Subgroups — C₆₇×#33136 #30712387

Order 17152.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2855	C₂	1	2	yes	maximal
4	C₆₇×#1971	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1944	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1935	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1892	C₂	1	2	yes	maximal
8	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#467×C₆₇	—	2	4	no	—
15	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×#103×C₆₇	C₂²	1	2	yes	—
18	C₂×#103×C₆₇	C₂²	1	4	yes	—
19	C₂×#97×C₆₇	—	2	4	no	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	#391×C₆₇	C₂²	1	4	yes	—
23	#389×C₆₇	C₂²	1	4	yes	—
24	#387×C₆₇	—	2	4	no	—
25	#385×C₆₇	—	2	2	no	—
26	#381×C₆₇	—	2	2	no	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	#340×C₆₇	C₂²	1	2	yes	—
29	#340×C₆₇	C₂²	1	4	yes	—
30	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×Q₁₆×C₆₇	—	2	2	no	—
35	C₂×Q₁₆×C₆₇	—	2	4	no	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	—	2	4	no	—
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	D₄	1	4	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₆₇	—	2	4	no	—
44	C₂²⋊Q₈×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
55	#107×C₆₇	—	2	2	no	—
56	#106×C₆₇	—	2	4	no	—
57	#103×C₆₇	—	2	4	no	—
58	#103×C₆₇	—	2	8	no	—
59	#97×C₆₇	—	2	4	no	—
60	#97×C₆₇	—	2	4	no	—
61	#95×C₆₇	—	2	2	no	—
62	#95×C₆₇	—	2	4	no	—
63	#95×C₆₇	—	2	4	no	—
64	#95×C₆₇	—	2	4	no	—
65	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₆₇	—	2	2	no	—
67	C₂×Q₈×C₆₇	—	2	4	no	—
68	C₂×Q₈×C₆₇	—	2	2	no	—
69	C₂×Q₈×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	Q₁₆×C₆₇	—	4	4	no	—
86	C₂×C₈×C₆₇	—	2	8	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	8	no	—
91	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	#41	1	4	yes	—
94	C₂×C₈×C₆₇	#41	1	4	yes	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₄⋊C₄×C₆₇	—	2	4	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	4	8	no	—
100	C₄⋊C₄×C₆₇	—	4	4	no	—
101	C₂²⋊C₄×C₆₇	—	4	4	no	—
102	C₂²⋊C₄×C₆₇	—	4	4	no	—
103	C₄²×C₆₇	—	2	4	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	#121	1	2	yes	—
109	C₂³×C₆₇	#137	1	4	yes	—
110	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₆₇	—	4	2	no	—
112	Q₈×C₆₇	—	4	2	no	—
113	Q₈×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	4	4	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	4	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	4	4	no	—
129	C₂×C₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	#121	1	2	yes	—
132	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
135	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₈×C₆₇	—	2	2	no	—
137	C₈×C₆₇	—	2	4	no	—
138	C₈×C₆₇	#123	1	4	yes	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	2	8	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	#470	1	4	yes	—
143	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
144	C₂²×C₆₇	#532	1	2	yes	—
145	C₂²×C₆₇	C₂×#137	1	2	yes	—
146	C₂²×C₆₇	#575	1	2	yes	—
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	—	4	2	no	—
150	C₄×C₆₇	—	8	2	no	—
151	C₄×C₆₇	—	8	4	no	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
154	C₄×C₆₇	#525	1	1	yes	characteristic
155	C₄×C₆₇	#529	1	2	yes	—
156	#33136	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₆₇	—	2	4	no	—
158	C₂×C₆₇	#2854	1	1	yes	characteristic
159	C₂×C₆₇	C₂×#470	1	1	yes	characteristic
160	C₂×C₆₇	#3054	1	1	yes	characteristic
161	C₂×C₆₇	#3181	1	4	yes	—
162	C₂×#467	C₂×C₆₇	1	1	yes	characteristic
163	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
164	#2855	C₂×C₆₇	1	2	yes	—
165	#1971	C₂×C₆₇	1	2	yes	complemented
