Subgroups — C₆₇×#33906 #30713157

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3339	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2234	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2233	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2228	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#553×C₆₇	—	2	2	no	—
13	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#499×C₆₇	C₂²	1	2	yes	—
17	#482×C₆₇	—	2	2	no	—
18	#481×C₆₇	—	2	2	no	—
19	#478×C₆₇	—	2	2	no	—
20	#466×C₆₇	C₂²	1	2	yes	—
21	#466×C₆₇	C₂²	1	2	yes	—
22	C₄×D₈×C₆₇	C₂²	1	2	yes	—
23	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	#427×C₆₇	C₂²	1	2	yes	—
25	C₂×#107×C₆₇	C₂²	1	2	yes	—
26	#367×C₆₇	C₂²	1	2	yes	—
27	#367×C₆₇	C₂²	1	2	yes	—
28	#366×C₆₇	C₂²	1	1	yes	characteristic
29	#366×C₆₇	C₂²	1	1	yes	characteristic
30	#366×C₆₇	C₂²	1	1	yes	characteristic
31	#366×C₆₇	C₂²	1	2	yes	—
32	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	#357×C₆₇	C₂²	1	2	yes	—
34	#357×C₆₇	C₂²	1	2	yes	—
35	#357×C₆₇	C₂²	1	2	yes	—
36	C₂²⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×D₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	#126×C₆₇	—	4	2	no	—
43	#125×C₆₇	—	2	2	no	—
44	#123×C₆₇	—	4	2	no	—
45	#121×C₆₇	—	2	2	no	—
46	#121×C₆₇	—	2	2	no	—
47	#121×C₆₇	—	4	2	no	—
48	C₄×D₄×C₆₇	—	2	2	no	—
49	C₄×D₄×C₆₇	—	4	2	no	—
50	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
51	#117×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	D₁₆×C₆₇	—	2	2	no	—
54	C₂×C₁₆×C₆₇	—	2	2	no	—
55	C₂×C₁₆×C₆₇	—	2	2	no	—
56	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
60	#107×C₆₇	—	2	2	no	—
61	#107×C₆₇	C₂³	1	2	yes	—
62	#107×C₆₇	C₂³	1	2	yes	—
63	#107×C₆₇	C₂³	1	2	yes	—
64	#106×C₆₇	—	2	2	no	—
65	#106×C₆₇	—	2	2	no	—
66	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
67	#102×C₆₇	—	2	2	no	—
68	#102×C₆₇	—	2	2	no	—
69	#102×C₆₇	—	2	2	no	—
70	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
71	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₆₇	—	2	2	no	—
74	C₂×D₄×C₆₇	—	4	2	no	—
75	C₂×D₄×C₆₇	—	4	2	no	—
76	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	—	4	2	no	—
78	C₂²×C₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	D₈×C₆₇	—	2	2	no	—
82	D₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	1	no	—
84	C₂×C₈×C₆₇	—	2	1	no	—
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₄⋊C₄×C₆₇	—	4	2	no	—
95	C₄⋊C₄×C₆₇	—	4	2	no	—
96	C₄⋊C₄×C₆₇	—	4	2	no	—
97	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
98	C₂²⋊C₄×C₆₇	—	4	2	no	—
99	C₂²⋊C₄×C₆₇	—	4	2	no	—
100	C₂²⋊C₄×C₆₇	—	4	2	no	—
101	C₂²⋊C₄×C₆₇	—	4	2	no	—
102	C₂²⋊C₄×C₆₇	D₈	1	1	yes	characteristic
103	C₂²⋊C₄×C₆₇	D₈	1	1	yes	characteristic
104	C₄²×C₆₇	—	4	2	no	—
105	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₆₇	—	2	1	no	—
107	C₁₆×C₆₇	—	2	1	no	—
108	C₁₆×C₆₇	—	2	1	no	—
109	C₁₆×C₆₇	#41	1	2	yes	—
110	C₂³×C₆₇	—	4	2	no	—
111	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
112	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
113	D₄×C₆₇	—	2	2	no	—
114	D₄×C₆₇	—	4	2	no	—
115	D₄×C₆₇	—	4	2	no	—
116	D₄×C₆₇	—	8	4	no	—
117	D₄×C₆₇	—	16	2	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	16	2	no	—
126	C₂×C₄×C₆₇	—	16	2	no	—
127	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₆₇	—	2	1	no	—
133	C₈×C₆₇	—	2	1	no	—
134	C₈×C₆₇	—	2	1	no	—
135	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	—	8	2	no	—
144	C₂²×C₆₇	—	8	2	no	—
145	C₂²×C₆₇	#508	1	2	yes	—
146	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	8	2	no	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	#547	1	1	yes	characteristic
155	C₄×C₆₇	#578	1	1	yes	characteristic
156	#33906	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₆₇	—	2	2	no	—
158	C₂×C₆₇	—	4	1	no	—
159	C₂×C₆₇	—	16	2	no	—
160	C₂×C₆₇	#3334	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
162	C₂×C₆₇	#3415	1	1	yes	characteristic
163	#3339	C₂×C₆₇	1	2	yes	complemented
164	#2253	C₂×C₆₇	1	1	yes	characteristic, complemented
165	#2234	C₂×C₆₇	1	2	yes	complemented
166	#2233	C₂×C₆₇	1	2	yes	complemented
167	#2228	C₂×C₆₇	1	2	yes	complemented
168	#2215	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#2215	C₂×C₆₇	1	1	yes	characteristic, complemented
