Subgroups — C₆₇×#33898 #30713149

Order 17152.

548 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3338	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3283	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2235	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2219	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2139	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₆₇	C₂²	1	2	yes	—
13	#554×C₆₇	—	2	1	no	—
14	#547×C₆₇	—	2	1	no	—
15	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	#500×C₆₇	C₂²	1	1	yes	characteristic
19	#494×C₆₇	C₂²	1	1	yes	characteristic
20	#484×C₆₇	—	2	2	no	—
21	#477×C₆₇	—	2	1	no	—
22	#467×C₆₇	C₂²	1	2	yes	—
23	#467×C₆₇	C₂²	1	2	yes	—
24	#461×C₆₇	—	2	1	no	—
25	#452×C₆₇	—	2	2	no	—
26	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#107×C₆₇	C₂²	1	2	yes	—
30	#367×C₆₇	C₂²	1	1	yes	characteristic
31	#367×C₆₇	C₂²	1	2	yes	—
32	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	#358×C₆₇	C₂²	1	2	yes	—
34	#358×C₆₇	C₂²	1	2	yes	—
35	#358×C₆₇	C₂²	1	2	yes	—
36	#358×C₆₇	C₂²	1	1	yes	characteristic
37	#357×C₆₇	C₂²	1	1	yes	characteristic
38	#357×C₆₇	C₂²	1	2	yes	—
39	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#41×C₆₇	—	2	2	no	—
42	#135×C₆₇	—	2	4	no	—
43	#135×C₆₇	—	4	2	no	—
44	C₂×Q₁₆×C₆₇	—	2	1	no	—
45	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂×SD₁₆×C₆₇	—	2	2	no	—
48	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	1	no	—
51	C₄⋊Q₈×C₆₇	—	2	1	no	—
52	#125×C₆₇	—	4	1	no	—
53	#123×C₆₇	—	4	2	no	—
54	C₂²⋊Q₈×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	—	2	2	no	—
56	C₂²⋊Q₈×C₆₇	—	4	2	no	—
57	#121×C₆₇	—	4	1	no	—
58	C₄×Q₈×C₆₇	—	2	1	no	—
59	C₄×D₄×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	—	4	1	no	—
61	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
64	Q₃₂×C₆₇	—	4	2	no	—
65	SD₃₂×C₆₇	—	2	2	no	—
66	M₃₂×C₆₇	—	4	2	no	—
67	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
70	#107×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
76	#103×C₆₇	—	2	2	no	—
77	#103×C₆₇	—	2	2	no	—
78	#103×C₆₇	—	2	1	no	—
79	#102×C₆₇	—	2	1	no	—
80	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	#41×C₆₇	—	4	4	no	—
83	#41×C₆₇	—	8	2	no	—
84	C₂×Q₈×C₆₇	—	2	1	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	—	2	1	no	—
88	C₂×D₄×C₆₇	—	4	2	no	—
89	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	—	4	1	no	—
92	C₂²×C₄×C₆₇	—	4	2	no	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
94	Q₁₆×C₆₇	—	2	1	no	—
95	Q₁₆×C₆₇	—	2	2	no	—
96	Q₁₆×C₆₇	—	2	1	no	—
97	Q₁₆×C₆₇	—	2	2	no	—
98	Q₁₆×C₆₇	—	8	1	no	—
99	SD₁₆×C₆₇	—	4	4	no	—
100	D₈×C₆₇	—	2	1	no	—
101	D₈×C₆₇	#41	1	2	yes	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₆₇	—	2	1	no	—
108	C₄⋊C₄×C₆₇	—	2	1	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₄⋊C₄×C₆₇	—	4	2	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
118	C₂²⋊C₄×C₆₇	—	4	2	no	—
119	C₂²⋊C₄×C₆₇	—	4	2	no	—
120	C₂²⋊C₄×C₆₇	—	4	1	no	—
121	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
122	C₄²×C₆₇	—	4	1	no	—
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₆₇	—	2	1	no	—
125	C₁₆×C₆₇	—	2	1	no	—
126	C₁₆×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	4	1	no	—
128	C₂³×C₆₇	C₂×D₈	1	2	yes	—
129	Q₈×C₆₇	—	4	2	no	—
130	Q₈×C₆₇	—	4	1	no	—
131	Q₈×C₆₇	—	4	2	no	—
132	Q₈×C₆₇	—	4	1	no	—
133	Q₈×C₆₇	—	16	1	no	—
134	D₄×C₆₇	—	2	2	no	—
135	D₄×C₆₇	—	2	2	no	—
136	D₄×C₆₇	—	4	1	no	—
137	D₄×C₆₇	—	8	4	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	—	4	1	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	8	4	no	—
146	C₂×C₄×C₆₇	—	8	2	no	—
147	C₂×C₄×C₆₇	—	16	2	no	—
148	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₆₇	—	2	2	no	—
153	C₈×C₆₇	—	2	1	no	—
154	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	4	2	no	—
160	C₂²×C₆₇	—	8	1	no	—
161	C₂²×C₆₇	—	8	1	no	—
162	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₆₇	—	2	1	no	—
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	8	1	no	—
166	C₄×C₆₇	—	8	1	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	—	8	1	no	—
170	C₄×C₆₇	#510	1	2	yes	—
171	C₄×C₆₇	#546	1	1	yes	characteristic
172	C₄×C₆₇	#577	1	1	yes	characteristic
173	#33898	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂×C₆₇	—	4	2	no	—
175	C₂×C₆₇	—	8	2	no	—
176	C₂×C₆₇	#3294	1	1	yes	characteristic
177	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
178	C₂×C₆₇	#3416	1	1	yes	characteristic
179	#3338	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#3283	C₂×C₆₇	1	1	yes	characteristic
181	#2241	C₂×C₆₇	1	1	yes	characteristic, complemented
182	#2235	C₂×C₆₇	1	2	yes	complemented
183	#2226	C₂×C₆₇	1	1	yes	characteristic, complemented
184	#2219	C₂×C₆₇	1	2	yes	complemented
185	#2209	C₂×C₆₇	1	1	yes	characteristic, complemented
