Subgroups — C₆₇×#10879 #30690130

Order 17152.

846 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic
2	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#339×C₆₇	C₂	1	2	yes	maximal
4	C₂×#339×C₆₇	C₂	1	2	yes	maximal
5	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
7	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#389×C₆₇	C₄	1	4	yes	complemented
12	#385×C₆₇	—	2	2	no	—
13	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
14	#376×C₆₇	—	2	4	no	—
15	C₂×#95×C₆₇	—	2	2	no	—
16	#339×C₆₇	—	2	4	no	—
17	#339×C₆₇	C₄	1	8	yes	complemented
18	#336×C₆₇	—	2	2	no	—
19	C₂³×C₄×C₆₇	—	2	2	no	—
20	C₂³×C₄×C₆₇	D₄	1	2	yes	—
21	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	#117×C₆₇	—	2	2	no	—
30	#117×C₆₇	—	2	2	no	—
31	#117×C₆₇	—	2	2	no	—
32	#117×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄²×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	—	2	2	no	—
40	C₂×C₄²×C₆₇	—	2	1	no	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	—	2	1	no	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	4	yes	—
49	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
50	#95×C₆₇	—	2	8	no	—
51	#95×C₆₇	—	2	4	no	—
52	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	—	2	4	no	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	4	4	no	—
61	C₂²×C₄×C₆₇	—	4	8	no	—
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₆₇	#41	1	2	yes	—
69	C₂²×C₄×C₆₇	#41	1	2	yes	—
70	C₂²×C₄×C₆₇	#41	1	2	yes	—
71	M₁₆×C₆₇	—	2	4	no	—
72	M₁₆×C₆₇	—	2	4	no	—
73	M₁₆×C₆₇	—	4	2	no	—
74	M₁₆×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	4	2	no	—
77	C₂×C₈×C₆₇	—	4	2	no	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	4	8	no	—
81	C₄⋊C₄×C₆₇	C₄²	1	4	yes	—
82	C₂²⋊C₄×C₆₇	—	2	4	no	—
83	C₂²⋊C₄×C₆₇	—	4	1	no	—
84	C₂²⋊C₄×C₆₇	—	4	1	no	—
85	C₄²×C₆₇	—	2	4	no	—
86	C₄²×C₆₇	—	2	4	no	—
87	C₄²×C₆₇	—	2	4	no	—
88	C₄²×C₆₇	—	2	4	no	—
89	C₄²×C₆₇	—	2	4	no	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	C₄⋊C₄	1	8	yes	complemented
93	C₂³×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₆₇	C₄×D₄	1	2	yes	—
96	C₂³×C₆₇	#120	1	1	yes	characteristic
97	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₆₇	#123	1	1	yes	characteristic
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	8	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	4	8	no	—
105	C₂×C₄×C₆₇	—	4	8	no	—
106	C₂×C₄×C₆₇	—	4	8	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₆₇	#117	1	2	yes	—
115	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₆₇	#121	1	2	yes	—
122	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₆₇	#123	1	2	yes	—
124	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#125	1	2	yes	—
126	C₂×C₄×C₆₇	#126	1	2	yes	—
127	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
128	C₈×C₆₇	—	4	4	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	—	2	4	no	—
133	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₆₇	#385	1	2	yes	—
135	C₂²×C₆₇	#386	1	1	yes	characteristic
136	C₂²×C₆₇	#387	1	2	yes	—
137	C₂²×C₆₇	#391	1	1	yes	characteristic
138	C₂²×C₆₇	#421	1	4	yes	—
139	C₄×C₆₇	—	4	8	no	—
140	C₄×C₆₇	—	4	4	no	—
141	C₄×C₆₇	—	4	4	no	—
142	C₄×C₆₇	—	8	2	no	—
143	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
144	C₄×C₆₇	#387	1	2	yes	—
145	C₄×C₆₇	#388	1	2	yes	—
146	C₄×C₆₇	#389	1	2	yes	—
147	#10879	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	#1436	1	1	yes	characteristic
151	C₂×C₆₇	#1759	1	2	yes	—
152	C₂×C₆₇	#1897	1	2	yes	—
153	C₂×C₆₇	#1995	1	2	yes	—
154	#2296	C₂×C₆₇	1	1	yes	characteristic
155	C₂×#376	C₂×C₆₇	1	1	yes	characteristic
156	C₂×#339	C₂×C₆₇	1	2	yes	—
157	C₂×#339	C₂×C₆₇	1	2	yes	—
