C₆₇⋊#338	—	2	2	no	—
11	C₆₇⋊#338	—	2	2	no	—
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	#405×C₆₇	—	2	2	no	—
16	#405×C₆₇	C₄	1	4	yes	—
17	C₄×M₁₆×C₆₇	—	2	2	no	—
18	C₂×#11761612	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	2	no	—
27	C₂×#11761612	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	C₂×C₄	1	4	yes	—
30	C₂²×C₆₇⋊C₈	—	2	2	no	—
31	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	1	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
40	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
41	C₂²×C₈×C₆₇	—	2	1	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	#117×C₆₇	—	2	1	no	—
44	#117×C₆₇	—	2	1	no	—
45	#117×C₆₇	C₂×C₄	1	2	yes	—
46	#117×C₆₇	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
49	C₂×C₄²×C₆₇	—	2	1	no	—
50	#97×C₆₇	—	2	4	no	—
51	C₄×C₈×C₆₇	—	2	4	no	—
52	#11761612	—	2	4	no	—
53	#11761612	—	2	4	no	—
54	#11761612	—	4	2	no	—
55	#11761612	—	4	2	no	—
56	#11761612	—	4	2	no	—
57	#11761612	—	4	2	no	—
58	#11761612	—	4	2	no	—
59	#11761612	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	2	yes	—
78	M₁₆×C₆₇	—	2	2	no	—
79	M₁₆×C₆₇	—	2	4	no	—
80	M₁₆×C₆₇	—	2	4	no	—
81	M₁₆×C₆₇	—	2	4	no	—
82	M₁₆×C₆₇	C₄²	1	4	yes	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₂²⋊C₄×C₆₇	—	2	2	no	—
91	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	8	1	no	—
95	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
97	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₆₇	#125	1	2	yes	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	8	2	no	—
102	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#117	1	2	yes	—
105	C₂×C₄×C₆₇	#117	1	2	yes	—
106	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#123	1	2	yes	—
116	C₂×C₄×C₆₇	#123	1	2	yes	—
117	C₂×C₄×C₆₇	#124	1	2	yes	—
118	C₂×C₄×C₆₇	#126	1	2	yes	—
119	C₂×C₄×C₆₇	#126	1	2	yes	—
120	C₈×C₆₇	—	2	4	no	—
121	C₈×C₆₇	—	2	4	no	—
122	C₆₇⋊C₈	—	4	4	no	—
123	C₆₇⋊C₈	—	4	4	no	—
124	C₂²×C₆₇	—	4	1	no	—
125	C₂²×C₆₇	—	4	1	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₆₇	#380	1	1	yes	characteristic
131	C₂²×C₆₇	#382	1	1	yes	characteristic
132	C₂²×C₆₇	#387	1	2	yes	—
133	C₂²×C₆₇	#387	1	2	yes	—
134	C₄×C₆₇	—	4	4	no	—
135	C₄×C₆₇	#376	1	1	yes	characteristic
136	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₆₇	#381	1	1	yes	characteristic
138	C₄×C₆₇	#383	1	1	yes	characteristic
139	C₄×C₆₇	#388	1	2	yes	—
140	C₄×C₆₇	#388	1	2	yes	—
141	#10812	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	2	1	no	—
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	#1434	1	1	yes	characteristic
148	#2873	D₆₇	1	1	yes	characteristic
149	C₂×#344	—	67	2	no	—
150	#1738	—	67	2	no	—
151	C₂×#338	—	67	1	no	—
152	C₂×#338	—	67	1	no	—
153	C₆₇	#10812	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	67	2	no	—
155	C₂²×M₁₆	—	67	2	no	—
156	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
157	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
158	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
159	#405	—	2	2	no	—
160	#405	Dic₆₇	1	4	yes	—
161	C₄×M₁₆	—	2	2	no	—
162	#344	—	134	4	no	—
163	#341	—	67	8	no	—
164	#338	—	134	2	no	—
165	#338	—	134	2	no	—
166	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	67	4	no	—
172	C₂×M₁₆	—	134	2	no	—
173	C₂×M₁₆	—	134	2	no	—
174	C₂×M₁₆	—	134	2	no	—
175	C₂×M₁₆	—	134	2	no	—
176	C₂×M₁₆	—	134	2	no	—