186	#2189	C₂×C₆₇	1	2	yes	complemented
187	C₄×SD₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
188	#2169	C₂×C₆₇	1	1	yes	characteristic, complemented
189	#2139	C₂×C₆₇	1	2	yes	complemented
190	C₆₇	#33898	1	1	yes	characteristic, direct product factor, Sylow
191	C₂×#135	C₂²×C₆₇	1	2	yes	—
192	#554	—	2	1	no	—
193	#547	—	2	1	no	—
194	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
195	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
197	#500	C₂²×C₆₇	1	1	yes	characteristic
198	#494	C₂²×C₆₇	1	1	yes	characteristic
199	#484	—	2	2	no	—
200	#477	—	2	1	no	—
201	#467	C₂²×C₆₇	1	2	yes	—
202	#467	C₂²×C₆₇	1	2	yes	—
203	#461	—	2	1	no	—
204	#452	—	2	2	no	—
205	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
206	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
207	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×#107	C₂²×C₆₇	1	2	yes	—
209	#367	C₂²×C₆₇	1	1	yes	characteristic
210	#367	C₂²×C₆₇	1	2	yes	—
211	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
212	#358	C₂²×C₆₇	1	2	yes	—
213	#358	C₂²×C₆₇	1	2	yes	—
214	#358	C₂²×C₆₇	1	1	yes	characteristic
215	#358	C₂²×C₆₇	1	2	yes	—
216	#357	C₂²×C₆₇	1	1	yes	characteristic
217	#357	C₂²×C₆₇	1	2	yes	—
218	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
219	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
220	C₂×#41	—	2	2	no	—
221	#135	—	2	4	no	—
222	#135	—	4	2	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
225	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
228	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	#125	—	4	1	no	—
232	#123	—	4	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	C₂²⋊Q₈	—	4	2	no	—
236	#121	—	4	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	4	1	no	—
240	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	—	4	1	no	—
243	Q₃₂	—	4	2	no	—
244	SD₃₂	—	2	2	no	—
245	M₃₂	—	4	2	no	—
246	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
247	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
248	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
249	#107	—	2	2	no	—
250	#107	—	2	1	no	—
251	#107	C₂³×C₆₇	1	1	yes	characteristic
252	#107	C₂³×C₆₇	1	2	yes	—
253	#107	C₂³×C₆₇	1	1	yes	characteristic
254	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
255	#103	—	2	1	no	—
256	#103	—	2	2	no	—
257	#103	—	2	2	no	—
258	#102	—	2	1	no	—
259	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
260	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
261	#41	—	4	4	no	—
262	#41	—	8	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
273	Q₁₆	—	2	1	no	—
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	8	1	no	—
278	SD₁₆	—	4	4	no	—
279	D₈	—	2	1	no	—
280	D₈	#41×C₆₇	1	2	yes	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
285	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
301	C₄²	—	4	1	no	—
302	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	1	no	—
305	C₁₆	—	2	2	no	—
306	C₂³	—	4	1	no	—
307	C₂³	C₂×D₈×C₆₇	1	2	yes	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	16	1	no	—
313	D₄	—	2	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	8	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	4	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	16	2	no	—
327	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
329	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	1	no	—
333	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
334	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	2	no	—
349	C₄	#510×C₆₇	1	2	yes	—
350	C₄	#546×C₆₇	1	1	yes	characteristic
351	C₄	#577×C₆₇	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	8	2	no	—
354	C₂	C₆₇×#3294	1	1	yes	characteristic
355	C₂	C₂×#510×C₆₇	1	1	yes	characteristic
356	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3283	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3338	C₂	semidirect
3. C₆₇×#2241	C₂	semidirect
4. C₆₇×#2235	C₂	semidirect
5. C₆₇×#2226	C₂	semidirect
6. C₆₇×#2219	C₂	semidirect
7. C₆₇×#2209	C₂	semidirect
8. C₆₇×#2189	C₂	semidirect
9. C₄×SD₃₂×C₆₇	C₂	semidirect
10. C₆₇×#2169	C₂	semidirect
11. C₆₇×#2139	C₂	semidirect
173. #33898	C₆₇	direct (N × H)
179. #3338	C₂×C₆₇	semidirect
181. #2241	C₂×C₆₇	semidirect
182. #2235	C₂×C₆₇	semidirect
183. #2226	C₂×C₆₇	semidirect
184. #2219	C₂×C₆₇	semidirect
185. #2209	C₂×C₆₇	semidirect
186. #2189	C₂×C₆₇	semidirect
187. C₄×SD₃₂	C₂×C₆₇	semidirect
188. #2169	C₂×C₆₇	semidirect
189. #2139	C₂×C₆₇	semidirect
190. C₆₇	#33898	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×#3338	2
C₆₇×#3283	2
C₆₇×#2241	2
C₆₇×#2235	2
C₆₇×#2226	2
C₆₇×#2219	2
C₆₇×#2209	2
C₆₇×#2189	2
C₄×SD₃₂×C₆₇	2
C₆₇×#2169	2
C₆₇×#2139	2
#33898	67

Subgroups — C67×#33898 #30713149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33898 #30713149