170	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
171	D₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
172	#2158	C₂×C₆₇	1	1	yes	characteristic, complemented
173	C₂×#366	C₂×C₆₇	1	1	yes	characteristic, complemented
174	C₆₇	#33906	1	1	yes	characteristic, direct product factor, Sylow
175	#553	—	2	2	no	—
176	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
177	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
178	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
179	#499	C₂²×C₆₇	1	2	yes	—
180	#482	—	2	2	no	—
181	#481	—	2	2	no	—
182	#478	—	2	2	no	—
183	#466	C₂²×C₆₇	1	2	yes	—
184	#466	C₂²×C₆₇	1	2	yes	—
185	C₄×D₈	C₂²×C₆₇	1	2	yes	—
186	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
187	#427	C₂²×C₆₇	1	2	yes	—
188	C₂×#107	C₂²×C₆₇	1	2	yes	—
189	#367	C₂²×C₆₇	1	2	yes	—
190	#367	C₂²×C₆₇	1	2	yes	—
191	#366	C₂²×C₆₇	1	1	yes	characteristic
192	#366	C₂²×C₆₇	1	1	yes	characteristic
193	#366	C₂²×C₆₇	1	2	yes	—
194	#366	C₂²×C₆₇	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
196	#357	C₂²×C₆₇	1	2	yes	—
197	#357	C₂²×C₆₇	1	2	yes	—
198	#357	C₂²×C₆₇	1	2	yes	—
199	C₂²⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
200	C₂²⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
201	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
202	C₂×D₈	C₂³×C₆₇	1	2	yes	—
203	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
205	#126	—	4	2	no	—
206	#125	—	2	2	no	—
207	#123	—	4	2	no	—
208	#121	—	2	2	no	—
209	#121	—	2	2	no	—
210	#121	—	4	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	4	2	no	—
213	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
214	#117	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	D₁₆	—	2	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	2	2	no	—
219	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
220	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
221	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
222	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
223	#107	—	2	2	no	—
224	#107	C₂³×C₆₇	1	2	yes	—
225	#107	C₂³×C₆₇	1	2	yes	—
226	#107	C₂³×C₆₇	1	2	yes	—
227	#106	—	2	2	no	—
228	#106	—	2	2	no	—
229	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
234	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	D₈×C₆₇	1	1	yes	characteristic
266	C₂²⋊C₄	D₈×C₆₇	1	1	yes	characteristic
267	C₄²	—	4	2	no	—
268	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	#41×C₆₇	1	2	yes	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
275	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
276	D₄	—	2	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	4	no	—
280	D₄	—	16	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	16	2	no	—
289	C₂×C₄	—	16	2	no	—
290	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
299	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	#508×C₆₇	1	2	yes	—
309	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	#547×C₆₇	1	1	yes	characteristic
318	C₄	#578×C₆₇	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	4	1	no	—
321	C₂	—	16	2	no	—
322	C₂	C₆₇×#3334	1	1	yes	characteristic
323	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
324	C₂	C₆₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3339	C₂	semidirect
2. C₆₇×#2253	C₂	semidirect
3. C₆₇×#2234	C₂	semidirect
4. C₆₇×#2233	C₂	semidirect
5. C₆₇×#2228	C₂	semidirect
6. C₆₇×#2215	C₂	semidirect
7. C₆₇×#2215	C₂	semidirect
8. C₄×D₁₆×C₆₇	C₂	semidirect
9. D₄×C₁₆×C₆₇	C₂	semidirect
10. C₆₇×#2158	C₂	semidirect
11. C₂×#366×C₆₇	C₂	semidirect
156. #33906	C₆₇	direct (N × H)
163. #3339	C₂×C₆₇	semidirect
164. #2253	C₂×C₆₇	semidirect
165. #2234	C₂×C₆₇	semidirect
166. #2233	C₂×C₆₇	semidirect
167. #2228	C₂×C₆₇	semidirect
168. #2215	C₂×C₆₇	semidirect
169. #2215	C₂×C₆₇	semidirect
170. C₄×D₁₆	C₂×C₆₇	semidirect
171. D₄×C₁₆	C₂×C₆₇	semidirect
172. #2158	C₂×C₆₇	semidirect
173. C₂×#366	C₂×C₆₇	semidirect
174. C₆₇	#33906	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×#3339	2
C₆₇×#2253	2
C₆₇×#2234	2
C₆₇×#2233	2
C₆₇×#2228	2
C₆₇×#2215	2
C₄×D₁₆×C₆₇	2
D₄×C₁₆×C₆₇	2
C₆₇×#2158	2
C₂×#366×C₆₇	2
#33906	67

Subgroups — C67×#33906 #30713157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33906 #30713157