Subgroups — #30764219

Order 17152.

602 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568773	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863166	C₂	1	2	yes	maximal
3	#22559906	C₂	1	2	yes	maximal
4	#22559906	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	1	yes	characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863423	—	2	4	no	—
10	#16863420	—	2	2	no	—
11	#16863166	—	2	4	no	—
12	#16863151	—	2	4	no	—
13	#16863151	—	2	2	no	—
14	C₂×#117×C₆₇	C₂²	1	2	yes	—
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	#342×C₆₇	C₄	1	4	yes	—
17	C₂×#11761612	—	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	1	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	D₄	1	2	yes	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×C₆₇⋊C₈	—	2	1	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	1	no	—
31	C₂²×C₆₇⋊C₈	—	2	2	no	—
32	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
33	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	—
34	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	#117×C₆₇	—	2	2	no	—
37	#117×C₆₇	—	2	2	no	—
38	#117×C₆₇	—	2	2	no	—
39	#117×C₆₇	—	2	2	no	—
40	#117×C₆₇	—	2	2	no	—
41	#117×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
44	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	#11761612	—	2	4	no	—
49	#11761612	—	4	2	no	—
50	#11761612	—	4	2	no	—
51	#11761612	—	4	2	no	—
52	#11761612	—	4	2	no	—
53	#11761612	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	4	1	no	—
61	C₂×C₆₇⋊C₈	—	4	1	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	4	1	no	—
64	C₂×C₆₇⋊C₈	—	4	1	no	—
65	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	#41	1	2	yes	—
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	1	yes	characteristic
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₄⋊C₄×C₆₇	—	4	2	no	—
81	C₂²⋊C₄×C₆₇	—	2	4	no	—
82	C₂²⋊C₄×C₆₇	—	2	4	no	—
83	C₂²⋊C₄×C₆₇	—	2	4	no	—
84	C₄²×C₆₇	—	2	2	no	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	—	4	2	no	—
88	C₂³×C₆₇	—	4	2	no	—
89	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₆₇	#117	1	2	yes	—
91	C₂³×C₆₇	C₄×D₄	1	2	yes	—
92	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	4	8	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#121	1	2	yes	—
100	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#123	1	2	yes	—
105	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#123	1	2	yes	—
107	C₂×C₄×C₆₇	#124	1	2	yes	—
108	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#125	1	2	yes	—
111	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#125	1	2	yes	—
113	C₂×C₄×C₆₇	#126	1	2	yes	—
114	C₂×C₄×C₆₇	#126	1	2	yes	—
115	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
116	C₆₇⋊C₈	—	4	2	no	—
117	C₆₇⋊C₈	—	4	4	no	—
118	C₆₇⋊C₈	—	4	2	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	2	4	no	—
121	C₂²×C₆₇	—	4	2	no	—
122	C₂²×C₆₇	—	4	1	no	—
123	C₂²×C₆₇	—	4	1	no	—
124	C₂²×C₆₇	#382	1	1	yes	characteristic
125	C₂²×C₆₇	#385	1	1	yes	characteristic
126	C₂²×C₆₇	#387	1	2	yes	—
127	C₂²×C₆₇	#388	1	2	yes	—
128	C₂²×C₆₇	#389	1	1	yes	characteristic
129	C₄×C₆₇	—	4	4	no	—
130	C₄×C₆₇	—	8	4	no	—
131	C₄×C₆₇	#393	1	1	yes	characteristic
132	C₄×C₆₇	#394	1	1	yes	characteristic
133	C₄×C₆₇	#397	1	1	yes	characteristic
134	C₄×C₆₇	#398	1	2	yes	—
135	C₄×C₆₇	#400	1	1	yes	characteristic
136	C₄×C₆₇	#400	1	2	yes	—
137	#11288	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	2	2	no	—
139	C₂×C₆₇	—	2	1	no	—
140	C₂×C₆₇	—	2	2	no	—
141	C₂×C₆₇	—	2	1	no	—
142	C₂×C₆₇	#1442	1	1	yes	characteristic
143	C₂×C₆₇	#1868	1	2	yes	—
144	#2905	—	67	1	no	—
145	C₂×#344	—	67	2	no	—
146	#1740	D₆₇	1	1	yes	characteristic
147	#1729	—	67	2	no	—
148	#1729	—	67	1	no	—
149	C₆₇	#11288	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	—	67	2	no	—
