Subgroups — C₆₇×#32864 #30712115

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2644	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2088	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2086	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2067	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2065	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2065	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2054	C₂	1	2	yes	maximal
8	C₆₇×#2054	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×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×#102×C₆₇	C₂²	1	2	yes	—
18	C₂×#102×C₆₇	C₂²	1	2	yes	—
19	C₂×#102×C₆₇	C₂²	1	2	yes	—
20	#397×C₆₇	—	2	2	no	—
21	#397×C₆₇	—	2	2	no	—
22	#397×C₆₇	—	2	2	no	—
23	#396×C₆₇	—	2	2	no	—
24	#390×C₆₇	C₂²	1	2	yes	—
25	#390×C₆₇	C₂²	1	2	yes	—
26	#390×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×D₄×C₆₇	C₂³	1	2	yes	—
36	C₂×D₈×C₆₇	—	4	2	no	—
37	C₂×D₈×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×C₂²⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
55	#107×C₆₇	—	4	1	no	—
56	#106×C₆₇	—	4	2	no	—
57	C₄⋊C₈×C₆₇	—	4	1	no	—
58	C₄⋊C₈×C₆₇	—	4	2	no	—
59	C₄⋊C₈×C₆₇	—	4	2	no	—
60	C₄⋊C₈×C₆₇	—	4	1	no	—
61	#102×C₆₇	—	4	2	no	—
62	#102×C₆₇	—	4	2	no	—
63	#102×C₆₇	—	4	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	2	yes	—
68	#95×C₆₇	D₄	1	1	yes	characteristic
69	#95×C₆₇	D₄	1	1	yes	characteristic
70	C₂⁴×C₆₇	—	2	2	no	—
71	C₂×D₄×C₆₇	—	2	2	no	—
72	C₂×D₄×C₆₇	—	2	2	no	—
73	C₂×D₄×C₆₇	—	2	2	no	—
74	C₂×D₄×C₆₇	—	2	2	no	—
75	C₂×D₄×C₆₇	—	2	2	no	—
76	C₂×D₄×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	D₈×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	C₂²⋊C₄×C₆₇	—	8	2	no	—
103	C₂²⋊C₄×C₆₇	—	8	2	no	—
104	C₂²⋊C₄×C₆₇	—	8	2	no	—
105	C₂²⋊C₄×C₆₇	—	8	2	no	—
106	C₂²⋊C₄×C₆₇	—	8	2	no	—
107	C₂²⋊C₄×C₆₇	—	8	2	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	1	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	1	no	—
112	C₂³×C₆₇	—	4	2	no	—
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	—	4	2	no	—
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	—	4	2	no	—
117	C₂³×C₆₇	—	4	2	no	—
118	C₂³×C₆₇	—	4	2	no	—
119	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
120	D₄×C₆₇	—	4	2	no	—
121	D₄×C₆₇	—	4	2	no	—
122	D₄×C₆₇	—	4	2	no	—
123	D₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
145	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
147	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₆₇	—	4	2	no	—
149	C₈×C₆₇	—	4	1	no	—
150	C₈×C₆₇	—	4	1	no	—
151	C₂²×C₆₇	—	8	2	no	—
152	C₂²×C₆₇	—	8	2	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	—	8	2	no	—
156	C₂²×C₆₇	—	8	2	no	—
157	C₂²×C₆₇	—	8	2	no	—
158	C₂²×C₆₇	#537	1	2	yes	—
159	C₂²×C₆₇	#537	1	1	yes	characteristic
160	C₂²×C₆₇	#576	1	1	yes	characteristic
161	C₂²×C₆₇	#576	1	1	yes	characteristic
162	C₂²×C₆₇	#577	1	2	yes	—
163	C₄×C₆₇	—	4	2	no	—
164	C₄×C₆₇	—	4	2	no	—
165	C₄×C₆₇	—	4	1	no	—
166	C₄×C₆₇	—	4	1	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	—	8	2	no	—
170	C₄×C₆₇	#560	1	1	yes	characteristic
171	C₄×C₆₇	#563	1	2	yes	—
172	C₄×C₆₇	#563	1	1	yes	characteristic
173	#32864	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂×C₆₇	—	16	2	no	—
175	C₂×C₆₇	#2823	1	1	yes	characteristic
176	C₂×C₆₇	#3257	1	2	yes	—
177	C₂×C₆₇	#3257	1	2	yes	—
178	C₂×C₆₇	#3259	1	1	yes	characteristic
179	C₂×C₆₇	#3263	1	1	yes	characteristic
180	#2644	C₂×C₆₇	1	2	yes	complemented
181	#2088	C₂×C₆₇	1	2	yes	complemented
182	#2086	C₂×C₆₇	1	1	yes	characteristic, complemented
