Subgroups — C₆₇×#29337 #30708588

Order 17152.

910 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2099	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2041	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1936	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1875	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1875	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#121×C₆₇	—	2	2	no	—
11	C₂×#121×C₆₇	C₂²	1	2	yes	—
12	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#102×C₆₇	C₂²	1	4	yes	—
16	C₂×#102×C₆₇	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#97×C₆₇	C₂²	1	2	yes	—
19	#390×C₆₇	—	2	2	no	—
20	#390×C₆₇	C₂²	1	2	yes	—
21	#389×C₆₇	C₂²	1	2	yes	—
22	#389×C₆₇	C₂²	1	2	yes	—
23	#385×C₆₇	—	2	2	no	—
24	#384×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
26	#340×C₆₇	C₂²	1	4	yes	—
27	#340×C₆₇	C₂²	1	4	yes	—
28	#336×C₆₇	C₂²	1	2	yes	—
29	C₂²×D₄×C₆₇	—	2	2	no	—
30	C₂²×D₄×C₆₇	—	2	2	no	—
31	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₆₇	D₄	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
35	#127×C₆₇	—	4	2	no	—
36	#121×C₆₇	—	2	8	no	—
37	#121×C₆₇	—	4	8	no	—
38	#117×C₆₇	—	2	4	no	—
39	#117×C₆₇	—	4	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	—	2	1	no	—
51	C₂×C₄²×C₆₇	—	2	1	no	—
52	C₂×C₄²×C₆₇	—	4	2	no	—
53	C₂×C₄²×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	#102×C₆₇	—	2	8	no	—
57	#102×C₆₇	—	4	4	no	—
58	C₂²⋊C₈×C₆₇	—	2	4	no	—
59	#97×C₆₇	—	4	2	no	—
60	#95×C₆₇	—	2	2	no	—
61	#95×C₆₇	—	2	2	no	—
62	C₂⁴×C₆₇	—	2	2	no	—
63	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×D₄×C₆₇	—	2	4	no	—
65	C₂×D₄×C₆₇	—	2	4	no	—
66	C₂×D₄×C₆₇	—	2	4	no	—
67	C₂×D₄×C₆₇	—	4	4	no	—
68	C₂×D₄×C₆₇	—	4	4	no	—
69	C₂×D₄×C₆₇	—	4	8	no	—
70	C₂×D₄×C₆₇	—	4	4	no	—
71	C₂×D₄×C₆₇	—	4	4	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	—	4	2	no	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	4	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₄⋊C₄×C₆₇	—	4	4	no	—
93	C₄⋊C₄×C₆₇	—	4	4	no	—
94	C₄⋊C₄×C₆₇	#41	1	4	yes	—
95	C₂²⋊C₄×C₆₇	—	2	4	no	—
96	C₂²⋊C₄×C₆₇	—	4	8	no	—
97	C₂²⋊C₄×C₆₇	—	4	2	no	—
98	C₂²⋊C₄×C₆₇	—	4	4	no	—
99	C₄²×C₆₇	—	2	4	no	—
100	C₄²×C₆₇	—	2	2	no	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	—	4	2	no	—
103	C₄²×C₆₇	—	4	2	no	—
104	C₂³×C₆₇	—	2	1	no	—
105	C₂³×C₆₇	—	2	1	no	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	—	4	2	no	—
109	C₂³×C₆₇	—	4	2	no	—
110	C₂³×C₆₇	—	4	2	no	—
111	C₂³×C₆₇	—	4	4	no	—
112	C₂³×C₆₇	—	4	4	no	—
113	C₂³×C₆₇	#124	1	2	yes	—
114	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₆₇	—	4	8	no	—
116	D₄×C₆₇	—	8	8	no	—
117	D₄×C₆₇	—	8	8	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	8	no	—
123	C₂×C₄×C₆₇	—	2	4	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	4	8	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	#124	1	2	yes	—
136	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
140	C₈×C₆₇	—	4	4	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	—	4	2	no	—
146	C₂²×C₆₇	—	8	2	no	—
147	C₂²×C₆₇	—	8	4	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	—	8	4	no	—
151	C₂²×C₆₇	#525	1	1	yes	characteristic
152	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
153	C₂²×C₆₇	#534	1	1	yes	characteristic
154	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
155	C₂²×C₆₇	#575	1	2	yes	—
156	C₄×C₆₇	—	2	2	no	—
157	C₄×C₆₇	—	2	4	no	—
158	C₄×C₆₇	—	4	4	no	—
159	C₄×C₆₇	—	4	2	no	—
160	C₄×C₆₇	—	8	2	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	#553	1	4	yes	—
163	#29337	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₆₇	—	4	2	no	—
165	C₂×C₆₇	—	16	2	no	—
166	C₂×C₆₇	#2591	1	1	yes	characteristic
167	C₂×C₆₇	#3115	1	2	yes	—
168	C₂×C₆₇	#3135	1	1	yes	characteristic
169	C₂×C₆₇	#3138	1	1	yes	characteristic
170	C₂×C₆₇	#3194	1	2	yes	—
171	#2856	C₂×C₆₇	1	1	yes	characteristic
172	#2296	C₂×C₆₇	1	1	yes	characteristic, complemented
173	#2099	C₂×C₆₇	1	2	yes	complemented
