Subgroups — C₆₇×#32866 #30712117

Order 17152.

562 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×#2645	C₂	1	2	yes	maximal
2	C₆₇×#2088	C₂	1	2	yes	maximal
3	C₆₇×#2087	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×#1843	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	#401×C₆₇	—	2	2	no	—
21	#400×C₆₇	—	2	2	no	—
22	#400×C₆₇	—	2	2	no	—
23	#400×C₆₇	—	2	2	no	—
24	#391×C₆₇	C₂²	1	2	yes	—
25	#391×C₆₇	C₂²	1	2	yes	—
26	#391×C₆₇	C₂²	1	2	yes	—
27	#389×C₆₇	C₂²	1	2	yes	—
28	#389×C₆₇	C₂²	1	2	yes	—
29	#389×C₆₇	C₂²	1	2	yes	—
30	#376×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	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
49	#107×C₆₇	—	4	1	no	—
50	#106×C₆₇	—	4	2	no	—
51	C₄⋊C₈×C₆₇	—	4	1	no	—
52	C₄⋊C₈×C₆₇	—	4	2	no	—
53	C₄⋊C₈×C₆₇	—	4	2	no	—
54	C₄⋊C₈×C₆₇	—	4	1	no	—
55	#103×C₆₇	—	4	2	no	—
56	#103×C₆₇	—	4	2	no	—
57	#103×C₆₇	—	4	2	no	—
58	#95×C₆₇	—	2	2	no	—
59	#95×C₆₇	—	2	2	no	—
60	#95×C₆₇	—	2	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	D₄	1	1	yes	characteristic
68	#95×C₆₇	D₄	1	1	yes	characteristic
69	#95×C₆₇	D₄	1	2	yes	—
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	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
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	—	8	2	no	—
100	C₄⋊C₄×C₆₇	—	8	2	no	—
101	C₄⋊C₄×C₆₇	—	8	2	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	1	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	—	2	1	no	—
106	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₆₇	—	4	2	no	—
108	Q₈×C₆₇	—	4	2	no	—
109	Q₈×C₆₇	—	4	2	no	—
110	Q₈×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	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	—	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	H(5,𝔽₂)	1	2	yes	—
135	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
137	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
138	C₈×C₆₇	—	4	2	no	—
139	C₈×C₆₇	—	4	1	no	—
140	C₈×C₆₇	—	4	1	no	—
141	C₂²×C₆₇	#537	1	2	yes	—
142	C₂²×C₆₇	#537	1	1	yes	characteristic
143	C₂²×C₆₇	#577	1	2	yes	—
144	C₂²×C₆₇	#578	1	1	yes	characteristic
145	C₂²×C₆₇	#578	1	1	yes	characteristic
146	C₄×C₆₇	—	4	2	no	—
147	C₄×C₆₇	—	4	2	no	—
148	C₄×C₆₇	—	4	1	no	—
149	C₄×C₆₇	—	4	1	no	—
150	C₄×C₆₇	—	8	2	no	—
151	C₄×C₆₇	—	8	2	no	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	#560	1	1	yes	characteristic
155	C₄×C₆₇	#563	1	2	yes	—
156	C₄×C₆₇	#563	1	1	yes	characteristic
157	#32866	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₆₇	#2823	1	1	yes	characteristic
159	C₂×C₆₇	#3258	1	2	yes	—
160	C₂×C₆₇	#3258	1	2	yes	—
161	C₂×C₆₇	#3261	1	1	yes	characteristic
162	C₂×C₆₇	#3263	1	1	yes	characteristic
163	#2645	C₂×C₆₇	1	2	yes	—
164	#2088	C₂×C₆₇	1	2	yes	—
165	#2087	C₂×C₆₇	1	1	yes	characteristic
166	#2068	C₂×C₆₇	1	2	yes	—
167	#2066	C₂×C₆₇	1	2	yes	—
168	#2066	C₂×C₆₇	1	2	yes	—
169	#2056	C₂×C₆₇	1	2	yes	—
170	#2056	C₂×C₆₇	1	1	yes	characteristic
171	#1843	C₂×C₆₇	1	1	yes	characteristic
172	C₆₇	#32866	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
174	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×#106	C₂²×C₆₇	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
178	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
180	C₂×#103	C₂²×C₆₇	1	2	yes	—
181	C₂×#103	C₂²×C₆₇	1	2	yes	—
182	C₂×#103	C₂²×C₆₇	1	2	yes	—
183	#401	—	2	2	no	—
184	#400	—	2	2	no	—
185	#400	—	2	2	no	—
186	#400	—	2	2	no	—
187	#391	C₂²×C₆₇	1	2	yes	—
188	#391	C₂²×C₆₇	1	2	yes	—
189	#391	C₂²×C₆₇	1	2	yes	—
190	#389	C₂²×C₆₇	1	2	yes	—
191	#389	C₂²×C₆₇	1	2	yes	—
192	#389	C₂²×C₆₇	1	2	yes	—
193	#376	C₂²×C₆₇	1	1	yes	characteristic
194	#340	C₂²×C₆₇	1	2	yes	—
195	#340	C₂²×C₆₇	1	1	yes	characteristic
196	#340	C₂²×C₆₇	1	1	yes	characteristic
197	#340	C₂²×C₆₇	1	2	yes	—
198	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
199	C₂×Q₁₆	—	4	2	no	—
200	C₂×Q₁₆	—	4	1	no	—
201	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
203	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
210	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
212	#107	—	4	1	no	—
213	#106	—	4	2	no	—
214	C₄⋊C₈	—	4	1	no	—
215	C₄⋊C₈	—	4	2	no	—
216	C₄⋊C₈	—	4	2	no	—
217	C₄⋊C₈	—	4	1	no	—
218	#103	—	4	2	no	—
219	#103	—	4	2	no	—
220	#103	—	4	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	D₄×C₆₇	1	1	yes	characteristic
231	#95	D₄×C₆₇	1	1	yes	characteristic
232	#95	D₄×C₆₇	1	2	yes	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
248	Q₁₆	—	4	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	8	2	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₄⋊C₄	—	8	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	1	no	—
269	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	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	—	4	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	H(5,𝔽₂)×C₆₇	1	2	yes	—
298	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
299	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
300	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
301	C₈	—	4	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	4	1	no	—
304	C₂²	#537×C₆₇	1	2	yes	—
305	C₂²	#537×C₆₇	1	1	yes	characteristic
306	C₂²	#577×C₆₇	1	2	yes	—
307	C₂²	#578×C₆₇	1	1	yes	characteristic
308	C₂²	#578×C₆₇	1	1	yes	characteristic
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	#560×C₆₇	1	1	yes	characteristic
318	C₄	#563×C₆₇	1	2	yes	—
319	C₄	#563×C₆₇	1	1	yes	characteristic
320	C₂	C₆₇×#2823	1	1	yes	characteristic
321	C₂	C₆₇×#3258	1	2	yes	—
322	C₂	C₆₇×#3258	1	2	yes	—
323	C₂	C₆₇×#3261	1	1	yes	characteristic
324	C₂	C₆₇×#3263	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 ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
157. #32866	C₆₇	direct (N × H)
172. C₆₇	#32866	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×#2645	2
C₆₇×#2088	2
C₆₇×#2087	2
C₆₇×#2068	2
C₆₇×#2066	2
C₆₇×#2056	2
C₆₇×#1843	2
#32866	67

Subgroups — C67×#32866 #30712117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32866 #30712117