183	#2067	C₂×C₆₇	1	2	yes	complemented
184	#2065	C₂×C₆₇	1	2	yes	complemented
185	#2065	C₂×C₆₇	1	2	yes	complemented
186	#2054	C₂×C₆₇	1	2	yes	—
187	#2054	C₂×C₆₇	1	1	yes	characteristic
188	#1843	C₂×C₆₇	1	1	yes	characteristic, complemented
189	C₆₇	#32864	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
191	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
192	C₂×#106	C₂²×C₆₇	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
195	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
197	C₂×#102	C₂²×C₆₇	1	2	yes	—
198	C₂×#102	C₂²×C₆₇	1	2	yes	—
199	C₂×#102	C₂²×C₆₇	1	2	yes	—
200	#397	—	2	2	no	—
201	#397	—	2	2	no	—
202	#397	—	2	2	no	—
203	#396	—	2	2	no	—
204	#390	C₂²×C₆₇	1	2	yes	—
205	#390	C₂²×C₆₇	1	2	yes	—
206	#390	C₂²×C₆₇	1	2	yes	—
207	#389	C₂²×C₆₇	1	2	yes	—
208	#389	C₂²×C₆₇	1	2	yes	—
209	#389	C₂²×C₆₇	1	2	yes	—
210	#376	C₂²×C₆₇	1	1	yes	characteristic
211	#340	C₂²×C₆₇	1	2	yes	—
212	#340	C₂²×C₆₇	1	1	yes	characteristic
213	#340	C₂²×C₆₇	1	1	yes	characteristic
214	#340	C₂²×C₆₇	1	2	yes	—
215	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
216	C₂×D₈	—	4	2	no	—
217	C₂×D₈	—	4	1	no	—
218	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
220	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
235	#107	—	4	1	no	—
236	#106	—	4	2	no	—
237	C₄⋊C₈	—	4	1	no	—
238	C₄⋊C₈	—	4	2	no	—
239	C₄⋊C₈	—	4	2	no	—
240	C₄⋊C₈	—	4	1	no	—
241	#102	—	4	2	no	—
242	#102	—	4	2	no	—
243	#102	—	4	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	D₄×C₆₇	1	2	yes	—
248	#95	D₄×C₆₇	1	1	yes	characteristic
249	#95	D₄×C₆₇	1	1	yes	characteristic
250	C₂⁴	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
265	D₈	—	4	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄⋊C₄	—	8	2	no	—
281	C₄⋊C₄	—	8	2	no	—
282	C₂²⋊C₄	—	8	2	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₂²⋊C₄	—	8	2	no	—
286	C₂²⋊C₄	—	8	2	no	—
287	C₂²⋊C₄	—	8	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
325	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
326	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
327	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
328	C₈	—	4	1	no	—
329	C₈	—	4	2	no	—
330	C₈	—	4	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	#537×C₆₇	1	2	yes	—
339	C₂²	#537×C₆₇	1	1	yes	characteristic
340	C₂²	#576×C₆₇	1	1	yes	characteristic
341	C₂²	#576×C₆₇	1	1	yes	characteristic
342	C₂²	#577×C₆₇	1	2	yes	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	#560×C₆₇	1	1	yes	characteristic
351	C₄	#563×C₆₇	1	2	yes	—
352	C₄	#563×C₆₇	1	1	yes	characteristic
353	C₂	—	16	2	no	—
354	C₂	C₆₇×#2823	1	1	yes	characteristic
355	C₂	C₆₇×#3257	1	2	yes	—
356	C₂	C₆₇×#3257	1	2	yes	—
357	C₂	C₆₇×#3259	1	1	yes	characteristic
358	C₂	C₆₇×#3263	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
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
1. C₆₇×#2644	C₂	semidirect
2. C₆₇×#2088	C₂	semidirect
3. C₆₇×#2086	C₂	semidirect
4. C₆₇×#2067	C₂	semidirect
5. C₆₇×#2065	C₂	semidirect
6. C₆₇×#2065	C₂	semidirect
9. C₆₇×#1843	C₂	semidirect
173. #32864	C₆₇	direct (N × H)
180. #2644	C₂×C₆₇	semidirect
181. #2088	C₂×C₆₇	semidirect
182. #2086	C₂×C₆₇	semidirect
183. #2067	C₂×C₆₇	semidirect
184. #2065	C₂×C₆₇	semidirect
185. #2065	C₂×C₆₇	semidirect
188. #1843	C₂×C₆₇	semidirect
189. C₆₇	#32864	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×#2644	2
C₆₇×#2088	2
C₆₇×#2086	2
C₆₇×#2067	2
C₆₇×#2065	2
C₆₇×#2054	2
C₆₇×#1843	2
#32864	67

Subgroups — C67×#32864 #30712115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32864 #30712115