158	#1728	C₂×C₆₇	1	1	yes	characteristic
159	C₆₇	#10879	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
161	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
162	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
163	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
164	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
165	#389	C₄×C₆₇	1	4	yes	complemented
166	#385	—	2	2	no	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	#376	—	2	4	no	—
169	C₂×#95	—	2	2	no	—
170	#339	—	2	4	no	—
171	#339	C₄×C₆₇	1	8	yes	complemented
172	#336	—	2	2	no	—
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	D₄×C₆₇	1	2	yes	—
175	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	C₂×C₄×C₆₇	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
190	C₂×C₂²⋊C₄	—	2	1	no	—
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
200	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
201	C₂×C₄²	D₄×C₆₇	1	2	yes	—
202	C₂×C₄²	D₄×C₆₇	1	4	yes	—
203	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
204	#95	—	2	8	no	—
205	#95	—	2	4	no	—
206	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	—	4	8	no	—
216	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
222	C₂²×C₄	#41×C₆₇	1	2	yes	—
223	C₂²×C₄	#41×C₆₇	1	2	yes	—
224	C₂²×C₄	#41×C₆₇	1	2	yes	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	8	no	—
235	C₄⋊C₄	C₄²×C₆₇	1	4	yes	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	1	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	4	2	no	—
246	C₄²	C₄⋊C₄×C₆₇	1	8	yes	complemented
247	C₂³	—	2	4	no	—
248	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
249	C₂³	C₄×D₄×C₆₇	1	2	yes	—
250	C₂³	#120×C₆₇	1	1	yes	characteristic
251	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
252	C₂³	#123×C₆₇	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
268	C₂×C₄	#117×C₆₇	1	2	yes	—
269	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
270	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
271	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
273	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
274	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
275	C₂×C₄	#121×C₆₇	1	2	yes	—
276	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
277	C₂×C₄	#123×C₆₇	1	2	yes	—
278	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
279	C₂×C₄	#125×C₆₇	1	2	yes	—
280	C₂×C₄	#126×C₆₇	1	2	yes	—
281	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
282	C₈	—	4	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
288	C₂²	#385×C₆₇	1	2	yes	—
289	C₂²	#386×C₆₇	1	1	yes	characteristic
290	C₂²	#387×C₆₇	1	2	yes	—
291	C₂²	#391×C₆₇	1	1	yes	characteristic
292	C₂²	#421×C₆₇	1	4	yes	—
293	C₄	—	4	8	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	4	no	—
296	C₄	—	8	2	no	—
297	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
298	C₄	#387×C₆₇	1	2	yes	—
299	C₄	#388×C₆₇	1	2	yes	—
300	C₄	#389×C₆₇	1	2	yes	—
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	C₆₇×#1436	1	1	yes	characteristic
304	C₂	C₆₇×#1759	1	2	yes	—
305	C₂	C₆₇×#1897	1	2	yes	—
306	C₂	C₆₇×#1995	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. #389×C₆₇	C₄	semidirect
17. #339×C₆₇	C₄	semidirect
92. C₄²×C₆₇	C₄⋊C₄	semidirect
147. #10879	C₆₇	direct (N × H)
159. C₆₇	#10879	direct (N × H)
165. #389	C₄×C₆₇	semidirect
171. #339	C₄×C₆₇	semidirect
246. C₄²	C₄⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2296	2
C₂×#376×C₆₇	2
C₂×#339×C₆₇	2
C₆₇×#1728	2
#10879	67

Subgroups — C67×#10879 #30690130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10879 #30690130