Subgroups — C₆₇×#19954 #30699205

Order 17152.

618 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3354	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3303	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1691	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1683	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1647	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1634	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#555×C₆₇	C₂²	1	2	yes	—
11	#547×C₆₇	C₂²	1	2	yes	—
12	#500×C₆₇	C₂²	1	1	yes	characteristic
13	#498×C₆₇	C₂²	1	2	yes	—
14	#494×C₆₇	C₂²	1	1	yes	characteristic
15	#487×C₆₇	C₂²	1	2	yes	—
16	#485×C₆₇	—	2	2	no	—
17	#482×C₆₇	—	2	2	no	—
18	#481×C₆₇	—	2	2	no	—
19	#476×C₆₇	C₂²	1	2	yes	—
20	#476×C₆₇	C₂²	1	2	yes	—
21	#474×C₆₇	C₂²	1	2	yes	—
22	#469×C₆₇	—	2	2	no	—
23	#465×C₆₇	—	2	2	no	—
24	#461×C₆₇	C₂²	1	2	yes	—
25	#449×C₆₇	—	2	2	no	—
26	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
28	#436×C₆₇	C₂²	1	2	yes	—
29	C₈×D₄×C₆₇	C₂²	1	2	yes	—
30	#428×C₆₇	—	2	2	no	—
31	#416×C₆₇	—	2	2	no	—
32	#335×C₆₇	C₂²	1	1	yes	characteristic
33	#335×C₆₇	C₂²	1	1	yes	characteristic
34	#333×C₆₇	C₂²	1	2	yes	—
35	#332×C₆₇	C₂²	1	2	yes	—
36	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	C₈²×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	4	1	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	#124×C₆₇	—	4	2	no	—
49	#123×C₆₇	—	4	2	no	—
50	C₂²⋊Q₈×C₆₇	—	2	2	no	—
51	#121×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	2	2	no	—
54	C₄×D₄×C₆₇	C₂³	1	2	yes	—
55	C₄×D₄×C₆₇	C₂³	1	2	yes	—
56	#117×C₆₇	—	2	2	no	—
57	#117×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	#107×C₆₇	—	2	2	no	—
60	#107×C₆₇	—	4	1	no	—
61	#107×C₆₇	D₄	1	1	yes	characteristic
62	#107×C₆₇	D₄	1	1	yes	characteristic
63	#106×C₆₇	—	2	2	no	—
64	#106×C₆₇	—	2	2	no	—
65	#106×C₆₇	—	2	2	no	—
66	#106×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
69	#103×C₆₇	—	2	2	no	—
70	#103×C₆₇	—	2	2	no	—
71	#103×C₆₇	—	2	2	no	—
72	#102×C₆₇	—	2	2	no	—
73	#102×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	D₄	1	2	yes	—
76	#102×C₆₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₂²⋊C₈×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
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	2	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	4	2	no	—
87	C₂×D₄×C₆₇	—	4	2	no	—
88	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	4	2	no	—
93	Q₁₆×C₆₇	—	8	2	no	—
94	SD₁₆×C₆₇	—	8	2	no	—
95	D₈×C₆₇	—	2	2	no	—
96	D₈×C₆₇	—	4	1	no	—
97	M₁₆×C₆₇	—	4	4	no	—
98	C₂×C₈×C₆₇	—	2	1	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	8	2	no	—
102	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₆₇	—	2	2	no	—
114	C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	2	no	—
116	C₂²⋊C₄×C₆₇	—	4	2	no	—
117	C₂²⋊C₄×C₆₇	—	4	2	no	—
118	C₄²×C₆₇	—	4	2	no	—
119	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂³×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	—	2	2	no	—
122	Q₈×C₆₇	—	4	2	no	—
123	Q₈×C₆₇	—	4	2	no	—
124	Q₈×C₆₇	—	8	2	no	—
125	D₄×C₆₇	—	2	2	no	—
126	D₄×C₆₇	—	2	2	no	—
127	D₄×C₆₇	—	2	2	no	—
128	D₄×C₆₇	—	2	2	no	—
129	D₄×C₆₇	—	8	4	no	—
130	D₄×C₆₇	—	16	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	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	—	8	4	no	—
140	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	2	1	no	—
146	C₈×C₆₇	—	2	1	no	—
147	C₈×C₆₇	—	4	2	no	—
148	C₈×C₆₇	—	4	2	no	—
149	C₈×C₆₇	#135	1	2	yes	—
150	C₂²×C₆₇	—	4	2	no	—
151	C₂²×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	—	4	2	no	—
153	C₂²×C₆₇	—	4	2	no	—
154	C₂²×C₆₇	—	4	2	no	—
155	C₂²×C₆₇	—	4	2	no	—
156	C₂²×C₆₇	D₄²	1	1	yes	characteristic
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	4	2	no	—
159	C₄×C₆₇	—	4	2	no	—
160	C₄×C₆₇	—	8	2	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
163	C₄×C₆₇	#575	1	1	yes	characteristic
164	C₄×C₆₇	#576	1	1	yes	characteristic
165	C₄×C₆₇	#578	1	1	yes	characteristic
166	#19954	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₆₇	—	8	2	no	—
168	C₂×C₆₇	—	8	2	no	—
169	C₂×C₆₇	#3273	1	1	yes	characteristic
170	C₂×C₆₇	#3283	1	1	yes	characteristic
171	C₂×C₆₇	#3291	1	1	yes	characteristic
172	#3354	C₂×C₆₇	1	2	yes	complemented
173	#3303	C₂×C₆₇	1	2	yes	complemented
174	#1707	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#1691	C₂×C₆₇	1	2	yes	complemented
176	#1683	C₂×C₆₇	1	2	yes	complemented
177	#1647	C₂×C₆₇	1	2	yes	complemented