166	#1944	C₂×C₆₇	1	2	yes	complemented
167	#1935	C₂×C₆₇	1	4	yes	complemented
168	#1892	C₂×C₆₇	1	2	yes	—
169	#1807	C₂×C₆₇	1	1	yes	characteristic
170	C₆₇	#33136	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
172	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
173	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
174	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
175	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
176	#467	—	2	4	no	—
177	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
178	C₂×#106	C₂²×C₆₇	1	2	yes	—
179	C₂×#103	C₂²×C₆₇	1	4	yes	—
180	C₂×#103	C₂²×C₆₇	1	2	yes	—
181	C₂×#97	—	2	4	no	—
182	C₂×#97	C₂²×C₆₇	1	2	yes	—
183	C₂×#97	C₂²×C₆₇	1	2	yes	—
184	#391	C₂²×C₆₇	1	4	yes	—
185	#389	C₂²×C₆₇	1	4	yes	—
186	#387	—	2	4	no	—
187	#385	—	2	2	no	—
188	#381	—	2	2	no	—
189	#380	C₂²×C₆₇	1	2	yes	—
190	#340	C₂²×C₆₇	1	2	yes	—
191	#340	C₂²×C₆₇	1	4	yes	—
192	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
193	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
194	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
195	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
196	C₂×Q₁₆	—	2	2	no	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	D₄×C₆₇	1	4	yes	—
202	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
204	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
205	C₂²⋊Q₈	—	2	4	no	—
206	C₂²⋊Q₈	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	D₄×C₆₇	1	2	yes	—
216	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
217	#107	—	2	2	no	—
218	#106	—	2	4	no	—
219	#103	—	2	4	no	—
220	#103	—	2	8	no	—
221	#97	—	2	4	no	—
222	#97	—	2	4	no	—
223	#95	—	2	2	no	—
224	#95	—	2	4	no	—
225	#95	—	2	4	no	—
226	#95	—	2	4	no	—
227	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
245	C₂²×C₄	#41×C₆₇	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
247	Q₁₆	—	4	4	no	—
248	C₂×C₈	—	2	8	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
255	C₂×C₈	#41×C₆₇	1	4	yes	—
256	C₂×C₈	#41×C₆₇	1	4	yes	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	8	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	#121×C₆₇	1	2	yes	—
271	C₂³	#137×C₆₇	1	4	yes	—
272	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	#121×C₆₇	1	2	yes	—
294	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
296	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
297	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	4	no	—
300	C₈	#123×C₆₇	1	4	yes	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	8	no	—
304	C₂²	#470×C₆₇	1	4	yes	—
305	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
306	C₂²	#532×C₆₇	1	2	yes	—
307	C₂²	C₂×#137×C₆₇	1	2	yes	—
308	C₂²	#575×C₆₇	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	4	no	—
315	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
316	C₄	#525×C₆₇	1	1	yes	characteristic
317	C₄	#529×C₆₇	1	2	yes	—
318	C₂	—	2	4	no	—
319	C₂	C₆₇×#2854	1	1	yes	characteristic
320	C₂	C₂×#470×C₆₇	1	1	yes	characteristic
321	C₂	C₆₇×#3054	1	1	yes	characteristic
322	C₂	C₆₇×#3181	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1971	C₂	semidirect
5. C₆₇×#1944	C₂	semidirect
6. C₆₇×#1935	C₂	semidirect
156. #33136	C₆₇	direct (N × H)
165. #1971	C₂×C₆₇	semidirect
166. #1944	C₂×C₆₇	semidirect
167. #1935	C₂×C₆₇	semidirect
170. C₆₇	#33136	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#467×C₆₇	2
C₂²×#97×C₆₇	2
C₆₇×#2855	2
C₆₇×#1971	2
C₆₇×#1944	2
C₆₇×#1935	2
C₆₇×#1892	2
C₆₇×#1807	2
#33136	67

Subgroups — C67×#33136 #30712387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33136 #30712387