Subgroups — C₆₇×#33927 #30713178

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3393	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2242	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2241	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2229	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2210	C₂	1	2	yes	maximal
8	C₆₇×#2208	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₃₂×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#556×C₆₇	—	2	1	no	—
13	#554×C₆₇	—	2	1	no	—
14	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
15	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#500×C₆₇	C₂²	1	1	yes	characteristic
17	#499×C₆₇	C₂²	1	2	yes	—
18	#495×C₆₇	C₂²	1	2	yes	—
19	#494×C₆₇	C₂²	1	1	yes	characteristic
20	#478×C₆₇	—	2	2	no	—
21	#476×C₆₇	—	2	1	no	—
22	#464×C₆₇	—	2	2	no	—
23	#461×C₆₇	—	2	1	no	—
24	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
26	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₆₇	C₂²	1	2	yes	—
28	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
29	#367×C₆₇	C₂²	1	2	yes	—
30	#366×C₆₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
33	#358×C₆₇	C₂²	1	2	yes	—
34	#358×C₆₇	C₂²	1	2	yes	—
35	#358×C₆₇	C₂²	1	2	yes	—
36	#357×C₆₇	C₂²	1	1	yes	characteristic
37	#357×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₆₇	—	2	1	no	—
42	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₆₇	—	2	1	no	—
47	C₄⋊Q₈×C₆₇	—	2	1	no	—
48	#126×C₆₇	—	4	2	no	—
49	#125×C₆₇	—	2	2	no	—
50	#124×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	4	1	no	—
52	C₂²⋊Q₈×C₆₇	—	4	1	no	—
53	C₂²⋊Q₈×C₆₇	—	4	2	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
56	C₄×D₄×C₆₇	—	2	1	no	—
57	C₄×D₄×C₆₇	—	2	1	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	#117×C₆₇	—	4	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
61	SD₃₂×C₆₇	—	2	4	no	—
62	D₁₆×C₆₇	—	2	2	no	—
63	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
65	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
66	#107×C₆₇	—	2	1	no	—
67	#107×C₆₇	C₂³	1	1	yes	characteristic
68	#107×C₆₇	C₂³	1	2	yes	—
69	#107×C₆₇	C₂³	1	1	yes	characteristic
70	#106×C₆₇	—	2	2	no	—
71	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
73	#103×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	1	no	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	—	2	1	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	4	1	no	—
81	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
82	C₂×D₄×C₆₇	—	2	1	no	—
83	C₂×D₄×C₆₇	—	2	1	no	—
84	C₂²×C₄×C₆₇	—	4	1	no	—
85	C₂²×C₄×C₆₇	—	4	1	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	Q₁₆×C₆₇	—	2	2	no	—
88	Q₁₆×C₆₇	—	2	2	no	—
89	Q₁₆×C₆₇	—	8	1	no	—
90	SD₁₆×C₆₇	—	8	2	no	—
91	D₈×C₆₇	—	2	1	no	—
92	D₈×C₆₇	—	2	1	no	—
93	D₈×C₆₇	—	2	1	no	—
94	D₈×C₆₇	#41	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	—	2	1	no	—
99	C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	4	2	no	—
102	C₄⋊C₄×C₆₇	—	4	1	no	—
103	C₄⋊C₄×C₆₇	—	4	1	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	—	4	2	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
108	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
109	C₂²⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	—	4	1	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₂²⋊C₄×C₆₇	—	4	1	no	—
113	C₂²⋊C₄×C₆₇	—	4	2	no	—
114	C₄²×C₆₇	—	4	2	no	—
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂×D₄	1	2	yes	—
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	—	2	2	no	—
119	C₁₆×C₆₇	—	2	1	no	—
120	C₂³×C₆₇	—	4	1	no	—
121	C₂³×C₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	2	2	no	—
123	Q₈×C₆₇	—	4	2	no	—
124	Q₈×C₆₇	—	4	2	no	—
125	Q₈×C₆₇	—	16	1	no	—
126	D₄×C₆₇	—	2	2	no	—
127	D₄×C₆₇	—	4	1	no	—
128	D₄×C₆₇	—	4	1	no	—
129	D₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	4	1	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	8	2	no	—
135	C₂×C₄×C₆₇	—	8	2	no	—
136	C₂×C₄×C₆₇	—	16	2	no	—
137	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₆₇	—	2	2	no	—
143	C₈×C₆₇	—	2	1	no	—
144	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂²×C₆₇	—	4	2	no	—
147	C₂²×C₆₇	—	8	1	no	—
148	C₂²×C₆₇	—	8	1	no	—
149	C₂²×C₆₇	—	8	1	no	—
150	C₂²×C₆₇	—	8	1	no	—
151	C₂²×C₆₇	—	8	1	no	—
152	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₆₇	—	2	2	no	—
154	C₄×C₆₇	—	2	2	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	—	8	1	no	—
158	C₄×C₆₇	—	8	1	no	—
159	C₄×C₆₇	—	8	2	no	—
160	C₄×C₆₇	#508	1	2	yes	—
161	C₄×C₆₇	#548	1	1	yes	characteristic
162	C₄×C₆₇	#577	1	1	yes	characteristic
163	#33927	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₆₇	—	8	2	no	—
165	C₂×C₆₇	—	16	1	no	—
166	C₂×C₆₇	#3358	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
