Subgroups — C₆₇×#23828 #30703079

Order 17152.

802 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2331	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2328	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2101	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2073	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2041	C₂	1	2	yes	maximal
6	C₆₇×#2023	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1762	C₂	1	2	yes	maximal, complemented
8	C₄×#102×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂×#102×C₆₇	C₂²	1	2	yes	—
14	C₂×#102×C₆₇	C₂²	1	2	yes	—
15	C₂×#102×C₆₇	C₂²	1	2	yes	—
16	C₂×#102×C₆₇	C₂²	1	2	yes	—
17	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#391×C₆₇	—	2	2	no	—
20	#390×C₆₇	C₂²	1	2	yes	—
21	#389×C₆₇	C₂²	1	2	yes	—
22	#387×C₆₇	—	2	2	no	—
23	#386×C₆₇	—	2	2	no	—
24	#386×C₆₇	—	2	2	no	—
25	#382×C₆₇	C₂²	1	1	yes	characteristic
26	#382×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	#376×C₆₇	C₂²	1	1	yes	characteristic
29	#340×C₆₇	C₂²	1	2	yes	—
30	#340×C₆₇	C₂²	1	2	yes	—
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×Q₈×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	—	2	2	no	—
37	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₆₇	—	2	4	no	—
42	#127×C₆₇	—	2	4	no	—
43	C₄×D₄×C₆₇	—	2	4	no	—
44	C₄×D₄×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	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×C₄²×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	#102×C₆₇	—	2	4	no	—
58	#102×C₆₇	—	2	4	no	—
59	#102×C₆₇	—	2	4	no	—
60	#102×C₆₇	—	2	4	no	—
61	#97×C₆₇	—	2	2	no	—
62	C₄×C₈×C₆₇	—	2	2	no	—
63	#95×C₆₇	—	2	2	no	—
64	#95×C₆₇	—	2	2	no	—
65	#95×C₆₇	—	2	1	no	—
66	#95×C₆₇	—	2	1	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×Q₈×C₆₇	—	2	4	no	—
72	C₂×Q₈×C₆₇	—	8	2	no	—
73	C₂×D₄×C₆₇	—	2	8	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	—	2	2	no	—
78	C₂×D₄×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	—	8	2	no	—
80	C₂×D₄×C₆₇	—	8	2	no	—
81	C₂×D₄×C₆₇	#41	1	4	yes	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	4	4	no	—
86	C₂²×C₄×C₆₇	—	8	2	no	—
87	C₂²×C₄×C₆₇	—	8	2	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	8	2	no	—
106	C₄⋊C₄×C₆₇	#41	1	4	yes	—
107	C₂²⋊C₄×C₆₇	—	4	4	no	—
108	C₂²⋊C₄×C₆₇	—	8	2	no	—
109	C₂²⋊C₄×C₆₇	—	8	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	8	2	no	—
113	C₄²×C₆₇	#41	1	2	yes	—
114	C₄²×C₆₇	#41	1	2	yes	—
115	C₂³×C₆₇	—	2	4	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	—	4	2	no	—
122	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₆₇	—	8	4	no	—
124	D₄×C₆₇	—	2	4	no	—
125	D₄×C₆₇	—	2	4	no	—
126	D₄×C₆₇	—	4	2	no	—
127	D₄×C₆₇	—	4	2	no	—
128	D₄×C₆₇	—	8	8	no	—
129	C₂×C₄×C₆₇	—	2	2	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	1	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	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	—	4	2	no	—
147	C₂×C₄×C₆₇	—	8	8	no	—
148	C₂×C₄×C₆₇	#124	1	4	yes	—
149	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₇	—	4	2	no	—
154	C₈×C₆₇	—	4	2	no	—
155	C₂²×C₆₇	—	4	4	no	—
156	C₂²×C₆₇	—	4	4	no	—
157	C₂²×C₆₇	—	4	4	no	—
158	C₂²×C₆₇	—	8	2	no	—
159	C₂²×C₆₇	—	8	2	no	—
160	C₂²×C₆₇	—	8	2	no	—
161	C₂²×C₆₇	—	8	2	no	—
162	C₂²×C₆₇	#525	1	1	yes	characteristic
163	C₂²×C₆₇	#525	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
165	C₂²×C₆₇	#576	1	2	yes	—
166	C₂²×C₆₇	#577	1	2	yes	—
167	C₄×C₆₇	—	2	4	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	—	4	1	no	—
170	C₄×C₆₇	—	4	4	no	—
171	C₄×C₆₇	—	4	1	no	—
172	C₄×C₆₇	—	8	2	no	—
173	C₄×C₆₇	—	8	2	no	—
174	C₄×C₆₇	#529	1	2	yes	—
175	C₄×C₆₇	#529	1	2	yes	—
176	#23828	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂×C₆₇	—	8	4	no	—
178	C₂×C₆₇	#2398	1	1	yes	characteristic
179	C₂×C₆₇	#3118	1	2	yes	—
180	C₂×C₆₇	#3122	1	1	yes	characteristic
181	C₂×C₆₇	#3126	1	1	yes	characteristic
182	C₂×C₆₇	#3136	1	2	yes	—
183	#2331	C₂×C₆₇	1	1	yes	characteristic, complemented
184	#2328	C₂×C₆₇	1	1	yes	characteristic
185	#2101	C₂×C₆₇	1	2	yes	complemented
186	#2073	C₂×C₆₇	1	2	yes	complemented
187	#2041	C₂×C₆₇	1	2	yes	—
188	#2023	C₂×C₆₇	1	2	yes	complemented
189	#1762	C₂×C₆₇	1	2	yes	complemented
190	C₄×#102	C₂×C₆₇	1	2	yes	complemented
