Subgroups — C₆₇×#32863 #30712114

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2863	C₂	1	2	yes	maximal
2	C₆₇×#2088	C₂	1	2	yes	maximal
3	C₆₇×#2086	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2068	C₂	1	2	yes	maximal
5	C₆₇×#2066	C₂	1	2	yes	maximal
6	C₆₇×#2066	C₂	1	2	yes	maximal
7	C₆₇×#2056	C₂	1	2	yes	maximal
8	C₆₇×#2056	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#106×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#103×C₆₇	C₂²	1	2	yes	—
18	C₂×#103×C₆₇	C₂²	1	2	yes	—
19	C₂×#103×C₆₇	C₂²	1	2	yes	—
20	#391×C₆₇	C₂²	1	2	yes	—
21	#391×C₆₇	C₂²	1	2	yes	—
22	#391×C₆₇	C₂²	1	2	yes	—
23	#389×C₆₇	C₂²	1	2	yes	—
24	#389×C₆₇	C₂²	1	2	yes	—
25	#389×C₆₇	C₂²	1	2	yes	—
26	#387×C₆₇	—	2	2	no	—
27	#387×C₆₇	—	2	2	no	—
28	#387×C₆₇	—	2	2	no	—
29	#383×C₆₇	—	2	2	no	—
30	C₄³×C₆₇	C₂²	1	1	yes	characteristic
31	#340×C₆₇	C₂²	1	2	yes	—
32	#340×C₆₇	C₂²	1	1	yes	characteristic
33	#340×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	2	yes	—
35	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
36	C₂×Q₁₆×C₆₇	—	4	2	no	—
37	C₂×Q₁₆×C₆₇	—	4	1	no	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	#107×C₆₇	—	4	1	no	—
53	#106×C₆₇	—	4	2	no	—
54	C₄⋊C₈×C₆₇	—	4	1	no	—
55	C₄⋊C₈×C₆₇	—	4	2	no	—
56	C₄⋊C₈×C₆₇	—	4	2	no	—
57	C₄⋊C₈×C₆₇	—	4	1	no	—
58	#103×C₆₇	—	4	2	no	—
59	#103×C₆₇	—	4	2	no	—
60	#103×C₆₇	—	4	2	no	—
61	#95×C₆₇	—	2	2	no	—
62	#95×C₆₇	—	2	2	no	—
63	#95×C₆₇	—	2	2	no	—
64	#95×C₆₇	—	2	2	no	—
65	#95×C₆₇	—	2	2	no	—
66	#95×C₆₇	—	2	2	no	—
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	—	2	2	no	—
69	#95×C₆₇	—	2	2	no	—
70	C₂×Q₈×C₆₇	—	2	2	no	—
71	C₂×Q₈×C₆₇	—	2	2	no	—
72	C₂×Q₈×C₆₇	—	2	2	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×Q₈×C₆₇	—	2	2	no	—
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
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	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	Q₁₆×C₆₇	—	4	4	no	—
86	C₂×C₈×C₆₇	—	2	2	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	1	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	4	4	no	—
100	C₄⋊C₄×C₆₇	—	4	4	no	—
101	C₄⋊C₄×C₆₇	—	4	4	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	4	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	—	2	1	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	1	no	—
108	C₄²×C₆₇	—	4	1	no	—
109	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₆₇	—	4	2	no	—
111	Q₈×C₆₇	—	4	2	no	—
112	Q₈×C₆₇	—	4	2	no	—
113	Q₈×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	#135	1	2	yes	—
136	C₂×C₄×C₆₇	#137	1	4	yes	—
137	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
140	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
141	C₈×C₆₇	—	4	2	no	—
142	C₈×C₆₇	—	4	1	no	—
143	C₈×C₆₇	—	4	1	no	—
144	C₂²×C₆₇	#537	1	2	yes	—
145	C₂²×C₆₇	#537	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
147	C₂²×C₆₇	C₂×#137	1	2	yes	—
148	C₂²×C₆₇	#575	1	1	yes	characteristic
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	1	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	#560	1	1	yes	characteristic
158	C₄×C₆₇	#563	1	2	yes	—
159	C₄×C₆₇	#563	1	1	yes	characteristic
160	#32863	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₆₇	#2823	1	1	yes	characteristic
162	C₂×C₆₇	#3248	1	2	yes	—
163	C₂×C₆₇	#3250	1	1	yes	characteristic
