Subgroups — C₆₇×#33923 #30713174

Order 17152.

470 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×#3342	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3338	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2237	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2235	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2219	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2169	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2159	C₂	1	2	yes	maximal, complemented
12	#556×C₆₇	—	2	1	no	—
13	#554×C₆₇	—	2	1	no	—
14	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
16	#500×C₆₇	C₂²	1	1	yes	characteristic
17	#500×C₆₇	C₂²	1	1	yes	characteristic
18	#485×C₆₇	—	2	2	no	—
19	#484×C₆₇	—	2	2	no	—
20	#477×C₆₇	—	2	1	no	—
21	#475×C₆₇	—	2	1	no	—
22	#467×C₆₇	C₂²	1	2	yes	—
23	#467×C₆₇	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	#427×C₆₇	C₂²	1	2	yes	—
28	C₂×#107×C₆₇	C₂²	1	2	yes	—
29	#367×C₆₇	C₂²	1	2	yes	—
30	#367×C₆₇	C₂²	1	2	yes	—
31	#367×C₆₇	C₂²	1	2	yes	—
32	#366×C₆₇	C₂²	1	1	yes	characteristic
33	#366×C₆₇	C₂²	1	2	yes	—
34	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	#358×C₆₇	C₂²	1	1	yes	characteristic
36	#358×C₆₇	C₂²	1	1	yes	characteristic
37	#358×C₆₇	C₂²	1	2	yes	—
38	#358×C₆₇	C₂²	1	2	yes	—
39	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₆₇	—	2	1	no	—
45	C₄⋊Q₈×C₆₇	—	2	1	no	—
46	#126×C₆₇	—	4	2	no	—
47	#125×C₆₇	—	4	1	no	—
48	#124×C₆₇	—	4	1	no	—
49	C₂²⋊Q₈×C₆₇	—	2	2	no	—
50	C₂²⋊Q₈×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	4	2	no	—
52	C₄×Q₈×C₆₇	—	2	1	no	—
53	C₄×Q₈×C₆₇	—	2	1	no	—
54	C₄×Q₈×C₆₇	—	4	1	no	—
55	C₄×D₄×C₆₇	—	4	1	no	—
56	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
57	#117×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	Q₃₂×C₆₇	—	2	2	no	—
60	M₃₂×C₆₇	—	2	4	no	—
61	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
62	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	—	2	2	no	—
65	#107×C₆₇	—	2	1	no	—
66	#107×C₆₇	—	2	1	no	—
67	#107×C₆₇	C₂³	1	2	yes	—
68	#107×C₆₇	C₂³	1	2	yes	—
69	#107×C₆₇	C₂³	1	1	yes	characteristic
70	#107×C₆₇	C₂³	1	1	yes	characteristic
71	#106×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
73	#103×C₆₇	—	2	2	no	—
74	#103×C₆₇	—	2	2	no	—
75	#103×C₆₇	—	2	1	no	—
76	#103×C₆₇	—	2	1	no	—
77	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₇	—	2	1	no	—
80	C₂×Q₈×C₆₇	—	2	1	no	—
81	C₂×Q₈×C₆₇	—	4	1	no	—
82	C₂×Q₈×C₆₇	—	4	1	no	—
83	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	—	4	2	no	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	Q₁₆×C₆₇	—	2	1	no	—
87	Q₁₆×C₆₇	—	2	1	no	—
88	Q₁₆×C₆₇	—	2	1	no	—
89	Q₁₆×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₆₇	—	2	1	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	4	2	no	—
100	C₄⋊C₄×C₆₇	—	4	1	no	—
101	C₄⋊C₄×C₆₇	—	4	1	no	—
102	C₄⋊C₄×C₆₇	—	4	1	no	—
103	C₄⋊C₄×C₆₇	—	4	2	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	—	4	1	no	—
106	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₆₇	—	4	2	no	—
108	C₂²⋊C₄×C₆₇	—	4	2	no	—
109	C₂²⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
111	C₄²×C₆₇	—	4	2	no	—
112	C₄²×C₆₇	—	4	1	no	—
113	C₄²×C₆₇	—	4	1	no	—
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₆₇	—	2	2	no	—
116	C₁₆×C₆₇	—	2	1	no	—
117	C₁₆×C₆₇	#41	1	2	yes	—
118	C₂³×C₆₇	C₂×D₈	1	2	yes	—
119	Q₈×C₆₇	—	4	1	no	—
120	Q₈×C₆₇	—	4	1	no	—
121	Q₈×C₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	4	1	no	—
123	Q₈×C₆₇	—	8	2	no	—
124	Q₈×C₆₇	—	16	1	no	—
125	D₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	1	no	—
131	C₂×C₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	16	2	no	—
135	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	8	1	no	—
150	C₄×C₆₇	—	8	1	no	—
151	C₄×C₆₇	—	8	2	no	—
152	C₄×C₆₇	—	8	1	no	—
153	C₄×C₆₇	—	8	1	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	#510	1	2	yes	—
156	C₄×C₆₇	#547	1	1	yes	characteristic
157	C₄×C₆₇	#578	1	1	yes	characteristic
158	#33923	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₆₇	—	4	2	no	—
160	C₂×C₆₇	#3334	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
162	C₂×C₆₇	#3416	1	1	yes	characteristic
163	#3342	C₂×C₆₇	1	1	yes	characteristic
