Subgroups — C₆₇×#9900 #30689151

Order 17152.

450 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1296	C₂	1	2	yes	maximal
5	C₆₇×#1296	C₂	1	2	yes	maximal
6	C₂×#125×C₆₇	—	2	1	no	—
7	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
8	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₆₇	—	2	1	no	—
12	#389×C₆₇	—	2	1	no	—
13	#389×C₆₇	—	2	1	no	—
14	#389×C₆₇	C₄	1	2	yes	—
15	#389×C₆₇	C₄	1	2	yes	—
16	#384×C₆₇	C₂²	1	1	yes	characteristic
17	#384×C₆₇	C₂²	1	1	yes	characteristic
18	#382×C₆₇	—	2	1	no	—
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	#336×C₆₇	—	2	1	no	—
21	C₂²×C₈×C₆₇	—	2	2	no	—
22	C₂²×C₈×C₆₇	—	2	2	no	—
23	#125×C₆₇	—	4	2	no	—
24	#125×C₆₇	—	4	2	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	2	4	no	—
42	C₄⋊C₈×C₆₇	—	4	2	no	—
43	C₄⋊C₈×C₆₇	—	4	2	no	—
44	C₄⋊C₈×C₆₇	Q₈	1	4	yes	—
45	C₄×C₈×C₆₇	—	4	2	no	—
46	#95×C₆₇	—	2	2	no	—
47	#95×C₆₇	—	2	2	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	—	4	2	no	—
59	C₂×C₈×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	—	4	2	no	—
61	C₂×C₈×C₆₇	—	4	2	no	—
62	C₂×C₈×C₆₇	—	4	2	no	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₄⋊C₄×C₆₇	—	4	4	no	—
67	C₄⋊C₄×C₆₇	—	4	1	no	—
68	C₄⋊C₄×C₆₇	—	4	1	no	—
69	C₄⋊C₄×C₆₇	—	4	1	no	—
70	C₄⋊C₄×C₆₇	—	4	1	no	—
71	C₄⋊C₄×C₆₇	—	4	2	no	—
72	C₄⋊C₄×C₆₇	—	4	1	no	—
73	C₄⋊C₄×C₆₇	—	4	1	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	1	no	—
76	C₄²×C₆₇	—	2	1	no	—
77	C₄²×C₆₇	—	4	1	no	—
78	C₄²×C₆₇	—	4	1	no	—
79	C₄²×C₆₇	D₈	1	2	yes	—
80	C₄²×C₆₇	SD₁₆	1	2	yes	—
81	C₄²×C₆₇	SD₁₆	1	2	yes	—
82	C₄²×C₆₇	Q₁₆	1	2	yes	—
83	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₆₇	#41	1	2	yes	—
85	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	#102	1	2	yes	—
103	C₂×C₄×C₆₇	#102	1	2	yes	—
104	C₂×C₄×C₆₇	#103	1	2	yes	—
105	C₂×C₄×C₆₇	#103	1	2	yes	—
106	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#135	1	2	yes	—
115	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
117	C₈×C₆₇	—	4	4	no	—
118	C₈×C₆₇	—	8	2	no	—
119	C₂²×C₆₇	#391	1	1	yes	characteristic
120	C₂²×C₆₇	#413	1	1	yes	characteristic
121	C₂²×C₆₇	#413	1	1	yes	characteristic
122	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
124	C₂²×C₆₇	#416	1	1	yes	characteristic
125	C₂²×C₆₇	#417	1	1	yes	characteristic
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	8	2	no	—
132	C₄×C₆₇	—	8	2	no	—
133	C₄×C₆₇	#478	1	2	yes	—
134	C₄×C₆₇	#479	1	2	yes	—
135	C₄×C₆₇	#486	1	1	yes	characteristic
136	C₄×C₆₇	#487	1	1	yes	characteristic
137	C₄×C₆₇	#498	1	1	yes	characteristic
138	C₄×C₆₇	#500	1	1	yes	characteristic
139	#9900	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₆₇	#1540	1	1	yes	characteristic
141	C₂×C₆₇	#1542	1	1	yes	characteristic
142	C₂×C₆₇	#1547	1	1	yes	characteristic
143	C₂×C₆₇	#1550	1	1	yes	characteristic
144	C₂×C₆₇	#1989	1	1	yes	characteristic
145	C₂×C₆₇	#1990	1	1	yes	characteristic
146	C₂×C₆₇	#1998	1	1	yes	characteristic
147	#2859	C₂×C₆₇	1	1	yes	characteristic