174	#2041	C₂×C₆₇	1	2	yes	complemented
175	#1936	C₂×C₆₇	1	4	yes	complemented
176	#1875	C₂×C₆₇	1	2	yes	complemented
177	#1875	C₂×C₆₇	1	2	yes	complemented
178	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
179	C₆₇	#29337	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
181	C₂×#121	—	2	2	no	—
182	C₂×#121	C₂²×C₆₇	1	2	yes	—
183	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
184	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
185	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
186	C₂×#102	C₂²×C₆₇	1	4	yes	—
187	C₂×#102	C₂²×C₆₇	1	4	yes	—
188	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
189	C₂×#97	C₂²×C₆₇	1	2	yes	—
190	#390	—	2	2	no	—
191	#390	C₂²×C₆₇	1	2	yes	—
192	#389	C₂²×C₆₇	1	2	yes	—
193	#389	C₂²×C₆₇	1	2	yes	—
194	#385	—	2	2	no	—
195	#384	C₂²×C₆₇	1	1	yes	characteristic
196	C₄×C₄⋊C₄	—	2	2	no	—
197	#340	C₂²×C₆₇	1	4	yes	—
198	#340	C₂²×C₆₇	1	4	yes	—
199	#336	C₂²×C₆₇	1	2	yes	—
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
203	C₂³×C₄	D₄×C₆₇	1	2	yes	—
204	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
206	#127	—	4	2	no	—
207	#121	—	2	8	no	—
208	#121	—	4	8	no	—
209	#117	—	2	4	no	—
210	#117	—	4	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
217	C₂×C₂²⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	—	2	1	no	—
219	C₂×C₂²⋊C₄	—	2	4	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	—	4	2	no	—
224	C₂×C₄²	D₄×C₆₇	1	2	yes	—
225	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
227	#102	—	2	8	no	—
228	#102	—	4	4	no	—
229	C₂²⋊C₈	—	2	4	no	—
230	#97	—	4	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	C₂⁴	—	2	2	no	—
234	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂×D₄	—	4	8	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
254	C₂²×C₄	#41×C₆₇	1	4	yes	—
255	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	#41×C₆₇	1	4	yes	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	4	8	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	4	no	—
283	C₂³	—	4	4	no	—
284	C₂³	#124×C₆₇	1	2	yes	—
285	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
286	D₄	—	4	8	no	—
287	D₄	—	8	8	no	—
288	D₄	—	8	8	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	#124×C₆₇	1	2	yes	—
307	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
308	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
309	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
310	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
311	C₈	—	4	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	4	no	—
322	C₂²	#525×C₆₇	1	1	yes	characteristic
323	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
324	C₂²	#534×C₆₇	1	1	yes	characteristic
325	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
326	C₂²	#575×C₆₇	1	2	yes	—
327	C₄	—	2	2	no	—
328	C₄	—	2	4	no	—
329	C₄	—	4	4	no	—
330	C₄	—	4	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	#553×C₆₇	1	4	yes	—
334	C₂	—	4	2	no	—
335	C₂	—	16	2	no	—
336	C₂	C₆₇×#2591	1	1	yes	characteristic
337	C₂	C₆₇×#3115	1	2	yes	—
338	C₂	C₆₇×#3135	1	1	yes	characteristic
339	C₂	C₆₇×#3138	1	1	yes	characteristic
340	C₂	C₆₇×#3194	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2296	C₂	semidirect
3. C₆₇×#2099	C₂	semidirect
4. C₆₇×#2041	C₂	semidirect
5. C₆₇×#1936	C₂	semidirect
6. C₆₇×#1875	C₂	semidirect
7. C₆₇×#1875	C₂	semidirect
8. C₆₇×#1749	C₂	semidirect
163. #29337	C₆₇	direct (N × H)
172. #2296	C₂×C₆₇	semidirect
173. #2099	C₂×C₆₇	semidirect
174. #2041	C₂×C₆₇	semidirect
175. #1936	C₂×C₆₇	semidirect
176. #1875	C₂×C₆₇	semidirect
177. #1875	C₂×C₆₇	semidirect
178. #1749	C₂×C₆₇	semidirect
179. C₆₇	#29337	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×#2856	2
C₆₇×#2296	2
C₆₇×#2099	2
C₆₇×#2041	2
C₆₇×#1936	2
C₆₇×#1875	2
C₆₇×#1749	2
#29337	67

Subgroups — C67×#29337 #30708588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#29337 #30708588