164	#3338	C₂×C₆₇	1	1	yes	characteristic
165	#2254	C₂×C₆₇	1	1	yes	characteristic, complemented
166	#2237	C₂×C₆₇	1	2	yes	complemented
167	#2235	C₂×C₆₇	1	2	yes	complemented
168	#2227	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#2225	C₂×C₆₇	1	1	yes	characteristic, complemented
170	#2219	C₂×C₆₇	1	2	yes	complemented
171	C₄×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
172	#2169	C₂×C₆₇	1	1	yes	characteristic
173	#2159	C₂×C₆₇	1	2	yes	complemented
174	C₆₇	#33923	1	1	yes	characteristic, direct product factor, Sylow
175	#556	—	2	1	no	—
176	#554	—	2	1	no	—
177	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
178	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
179	#500	C₂²×C₆₇	1	1	yes	characteristic
180	#500	C₂²×C₆₇	1	1	yes	characteristic
181	#485	—	2	2	no	—
182	#484	—	2	2	no	—
183	#477	—	2	1	no	—
184	#475	—	2	1	no	—
185	#467	C₂²×C₆₇	1	2	yes	—
186	#467	C₂²×C₆₇	1	2	yes	—
187	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
188	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
189	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
190	#427	C₂²×C₆₇	1	2	yes	—
191	C₂×#107	C₂²×C₆₇	1	2	yes	—
192	#367	C₂²×C₆₇	1	2	yes	—
193	#367	C₂²×C₆₇	1	2	yes	—
194	#367	C₂²×C₆₇	1	2	yes	—
195	#366	C₂²×C₆₇	1	1	yes	characteristic
196	#366	C₂²×C₆₇	1	2	yes	—
197	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
198	#358	C₂²×C₆₇	1	1	yes	characteristic
199	#358	C₂²×C₆₇	1	1	yes	characteristic
200	#358	C₂²×C₆₇	1	2	yes	—
201	#358	C₂²×C₆₇	1	2	yes	—
202	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
203	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
204	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
205	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	#126	—	4	2	no	—
210	#125	—	4	1	no	—
211	#124	—	4	1	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	4	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	4	1	no	—
218	C₄×D₄	—	4	1	no	—
219	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	Q₃₂	—	2	2	no	—
223	M₃₂	—	2	4	no	—
224	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
226	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
227	#107	—	2	2	no	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂³×C₆₇	1	1	yes	characteristic
231	#107	C₂³×C₆₇	1	1	yes	characteristic
232	#107	C₂³×C₆₇	1	2	yes	—
233	#107	C₂³×C₆₇	1	2	yes	—
234	#106	—	2	2	no	—
235	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
236	#103	—	2	1	no	—
237	#103	—	2	1	no	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
241	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
257	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
274	C₄²	—	4	2	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	#41×C₆₇	1	2	yes	—
281	C₂³	C₂×D₈×C₆₇	1	2	yes	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	16	1	no	—
288	D₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
305	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	1	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	#510×C₆₇	1	2	yes	—
319	C₄	#547×C₆₇	1	1	yes	characteristic
320	C₄	#578×C₆₇	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	C₆₇×#3334	1	1	yes	characteristic
323	C₂	C₂×#510×C₆₇	1	1	yes	characteristic
324	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×D₄×C₆₇	D₈
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
3. C₆₇×#2254	C₂	semidirect
4. C₆₇×#2237	C₂	semidirect
5. C₆₇×#2235	C₂	semidirect
6. C₆₇×#2227	C₂	semidirect
7. C₆₇×#2225	C₂	semidirect
8. C₆₇×#2219	C₂	semidirect
9. C₄×Q₃₂×C₆₇	C₂	semidirect
11. C₆₇×#2159	C₂	semidirect
158. #33923	C₆₇	direct (N × H)
165. #2254	C₂×C₆₇	semidirect
166. #2237	C₂×C₆₇	semidirect
167. #2235	C₂×C₆₇	semidirect
168. #2227	C₂×C₆₇	semidirect
169. #2225	C₂×C₆₇	semidirect
170. #2219	C₂×C₆₇	semidirect
171. C₄×Q₃₂	C₂×C₆₇	semidirect
173. #2159	C₂×C₆₇	semidirect
174. C₆₇	#33923	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×#3342	2
C₆₇×#3338	2
C₆₇×#2254	2
C₆₇×#2237	2
C₆₇×#2235	2
C₆₇×#2227	2
C₆₇×#2225	2
C₆₇×#2219	2
C₄×Q₃₂×C₆₇	2
C₆₇×#2169	2
C₆₇×#2159	2
#33923	67

Subgroups — C67×#33923 #30713174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33923 #30713174