148	#1842	C₂×C₆₇	1	1	yes	characteristic
149	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
150	#1296	C₂×C₆₇	1	2	yes	—
151	#1296	C₂×C₆₇	1	2	yes	—
152	C₆₇	#9900	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#125	—	2	1	no	—
154	C₂×C₄⋊C₈	—	2	1	no	—
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
158	C₂×C₄×C₈	—	2	1	no	—
159	#389	—	2	1	no	—
160	#389	—	2	1	no	—
161	#389	C₄×C₆₇	1	2	yes	—
162	#389	C₄×C₆₇	1	2	yes	—
163	#384	C₂²×C₆₇	1	1	yes	characteristic
164	#384	C₂²×C₆₇	1	1	yes	characteristic
165	#382	—	2	1	no	—
166	C₄³	C₂²×C₆₇	1	1	yes	characteristic
167	#336	—	2	1	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	#125	—	4	2	no	—
171	#125	—	4	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
185	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	4	2	no	—
190	C₄⋊C₈	—	4	2	no	—
191	C₄⋊C₈	Q₈×C₆₇	1	4	yes	—
192	C₄×C₈	—	4	2	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	4	1	no	—
225	C₄²	—	4	1	no	—
226	C₄²	D₈×C₆₇	1	2	yes	—
227	C₄²	SD₁₆×C₆₇	1	2	yes	—
228	C₄²	SD₁₆×C₆₇	1	2	yes	—
229	C₄²	Q₁₆×C₆₇	1	2	yes	—
230	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
231	C₄²	#41×C₆₇	1	2	yes	—
232	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	#102×C₆₇	1	2	yes	—
250	C₂×C₄	#102×C₆₇	1	2	yes	—
251	C₂×C₄	#103×C₆₇	1	2	yes	—
252	C₂×C₄	#103×C₆₇	1	2	yes	—
253	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
254	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
255	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
256	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
260	C₂×C₄	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
261	C₂×C₄	#135×C₆₇	1	2	yes	—
262	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
263	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₈	—	8	2	no	—
266	C₂²	#391×C₆₇	1	1	yes	characteristic
267	C₂²	#413×C₆₇	1	1	yes	characteristic
268	C₂²	#413×C₆₇	1	1	yes	characteristic
269	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
270	C₂²	C₂×#103×C₆₇	1	1	yes	characteristic
271	C₂²	#416×C₆₇	1	1	yes	characteristic
272	C₂²	#417×C₆₇	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	#478×C₆₇	1	2	yes	—
281	C₄	#479×C₆₇	1	2	yes	—
282	C₄	#486×C₆₇	1	1	yes	characteristic
283	C₄	#487×C₆₇	1	1	yes	characteristic
284	C₄	#498×C₆₇	1	1	yes	characteristic
285	C₄	#500×C₆₇	1	1	yes	characteristic
286	C₂	C₆₇×#1540	1	1	yes	characteristic
287	C₂	C₆₇×#1542	1	1	yes	characteristic
288	C₂	C₆₇×#1547	1	1	yes	characteristic
289	C₂	C₆₇×#1550	1	1	yes	characteristic
290	C₂	C₆₇×#1989	1	1	yes	characteristic
291	C₂	C₆₇×#1990	1	1	yes	characteristic
292	C₂	C₆₇×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
139. #9900	C₆₇	direct (N × H)
152. C₆₇	#9900	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×#2859	2
C₆₇×#1842	2
C₄×C₄⋊C₈×C₆₇	2
C₆₇×#1296	2
#9900	67

Subgroups — C67×#9900 #30689151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9900 #30689151