191	#1746	C₂×C₆₇	1	1	yes	characteristic, complemented
192	C₆₇	#23828	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
195	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
196	C₂×#102	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	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
202	#391	—	2	2	no	—
203	#390	C₂²×C₆₇	1	2	yes	—
204	#389	C₂²×C₆₇	1	2	yes	—
205	#387	—	2	2	no	—
206	#386	—	2	2	no	—
207	#386	—	2	2	no	—
208	#382	C₂²×C₆₇	1	1	yes	characteristic
209	#382	C₂²×C₆₇	1	1	yes	characteristic
210	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
211	#376	C₂²×C₆₇	1	1	yes	characteristic
212	#340	C₂²×C₆₇	1	2	yes	—
213	#340	C₂²×C₆₇	1	2	yes	—
214	#340	C₂²×C₆₇	1	2	yes	—
215	#340	C₂²×C₆₇	1	2	yes	—
216	#336	C₂²×C₆₇	1	2	yes	—
217	#336	C₂²×C₆₇	1	2	yes	—
218	C₂²×Q₈	—	2	1	no	—
219	C₂²×D₄	—	2	2	no	—
220	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
221	C₂³×C₄	—	2	2	no	—
222	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
223	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
224	C₄⋊Q₈	—	2	4	no	—
225	#127	—	2	4	no	—
226	C₄×D₄	—	2	4	no	—
227	C₄×D₄	—	2	4	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
232	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₄²	—	2	2	no	—
237	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
240	#102	—	2	4	no	—
241	#102	—	2	4	no	—
242	#102	—	2	4	no	—
243	#102	—	2	4	no	—
244	#97	—	2	2	no	—
245	C₄×C₈	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	—	2	1	no	—
249	#95	—	2	1	no	—
250	#95	—	2	2	no	—
251	#95	D₄×C₆₇	1	2	yes	—
252	#95	D₄×C₆₇	1	2	yes	—
253	C₂⁴	—	2	2	no	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	—	8	2	no	—
256	C₂×D₄	—	2	8	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	—	2	2	no	—
262	C₂×D₄	—	8	2	no	—
263	C₂×D₄	—	8	2	no	—
264	C₂×D₄	#41×C₆₇	1	4	yes	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	4	4	no	—
269	C₂²×C₄	—	8	2	no	—
270	C₂²×C₄	—	8	2	no	—
271	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
272	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
275	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	8	2	no	—
289	C₄⋊C₄	#41×C₆₇	1	4	yes	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₂²⋊C₄	—	8	2	no	—
292	C₂²⋊C₄	—	8	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	8	2	no	—
296	C₄²	#41×C₆₇	1	2	yes	—
297	C₄²	#41×C₆₇	1	2	yes	—
298	C₂³	—	2	4	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	Q₈	—	8	4	no	—
307	D₄	—	2	4	no	—
308	D₄	—	2	4	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	8	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	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	—	8	8	no	—
331	C₂×C₄	#124×C₆₇	1	4	yes	—
332	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
333	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
334	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	#525×C₆₇	1	1	yes	characteristic
346	C₂²	#525×C₆₇	1	1	yes	characteristic
347	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
348	C₂²	#576×C₆₇	1	2	yes	—
349	C₂²	#577×C₆₇	1	2	yes	—
350	C₄	—	2	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	4	no	—
354	C₄	—	4	1	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	#529×C₆₇	1	2	yes	—
358	C₄	#529×C₆₇	1	2	yes	—
359	C₂	—	8	4	no	—
360	C₂	C₆₇×#2398	1	1	yes	characteristic
361	C₂	C₆₇×#3118	1	2	yes	—
362	C₂	C₆₇×#3122	1	1	yes	characteristic
363	C₂	C₆₇×#3126	1	1	yes	characteristic
364	C₂	C₆₇×#3136	1	2	yes	—

	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×#2331	C₂	semidirect
3. C₆₇×#2101	C₂	semidirect
4. C₆₇×#2073	C₂	semidirect
6. C₆₇×#2023	C₂	semidirect
7. C₆₇×#1762	C₂	semidirect
8. C₄×#102×C₆₇	C₂	semidirect
9. C₆₇×#1746	C₂	semidirect
176. #23828	C₆₇	direct (N × H)
183. #2331	C₂×C₆₇	semidirect
185. #2101	C₂×C₆₇	semidirect
186. #2073	C₂×C₆₇	semidirect
188. #2023	C₂×C₆₇	semidirect
189. #1762	C₂×C₆₇	semidirect
190. C₄×#102	C₂×C₆₇	semidirect
191. #1746	C₂×C₆₇	semidirect
192. C₆₇	#23828	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×#2331	2
C₆₇×#2328	2
C₆₇×#2101	2
C₆₇×#2073	2
C₆₇×#2041	2
C₆₇×#2023	2
C₆₇×#1762	2
C₄×#102×C₆₇	2
C₆₇×#1746	2
#23828	67

Subgroups — C67×#23828 #30703079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#23828 #30703079