164	C₂×C₆₇	#3252	1	2	yes	—
165	C₂×C₆₇	#3255	1	1	yes	characteristic
166	#2863	C₂×C₆₇	1	2	yes	—
167	#2088	C₂×C₆₇	1	2	yes	—
168	#2086	C₂×C₆₇	1	1	yes	characteristic
169	#2068	C₂×C₆₇	1	2	yes	—
170	#2066	C₂×C₆₇	1	2	yes	—
171	#2066	C₂×C₆₇	1	2	yes	—
172	#2056	C₂×C₆₇	1	2	yes	—
173	#2056	C₂×C₆₇	1	1	yes	characteristic
174	#1842	C₂×C₆₇	1	1	yes	characteristic
175	C₆₇	#32863	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
177	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
178	C₂×#106	C₂²×C₆₇	1	2	yes	—
179	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
180	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
181	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
183	C₂×#103	C₂²×C₆₇	1	2	yes	—
184	C₂×#103	C₂²×C₆₇	1	2	yes	—
185	C₂×#103	C₂²×C₆₇	1	2	yes	—
186	#391	C₂²×C₆₇	1	2	yes	—
187	#391	C₂²×C₆₇	1	2	yes	—
188	#391	C₂²×C₆₇	1	2	yes	—
189	#389	C₂²×C₆₇	1	2	yes	—
190	#389	C₂²×C₆₇	1	2	yes	—
191	#389	C₂²×C₆₇	1	2	yes	—
192	#387	—	2	2	no	—
193	#387	—	2	2	no	—
194	#387	—	2	2	no	—
195	#383	—	2	2	no	—
196	C₄³	C₂²×C₆₇	1	1	yes	characteristic
197	#340	C₂²×C₆₇	1	2	yes	—
198	#340	C₂²×C₆₇	1	1	yes	characteristic
199	#340	C₂²×C₆₇	1	1	yes	characteristic
200	#340	C₂²×C₆₇	1	2	yes	—
201	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
202	C₂×Q₁₆	—	4	2	no	—
203	C₂×Q₁₆	—	4	1	no	—
204	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
206	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
213	C₂×C₄²	D₄×C₆₇	1	2	yes	—
214	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
215	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
218	#107	—	4	1	no	—
219	#106	—	4	2	no	—
220	C₄⋊C₈	—	4	1	no	—
221	C₄⋊C₈	—	4	2	no	—
222	C₄⋊C₈	—	4	2	no	—
223	C₄⋊C₈	—	4	1	no	—
224	#103	—	4	2	no	—
225	#103	—	4	2	no	—
226	#103	—	4	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
250	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
251	Q₁₆	—	4	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	1	no	—
275	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	#135×C₆₇	1	2	yes	—
302	C₂×C₄	#137×C₆₇	1	4	yes	—
303	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
304	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
305	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
306	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
307	C₈	—	4	1	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	1	no	—
310	C₂²	#537×C₆₇	1	2	yes	—
311	C₂²	#537×C₆₇	1	1	yes	characteristic
312	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
313	C₂²	C₂×#137×C₆₇	1	2	yes	—
314	C₂²	#575×C₆₇	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	#560×C₆₇	1	1	yes	characteristic
324	C₄	#563×C₆₇	1	2	yes	—
325	C₄	#563×C₆₇	1	1	yes	characteristic
326	C₂	C₆₇×#2823	1	1	yes	characteristic
327	C₂	C₆₇×#3248	1	2	yes	—
328	C₂	C₆₇×#3250	1	1	yes	characteristic
329	C₂	C₆₇×#3252	1	2	yes	—
330	C₂	C₆₇×#3255	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
160. #32863	C₆₇	direct (N × H)
175. C₆₇	#32863	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×#2863	2
C₆₇×#2088	2
C₆₇×#2086	2
C₆₇×#2068	2
C₆₇×#2066	2
C₆₇×#2056	2
C₆₇×#1842	2
#32863	67

Subgroups — C67×#32863 #30712114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32863 #30712114