178	#1634	C₂×C₆₇	1	2	yes	complemented
179	#1589	C₂×C₆₇	1	1	yes	characteristic, complemented
180	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
181	C₆₇	#19954	1	1	yes	characteristic, direct product factor, Sylow
182	#555	C₂²×C₆₇	1	2	yes	—
183	#547	C₂²×C₆₇	1	2	yes	—
184	#500	C₂²×C₆₇	1	1	yes	characteristic
185	#498	C₂²×C₆₇	1	2	yes	—
186	#494	C₂²×C₆₇	1	1	yes	characteristic
187	#487	C₂²×C₆₇	1	2	yes	—
188	#485	—	2	2	no	—
189	#482	—	2	2	no	—
190	#481	—	2	2	no	—
191	#476	C₂²×C₆₇	1	2	yes	—
192	#476	C₂²×C₆₇	1	2	yes	—
193	#474	C₂²×C₆₇	1	2	yes	—
194	#469	—	2	2	no	—
195	#465	—	2	2	no	—
196	#461	C₂²×C₆₇	1	2	yes	—
197	#449	—	2	2	no	—
198	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
199	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
200	#436	C₂²×C₆₇	1	2	yes	—
201	C₈×D₄	C₂²×C₆₇	1	2	yes	—
202	#428	—	2	2	no	—
203	#416	—	2	2	no	—
204	#335	C₂²×C₆₇	1	1	yes	characteristic
205	#335	C₂²×C₆₇	1	1	yes	characteristic
206	#333	C₂²×C₆₇	1	2	yes	—
207	#332	C₂²×C₆₇	1	2	yes	—
208	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
209	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
210	C₈²	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×#41	—	2	2	no	—
212	C₂×Q₁₆	—	4	1	no	—
213	C₂×SD₁₆	—	2	2	no	—
214	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
215	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
216	C₂×M₁₆	—	2	2	no	—
217	C₂²×C₈	—	2	2	no	—
218	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
219	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
220	#124	—	4	2	no	—
221	#123	—	4	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	#121	—	2	2	no	—
226	C₄×D₄	C₂³×C₆₇	1	2	yes	—
227	C₄×D₄	C₂³×C₆₇	1	2	yes	—
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	#107	—	2	2	no	—
232	#107	—	4	1	no	—
233	#107	D₄×C₆₇	1	1	yes	characteristic
234	#107	D₄×C₆₇	1	1	yes	characteristic
235	#106	—	2	2	no	—
236	#106	—	2	2	no	—
237	#106	—	2	2	no	—
238	#106	—	2	2	no	—
239	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
240	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
241	#103	—	2	2	no	—
242	#103	—	2	2	no	—
243	#103	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	#102	D₄×C₆₇	1	2	yes	—
248	#102	D₄×C₆₇	1	2	yes	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₂²⋊C₈	—	2	2	no	—
251	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
254	#41	—	4	4	no	—
255	#41	—	8	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
261	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	Q₁₆	—	8	2	no	—
266	SD₁₆	—	8	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	4	1	no	—
269	M₁₆	—	4	4	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	8	2	no	—
274	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
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	—	4	2	no	—
283	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
284	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
292	C₂³	—	2	2	no	—
293	C₂³	—	2	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	2	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	16	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	4	2	no	—
320	C₈	—	4	2	no	—
321	C₈	#135×C₆₇	1	2	yes	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	D₄²×C₆₇	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
335	C₄	#575×C₆₇	1	1	yes	characteristic
336	C₄	#576×C₆₇	1	1	yes	characteristic
337	C₄	#578×C₆₇	1	1	yes	characteristic
338	C₂	—	8	2	no	—
339	C₂	—	8	2	no	—
340	C₂	C₆₇×#3273	1	1	yes	characteristic
341	C₂	C₆₇×#3283	1	1	yes	characteristic
342	C₂	C₆₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3354	C₂	semidirect
2. C₆₇×#3303	C₂	semidirect
3. C₆₇×#1707	C₂	semidirect
4. C₆₇×#1691	C₂	semidirect
5. C₆₇×#1683	C₂	semidirect
6. C₆₇×#1647	C₂	semidirect
7. C₆₇×#1634	C₂	semidirect
8. C₆₇×#1589	C₂	semidirect
9. C₈×D₈×C₆₇	C₂	semidirect
166. #19954	C₆₇	direct (N × H)
172. #3354	C₂×C₆₇	semidirect
173. #3303	C₂×C₆₇	semidirect
174. #1707	C₂×C₆₇	semidirect
175. #1691	C₂×C₆₇	semidirect
176. #1683	C₂×C₆₇	semidirect
177. #1647	C₂×C₆₇	semidirect
178. #1634	C₂×C₆₇	semidirect
179. #1589	C₂×C₆₇	semidirect
180. C₈×D₈	C₂×C₆₇	semidirect
181. C₆₇	#19954	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×#3354	2
C₆₇×#3303	2
C₆₇×#1707	2
C₆₇×#1691	2
C₆₇×#1683	2
C₆₇×#1647	2
C₆₇×#1634	2
C₆₇×#1589	2
C₈×D₈×C₆₇	2
#19954	67

Subgroups — C67×#19954 #30699205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19954 #30699205