151	C₂²×M₁₆	—	67	1	no	—
152	C₂³×C₈	—	67	1	no	—
153	C₂×#117	C₂×D₆₇	1	2	yes	—
154	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
155	#424	—	134	4	no	—
156	#423	—	134	2	no	—
157	#344	—	134	4	no	—
158	#342	Dic₆₇	1	4	yes	—
159	#336	—	134	4	no	—
160	#336	—	134	2	no	—
161	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
162	C₂×M₁₆	—	67	2	no	—
163	C₂×M₁₆	—	134	2	no	—
164	C₂×M₁₆	—	134	2	no	—
165	C₂×M₁₆	—	134	2	no	—
166	C₂×M₁₆	—	134	2	no	—
167	C₂×M₁₆	—	134	2	no	—
168	C₂×M₁₆	—	134	2	no	—
169	C₂×M₁₆	—	134	2	no	—
170	C₂×M₁₆	—	134	1	no	—
171	C₂×M₁₆	—	134	2	no	—
172	C₂²×C₈	—	67	4	no	—
173	C₂²×C₈	—	67	2	no	—
174	C₂²×C₈	—	134	2	no	—
175	C₂²×C₈	—	134	1	no	—
176	C₂²×C₈	—	134	1	no	—
177	C₂²×C₈	—	134	1	no	—
178	C₂²×C₈	—	134	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
188	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	D₂₆₈	1	2	yes	—
191	C₄⋊C₈	—	134	8	no	—
192	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
195	C₂²×C₄	D₄×D₆₇	1	2	yes	—
196	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
198	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
200	C₂²×C₄	#11761627	1	2	yes	—
201	C₂²×C₄	#11761627	1	1	yes	characteristic
202	C₂²×C₄	#11761628	1	2	yes	—
203	C₂²×C₄	#11761628	1	2	yes	—
204	C₂²×C₄	#11761629	1	1	yes	characteristic
205	M₁₆	—	268	2	no	—
206	M₁₆	—	268	2	no	—
207	M₁₆	—	268	2	no	—
208	M₁₆	—	268	2	no	—
209	M₁₆	—	268	2	no	—
210	M₁₆	—	134	4	no	—
211	C₂×C₈	—	268	1	no	—
212	C₂×C₈	—	268	1	no	—
213	C₂×C₈	—	268	2	no	—
214	C₂×C₈	—	268	1	no	—
215	C₂×C₈	—	268	1	no	—
216	C₂×C₈	—	134	4	no	—
217	C₂×C₈	—	134	4	no	—
218	C₂×C₈	—	134	2	no	—
219	C₂×C₈	—	134	4	no	—
220	C₂×C₈	—	134	2	no	—
221	C₂×C₈	—	134	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
234	C₂³	#14524002	1	2	yes	—
235	C₂³	D₄×Dic₆₇	1	2	yes	—
236	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
237	C₂³	Q₈×Dic₆₇	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	#14524013	1	1	yes	characteristic
243	C₂×C₄	#14524016	1	2	yes	—
244	C₂×C₄	#14524018	1	1	yes	characteristic
245	C₂×C₄	#14524020	1	2	yes	—
246	C₂×C₄	#14524021	1	2	yes	—
247	C₂×C₄	#14524023	1	1	yes	characteristic
248	C₂×C₄	#14524025	1	2	yes	—
249	C₂×C₄	#14524027	1	1	yes	characteristic
250	C₂×C₄	#14524028	1	2	yes	—
251	C₂×C₄	#14524029	1	2	yes	—
252	C₂×C₄	#14524031	1	1	yes	characteristic
253	C₂×C₄	#14524031	1	1	yes	characteristic
254	C₂×C₄	#14524033	1	2	yes	—
255	C₂×C₄	#14524034	1	1	yes	characteristic
256	C₂×C₄	#14524035	1	2	yes	—
257	C₂×C₄	#14524036	1	2	yes	—
258	C₂×C₄	#14524038	1	2	yes	—
259	C₂×C₄	#14524042	1	1	yes	characteristic
260	C₈	—	268	2	no	—
261	C₈	—	268	4	no	—
262	C₈	—	268	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#16863262	1	1	yes	characteristic
269	C₂²	#16863271	1	1	yes	characteristic
270	C₂²	#16863282	1	2	yes	—
271	C₂²	#16863289	1	2	yes	—
272	C₂²	#16863295	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	8	4	no	—
275	C₄	#16863309	1	1	yes	characteristic
276	C₄	#16863314	1	1	yes	characteristic
277	C₄	#16863323	1	1	yes	characteristic
278	C₄	#16863330	1	2	yes	—
279	C₄	#16863336	1	1	yes	characteristic
280	C₄	#16863338	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	#22558466	1	1	yes	characteristic
286	C₂	#22560497	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524013
commutator, prime radical	C₂⁴×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#1740	C₂
socle	C₂²×C₆₇	#385
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#1740	D₆₇
hypocenter	C₆₇	#11288
LCS	G ▷ C₂⁴×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
149. C₆₇	#11288	semidirect

Maximal subgroups

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

subgroup	index
#22568773	2
C₂×#16863166	2
#22559906	2
C₆₇×#1740	2
#11288	67