168	C₂×C₆₇	#3416	1	1	yes	characteristic
169	#3393	C₂×C₆₇	1	1	yes	characteristic, complemented
170	#3380	C₂×C₆₇	1	1	yes	characteristic, complemented
171	#2242	C₂×C₆₇	1	2	yes	complemented
172	#2241	C₂×C₆₇	1	1	yes	characteristic, complemented
173	#2229	C₂×C₆₇	1	2	yes	complemented
174	#2226	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#2210	C₂×C₆₇	1	2	yes	—
176	#2208	C₂×C₆₇	1	1	yes	characteristic, complemented
177	Q₈×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
178	C₄×SD₃₂	C₂×C₆₇	1	2	yes	complemented
179	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic
180	C₆₇	#33927	1	1	yes	characteristic, direct product factor, Sylow
181	#556	—	2	1	no	—
182	#554	—	2	1	no	—
183	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
184	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
185	#500	C₂²×C₆₇	1	1	yes	characteristic
186	#499	C₂²×C₆₇	1	2	yes	—
187	#495	C₂²×C₆₇	1	2	yes	—
188	#494	C₂²×C₆₇	1	1	yes	characteristic
189	#478	—	2	2	no	—
190	#476	—	2	1	no	—
191	#464	—	2	2	no	—
192	#461	—	2	1	no	—
193	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
195	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
196	C₄×D₈	C₂²×C₆₇	1	2	yes	—
197	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
198	#367	C₂²×C₆₇	1	2	yes	—
199	#366	C₂²×C₆₇	1	1	yes	characteristic
200	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
202	#358	C₂²×C₆₇	1	2	yes	—
203	#358	C₂²×C₆₇	1	2	yes	—
204	#358	C₂²×C₆₇	1	2	yes	—
205	#357	C₂²×C₆₇	1	1	yes	characteristic
206	#357	C₂²×C₆₇	1	1	yes	characteristic
207	#357	C₂²×C₆₇	1	2	yes	—
208	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
209	C₄×C₁₆	C₂²×C₆₇	1	2	yes	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
214	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	#126	—	4	2	no	—
218	#125	—	2	2	no	—
219	#124	—	2	2	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	C₂²⋊Q₈	—	4	1	no	—
222	C₂²⋊Q₈	—	4	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	2	1	no	—
228	#117	—	4	1	no	—
229	C₂×C₄⋊C₄	—	4	1	no	—
230	SD₃₂	—	2	4	no	—
231	D₁₆	—	2	2	no	—
232	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
233	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
234	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
235	#107	—	2	1	no	—
236	#107	C₂³×C₆₇	1	1	yes	characteristic
237	#107	C₂³×C₆₇	1	2	yes	—
238	#107	C₂³×C₆₇	1	1	yes	characteristic
239	#106	—	2	2	no	—
240	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
241	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
242	#103	—	2	2	no	—
243	#102	—	2	1	no	—
244	#102	—	2	2	no	—
245	#102	—	2	1	no	—
246	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₆₇	1	2	yes	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	8	1	no	—
259	SD₁₆	—	8	2	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	D₈	—	2	1	no	—
263	D₈	#41×C₆₇	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
266	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
277	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₆₇	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	2	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	16	2	no	—
306	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
307	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
308	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
314	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
315	C₂²	—	4	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	#508×C₆₇	1	2	yes	—
330	C₄	#548×C₆₇	1	1	yes	characteristic
331	C₄	#577×C₆₇	1	1	yes	characteristic
332	C₂	—	8	2	no	—
333	C₂	—	16	1	no	—
334	C₂	C₆₇×#3358	1	1	yes	characteristic
335	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
336	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×D₁₆×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3393	C₂	semidirect
2. C₆₇×#3380	C₂	semidirect
3. C₆₇×#2242	C₂	semidirect
4. C₆₇×#2241	C₂	semidirect
5. C₆₇×#2229	C₂	semidirect
6. C₆₇×#2226	C₂	semidirect
8. C₆₇×#2208	C₂	semidirect
9. Q₈×C₁₆×C₆₇	C₂	semidirect
10. C₄×SD₃₂×C₆₇	C₂	semidirect
163. #33927	C₆₇	direct (N × H)
169. #3393	C₂×C₆₇	semidirect
170. #3380	C₂×C₆₇	semidirect
171. #2242	C₂×C₆₇	semidirect
172. #2241	C₂×C₆₇	semidirect
173. #2229	C₂×C₆₇	semidirect
174. #2226	C₂×C₆₇	semidirect
176. #2208	C₂×C₆₇	semidirect
177. Q₈×C₁₆	C₂×C₆₇	semidirect
178. C₄×SD₃₂	C₂×C₆₇	semidirect
180. C₆₇	#33927	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×#3393	2
C₆₇×#3380	2
C₆₇×#2242	2
C₆₇×#2241	2
C₆₇×#2229	2
C₆₇×#2226	2
C₆₇×#2210	2
C₆₇×#2208	2
Q₈×C₁₆×C₆₇	2
C₄×SD₃₂×C₆₇	2
C₄×D₁₆×C₆₇	2
#33927	67

Subgroups — C67×#33927 #30713178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33927 #30713178