Subgroups — C₆₇×#32827 #30712078

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2641	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2092	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2071	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2067	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2065	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2059	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2041	C₂	1	2	yes	maximal
9	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#107×C₆₇	C₂²	1	2	yes	—
13	C₂×#106×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#102×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	#400×C₆₇	—	2	2	no	—
21	#398×C₆₇	—	2	2	no	—
22	#394×C₆₇	—	2	2	no	—
23	#394×C₆₇	—	2	2	no	—
24	#390×C₆₇	C₂²	1	2	yes	—
25	#390×C₆₇	C₂²	1	2	yes	—
26	#389×C₆₇	C₂²	1	2	yes	—
27	#389×C₆₇	C₂²	1	2	yes	—
28	#384×C₆₇	C₂²	1	1	yes	characteristic
29	#382×C₆₇	C₂²	1	1	yes	characteristic
30	#376×C₆₇	C₂²	1	1	yes	characteristic
31	#340×C₆₇	C₂²	1	2	yes	—
32	#340×C₆₇	C₂²	1	2	yes	—
33	#336×C₆₇	C₂²	1	2	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	C₂²×D₄×C₆₇	—	2	2	no	—
36	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
39	#127×C₆₇	—	4	1	no	—
40	#127×C₆₇	—	4	1	no	—
41	#125×C₆₇	—	4	1	no	—
42	#125×C₆₇	—	4	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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×C₂²⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#107×C₆₇	—	4	2	no	—
58	#106×C₆₇	—	4	2	no	—
59	C₄⋊C₈×C₆₇	—	4	1	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	#102×C₆₇	—	4	2	no	—
65	#95×C₆₇	—	2	2	no	—
66	#95×C₆₇	—	2	2	no	—
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	D₄	1	2	yes	—
69	#95×C₆₇	D₄	1	2	yes	—
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	C₂×D₄×C₆₇	—	4	4	no	—
78	C₂×D₄×C₆₇	—	8	2	no	—
79	C₂×D₄×C₆₇	—	8	2	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×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	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	4	4	no	—
101	C₄⋊C₄×C₆₇	—	8	2	no	—
102	C₄⋊C₄×C₆₇	—	8	2	no	—
103	C₄⋊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	1	no	—
109	C₄²×C₆₇	—	2	1	no	—
110	C₄²×C₆₇	—	2	1	no	—
111	C₄²×C₆₇	—	2	1	no	—
112	C₄²×C₆₇	—	2	1	no	—
113	C₄²×C₆₇	—	2	1	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	—	4	2	no	—
120	C₂³×C₆₇	—	4	2	no	—
121	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	D₄×C₆₇	—	4	2	no	—
123	D₄×C₆₇	—	4	2	no	—
124	D₄×C₆₇	—	4	2	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	8	8	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	2	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	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	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	—	4	2	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
152	C₈×C₆₇	—	4	2	no	—
153	C₈×C₆₇	—	4	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	—	8	2	no	—
159	C₂²×C₆₇	—	8	2	no	—
160	C₂²×C₆₇	—	8	2	no	—
161	C₂²×C₆₇	#535	1	1	yes	characteristic
162	C₂²×C₆₇	#536	1	1	yes	characteristic
163	C₂²×C₆₇	#537	1	1	yes	characteristic
164	C₂²×C₆₇	#576	1	2	yes	—
165	C₂²×C₆₇	#577	1	2	yes	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	4	2	no	—
168	C₄×C₆₇	—	4	1	no	—
169	C₄×C₆₇	—	4	1	no	—
170	C₄×C₆₇	—	8	2	no	—
171	C₄×C₆₇	—	8	2	no	—
172	C₄×C₆₇	—	8	2	no	—
173	C₄×C₆₇	#562	1	1	yes	characteristic
174	C₄×C₆₇	#562	1	1	yes	characteristic
175	C₄×C₆₇	#563	1	1	yes	characteristic
176	C₄×C₆₇	#563	1	1	yes	characteristic
177	#32827	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₆₇	—	16	2	no	—
179	C₂×C₆₇	#2820	1	1	yes	characteristic
180	C₂×C₆₇	#3224	1	2	yes	—
181	C₂×C₆₇	#3241	1	2	yes	—
182	C₂×C₆₇	#3259	1	1	yes	characteristic
183	C₂×C₆₇	#3260	1	1	yes	characteristic
184	#2647	C₂×C₆₇	1	1	yes	characteristic, complemented
185	#2641	C₂×C₆₇	1	1	yes	characteristic
186	#2092	C₂×C₆₇	1	2	yes	complemented
187	#2071	C₂×C₆₇	1	2	yes	complemented
188	#2067	C₂×C₆₇	1	2	yes	complemented