177	C₂×M₁₆	—	134	2	no	—
178	C₂×M₁₆	—	134	2	no	—
179	C₂×M₁₆	—	134	2	no	—
180	C₂×M₁₆	—	134	2	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
184	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
185	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
186	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	67	4	no	—
189	C₂²×C₈	—	134	2	no	—
190	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
191	#117	—	2	1	no	—
192	#117	—	2	1	no	—
193	#117	D₂₆₈	1	2	yes	—
194	#117	C₂×Dic₆₇	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	Dic₁₃₄	1	2	yes	—
197	C₂×C₄²	—	2	1	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	4	no	—
200	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
207	C₂²×C₄	#11761628	1	1	yes	characteristic
208	C₂²×C₄	#11761628	1	1	yes	characteristic
209	C₂²×C₄	#11761628	1	2	yes	—
210	C₂²×C₄	#11761628	1	2	yes	—
211	C₂²×C₄	#11761628	1	2	yes	—
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	268	2	no	—
217	M₁₆	—	268	2	no	—
218	M₁₆	—	268	2	no	—
219	M₁₆	—	268	2	no	—
220	M₁₆	—	268	2	no	—
221	M₁₆	—	268	2	no	—
222	M₁₆	—	134	4	no	—
223	M₁₆	—	134	4	no	—
224	M₁₆	C₄×Dic₆₇	1	4	yes	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	268	2	no	—
230	C₂×C₈	—	268	2	no	—
231	C₂×C₈	—	134	4	no	—
232	C₂×C₈	—	134	4	no	—
233	C₂×C₈	—	134	4	no	—
234	C₂×C₈	—	134	4	no	—
235	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
244	C₂³	C₄×Dic₁₃₄	1	2	yes	—
245	C₂³	#14524022	1	1	yes	characteristic
246	C₂³	#14524022	1	1	yes	characteristic
247	C₂³	#14524035	1	2	yes	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
252	C₂×C₄	#14524000	1	2	yes	—
253	C₂×C₄	#14524000	1	2	yes	—
254	C₂×C₄	#14524002	1	1	yes	characteristic
255	C₂×C₄	#14524002	1	1	yes	characteristic
256	C₂×C₄	#14524002	1	1	yes	characteristic
257	C₂×C₄	#14524002	1	1	yes	characteristic
258	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
259	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
260	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
261	C₂×C₄	#14524017	1	1	yes	characteristic
262	C₂×C₄	#14524017	1	1	yes	characteristic
263	C₂×C₄	#14524026	1	2	yes	—
264	C₂×C₄	#14524026	1	2	yes	—
265	C₂×C₄	#14524031	1	2	yes	—
266	C₂×C₄	#14524037	1	2	yes	—
267	C₂×C₄	#14524037	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	2	4	no	—
270	C₈	—	268	4	no	—
271	C₈	—	268	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
278	C₂²	#16863256	1	1	yes	characteristic
279	C₂²	#16863262	1	1	yes	characteristic
280	C₂²	#16863280	1	2	yes	—
281	C₂²	#16863280	1	2	yes	—
282	C₄	—	4	4	no	—
283	C₄	#16863242	1	1	yes	characteristic
284	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
285	C₄	#16863259	1	1	yes	characteristic
286	C₄	#16863264	1	1	yes	characteristic
287	C₄	#16863287	1	2	yes	—
288	C₄	#16863287	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	#22558436	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#2873	C₂
socle	C₂×C₆₇	#1434
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#2873	D₆₇
hypocenter	C₆₇	#10812
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ #2873

Complements

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

N	H ≅ G/N	type
9. #16863160	C₄	semidirect
153. C₆₇	#10812	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863166	2
#22559935	2
C₂×C₆₇⋊#338	2
C₆₇×#2873	2
#10812	67