Subgroups — C₆₇×#33099 #30712350

Order 17152.

858 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2855	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1928	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1921	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1872	C₂	1	2	yes	maximal
6	C₆₇×#1852	C₂	1	2	yes	maximal
7	C₆₇×#1817	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	—	2	1	no	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#107×C₆₇	C₂²	1	4	yes	—
13	C₂×#103×C₆₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
15	#391×C₆₇	C₂²	1	2	yes	—
16	#389×C₆₇	C₂²	1	2	yes	—
17	#387×C₆₇	—	2	2	no	—
18	#385×C₆₇	—	2	1	no	—
19	#381×C₆₇	—	2	1	no	—
20	#380×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
23	C₂×#95×C₆₇	—	2	1	no	—
24	#340×C₆₇	C₂²	1	4	yes	—
25	#340×C₆₇	C₂²	1	4	yes	—
26	#336×C₆₇	C₂²	1	4	yes	—
27	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	—	2	2	no	—
29	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
33	C₂²⋊Q₈×C₆₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
41	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄²×C₆₇	—	2	4	no	—
44	C₂×C₄²×C₆₇	—	2	4	no	—
45	C₂×C₄²×C₆₇	—	4	4	no	—
46	C₂×C₄²×C₆₇	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
48	#107×C₆₇	—	2	8	no	—
49	#103×C₆₇	—	2	8	no	—
50	C₂²⋊C₈×C₆₇	—	2	4	no	—
51	#95×C₆₇	—	2	4	no	—
52	#95×C₆₇	—	2	1	no	—
53	#95×C₆₇	—	2	2	no	—
54	#95×C₆₇	—	2	2	no	—
55	#95×C₆₇	—	2	2	no	—
56	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₆₇	—	2	4	no	—
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	—	2	8	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	—	4	4	no	—
70	C₂²×C₄×C₆₇	—	4	4	no	—
71	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₆₇	#41	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₄⋊C₄×C₆₇	—	2	4	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	4	4	no	—
85	C₄⋊C₄×C₆₇	—	4	2	no	—
86	C₄⋊C₄×C₆₇	#41	1	8	yes	—
87	C₂²⋊C₄×C₆₇	—	4	2	no	—
88	C₂²⋊C₄×C₆₇	—	4	2	no	—
89	C₄²×C₆₇	—	2	4	no	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	4	8	no	—
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	#123	1	2	yes	—
95	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
96	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	8	no	—
103	C₂×C₄×C₆₇	—	2	8	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	8	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	16	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	1	no	—
113	C₂×C₄×C₆₇	—	4	1	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	#123	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
121	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
123	C₈×C₆₇	—	4	4	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	2	4	no	—
128	C₂²×C₆₇	#484	1	4	yes	—
129	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
130	C₂²×C₆₇	#525	1	2	yes	—
131	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂²×C₆₇	#575	1	2	yes	—
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	4	2	no	—
137	C₄×C₆₇	—	4	8	no	—
138	C₄×C₆₇	—	8	1	no	—
139	C₄×C₆₇	—	8	1	no	—
140	C₄×C₆₇	—	8	2	no	—
141	C₄×C₆₇	#529	1	4	yes	—
142	#33099	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₆₇	—	2	4	no	—
144	C₂×C₆₇	#2853	1	1	yes	characteristic
145	C₂×C₆₇	C₂×#484	1	1	yes	characteristic
146	C₂×C₆₇	#3093	1	1	yes	characteristic
147	C₂×C₆₇	#3104	1	2	yes	—
148	C₂×C₆₇	#3114	1	2	yes	—
149	#2855	C₂×C₆₇	1	1	yes	characteristic
150	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
151	#1928	C₂×C₆₇	1	4	yes	complemented
152	#1921	C₂×C₆₇	1	4	yes	complemented
153	#1872	C₂×C₆₇	1	2	yes	—
154	#1852	C₂×C₆₇	1	2	yes	—
155	#1817	C₂×C₆₇	1	1	yes	characteristic
156	C₆₇	#33099	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
159	C₂²×C₄²	—	2	1	no	—
160	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
161	C₂×#107	C₂²×C₆₇	1	4	yes	—
162	C₂×#103	C₂²×C₆₇	1	4	yes	—
163	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
164	#391	C₂²×C₆₇	1	2	yes	—
165	#389	C₂²×C₆₇	1	2	yes	—
166	#387	—	2	2	no	—
167	#385	—	2	1	no	—
168	#381	—	2	1	no	—
169	#380	C₂²×C₆₇	1	1	yes	characteristic
170	C₄×C₄⋊C₄	—	2	2	no	—
171	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
172	C₂×#95	—	2	1	no	—
173	#340	C₂²×C₆₇	1	4	yes	—
174	#340	C₂²×C₆₇	1	4	yes	—
175	#336	C₂²×C₆₇	1	4	yes	—
176	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
179	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
180	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
182	C₂²⋊Q₈	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	8	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	1	no	—
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₄²	—	2	4	no	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	—	4	4	no	—
195	C₂×C₄²	D₄×C₆₇	1	2	yes	—
196	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
197	#107	—	2	8	no	—
198	#103	—	2	8	no	—
199	C₂²⋊C₈	—	2	4	no	—
200	#95	—	2	4	no	—
201	#95	—	2	1	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
221	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
224	C₂²×C₄	#41×C₆₇	1	4	yes	—
225	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	#41×C₆₇	1	8	yes	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	4	8	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	#123×C₆₇	1	2	yes	—
244	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
245	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
246	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
247	Q₈	—	4	4	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	#123×C₆₇	1	2	yes	—
269	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
270	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
271	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
272	C₈	—	4	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	#484×C₆₇	1	4	yes	—
278	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
279	C₂²	#525×C₆₇	1	2	yes	—
280	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
281	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
282	C₂²	#575×C₆₇	1	2	yes	—
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	8	no	—
287	C₄	—	8	1	no	—
288	C₄	—	8	1	no	—
289	C₄	—	8	2	no	—
290	C₄	#529×C₆₇	1	4	yes	—
291	C₂	—	2	4	no	—
292	C₂	C₆₇×#2853	1	1	yes	characteristic
293	C₂	C₂×#484×C₆₇	1	1	yes	characteristic
294	C₂	C₆₇×#3093	1	1	yes	characteristic
295	C₂	C₆₇×#3104	1	2	yes	—
296	C₂	C₆₇×#3114	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1928	C₂	semidirect
4. C₆₇×#1921	C₂	semidirect
142. #33099	C₆₇	direct (N × H)
151. #1928	C₂×C₆₇	semidirect
152. #1921	C₂×C₆₇	semidirect
156. C₆₇	#33099	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2855	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1928	2
C₆₇×#1921	2
C₆₇×#1872	2
C₆₇×#1852	2
C₆₇×#1817	2
#33099	67

Subgroups — C67×#33099 #30712350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33099 #30712350