189	#2065	C₂×C₆₇	1	2	yes	complemented
190	#2059	C₂×C₆₇	1	2	yes	complemented
191	#2041	C₂×C₆₇	1	2	yes	—
192	#1832	C₂×C₆₇	1	1	yes	characteristic, complemented
193	C₆₇	#32827	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
195	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×#107	C₂²×C₆₇	1	2	yes	—
197	C₂×#106	C₂²×C₆₇	1	2	yes	—
198	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
199	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
200	C₂×#102	C₂²×C₆₇	1	2	yes	—
201	C₂×#102	C₂²×C₆₇	1	2	yes	—
202	C₂×#102	C₂²×C₆₇	1	2	yes	—
203	C₂×#102	C₂²×C₆₇	1	2	yes	—
204	#400	—	2	2	no	—
205	#398	—	2	2	no	—
206	#394	—	2	2	no	—
207	#394	—	2	2	no	—
208	#390	C₂²×C₆₇	1	2	yes	—
209	#390	C₂²×C₆₇	1	2	yes	—
210	#389	C₂²×C₆₇	1	2	yes	—
211	#389	C₂²×C₆₇	1	2	yes	—
212	#384	C₂²×C₆₇	1	1	yes	characteristic
213	#382	C₂²×C₆₇	1	1	yes	characteristic
214	#376	C₂²×C₆₇	1	1	yes	characteristic
215	#340	C₂²×C₆₇	1	2	yes	—
216	#340	C₂²×C₆₇	1	2	yes	—
217	#336	C₂²×C₆₇	1	2	yes	—
218	#336	C₂²×C₆₇	1	2	yes	—
219	C₂²×D₄	—	2	2	no	—
220	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
221	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
222	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
223	#127	—	4	1	no	—
224	#127	—	4	1	no	—
225	#125	—	4	1	no	—
226	#125	—	4	1	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
232	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
233	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
241	#107	—	4	2	no	—
242	#106	—	4	2	no	—
243	C₄⋊C₈	—	4	1	no	—
244	C₄⋊C₈	—	4	1	no	—
245	#102	—	4	2	no	—
246	#102	—	4	2	no	—
247	#102	—	4	2	no	—
248	#102	—	4	2	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	2	no	—
252	#95	D₄×C₆₇	1	2	yes	—
253	#95	D₄×C₆₇	1	2	yes	—
254	C₂⁴	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	4	no	—
262	C₂×D₄	—	8	2	no	—
263	C₂×D₄	—	8	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
270	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	—	8	2	no	—
287	C₄⋊C₄	—	8	2	no	—
288	C₂²⋊C₄	—	8	2	no	—
289	C₂²⋊C₄	—	8	2	no	—
290	C₂²⋊C₄	—	8	2	no	—
291	C₂²⋊C₄	—	8	2	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	4	2	no	—
305	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	8	8	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
331	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
332	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
333	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
334	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
335	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	#535×C₆₇	1	1	yes	characteristic
346	C₂²	#536×C₆₇	1	1	yes	characteristic
347	C₂²	#537×C₆₇	1	1	yes	characteristic
348	C₂²	#576×C₆₇	1	2	yes	—
349	C₂²	#577×C₆₇	1	2	yes	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	#562×C₆₇	1	1	yes	characteristic
358	C₄	#562×C₆₇	1	1	yes	characteristic
359	C₄	#563×C₆₇	1	1	yes	characteristic
360	C₄	#563×C₆₇	1	1	yes	characteristic
361	C₂	—	16	2	no	—
362	C₂	C₆₇×#2820	1	1	yes	characteristic
363	C₂	C₆₇×#3224	1	2	yes	—
364	C₂	C₆₇×#3241	1	2	yes	—
365	C₂	C₆₇×#3259	1	1	yes	characteristic
366	C₂	C₆₇×#3260	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×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×#2647	C₂	semidirect
3. C₆₇×#2092	C₂	semidirect
4. C₆₇×#2071	C₂	semidirect
5. C₆₇×#2067	C₂	semidirect
6. C₆₇×#2065	C₂	semidirect
7. C₆₇×#2059	C₂	semidirect
9. C₆₇×#1832	C₂	semidirect
177. #32827	C₆₇	direct (N × H)
184. #2647	C₂×C₆₇	semidirect
186. #2092	C₂×C₆₇	semidirect
187. #2071	C₂×C₆₇	semidirect
188. #2067	C₂×C₆₇	semidirect
189. #2065	C₂×C₆₇	semidirect
190. #2059	C₂×C₆₇	semidirect
192. #1832	C₂×C₆₇	semidirect
193. C₆₇	#32827	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×#2647	2
C₆₇×#2641	2
C₆₇×#2092	2
C₆₇×#2071	2
C₆₇×#2067	2
C₆₇×#2065	2
C₆₇×#2059	2
C₆₇×#2041	2
C₆₇×#1832	2
#32827	67

Subgroups — C67×#32827 #30712078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32827 #30712078