Subgroups — C₆₇×#24988 #30704239

Order 17152.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#472×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2081	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2048	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1904	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1833	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1829	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1814	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	#518×C₆₇	C₂²	1	2	yes	—
16	#472×C₆₇	—	2	2	no	—
17	#472×C₆₇	—	2	2	no	—
18	#442×C₆₇	—	2	2	no	—
19	C₄×D₈×C₆₇	—	2	2	no	—
20	#430×C₆₇	C₂²	1	2	yes	—
21	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
23	#427×C₆₇	C₂²	1	1	yes	characteristic
24	#421×C₆₇	C₂²	1	2	yes	—
25	#418×C₆₇	C₂²	1	2	yes	—
26	#418×C₆₇	C₂²	1	2	yes	—
27	#413×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#100×C₆₇	C₂²	1	2	yes	—
29	C₂×#100×C₆₇	C₂²	1	2	yes	—
30	#410×C₆₇	C₂²	1	1	yes	characteristic
31	#405×C₆₇	C₂²	1	1	yes	characteristic
32	#405×C₆₇	C₂²	1	1	yes	characteristic
33	#344×C₆₇	C₂²	1	1	yes	characteristic
34	#344×C₆₇	C₂²	1	1	yes	characteristic
35	#344×C₆₇	C₂²	1	2	yes	—
36	#344×C₆₇	C₂²	1	2	yes	—
37	#344×C₆₇	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₆₇	C₂²	1	2	yes	—
39	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
41	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
42	AGL(1,ℤ/8)×C₆₇	—	4	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×D₈×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	—	2	1	no	—
46	C₂×D₈×C₆₇	—	2	1	no	—
47	C₂×D₈×C₆₇	—	2	1	no	—
48	C₂×D₈×C₆₇	—	2	1	no	—
49	C₂×D₈×C₆₇	D₄	1	2	yes	—
50	C₂×D₈×C₆₇	D₄	1	2	yes	—
51	#131×C₆₇	—	2	1	no	—
52	#131×C₆₇	—	2	1	no	—
53	#131×C₆₇	—	2	1	no	—
54	#131×C₆₇	—	2	1	no	—
55	C₂×M₁₆×C₆₇	—	2	1	no	—
56	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
58	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
60	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	—	2	1	no	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₆₇	—	2	4	no	—
64	C₄×D₄×C₆₇	—	4	2	no	—
65	#117×C₆₇	C₂³	1	2	yes	—
66	#117×C₆₇	C₂³	1	1	yes	characteristic
67	#117×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
69	#108×C₆₇	—	2	1	no	—
70	#108×C₆₇	—	2	1	no	—
71	#108×C₆₇	—	2	1	no	—
72	#108×C₆₇	—	2	1	no	—
73	#108×C₆₇	—	4	2	no	—
74	#107×C₆₇	—	2	1	no	—
75	#106×C₆₇	—	2	1	no	—
76	#104×C₆₇	—	4	2	no	—
77	#102×C₆₇	—	2	4	no	—
78	#102×C₆₇	—	4	2	no	—
79	#101×C₆₇	—	2	1	no	—
80	#100×C₆₇	—	2	4	no	—
81	#100×C₆₇	—	2	1	no	—
82	#100×C₆₇	—	4	2	no	—
83	#99×C₆₇	—	2	2	no	—
84	#97×C₆₇	—	2	1	no	—
85	#97×C₆₇	—	2	1	no	—
86	C₄×C₈×C₆₇	—	2	1	no	—
87	C₄×C₈×C₆₇	—	2	1	no	—
88	C₂⁴×C₆₇	—	2	2	no	—
89	#41×C₆₇	—	2	1	no	—
90	#41×C₆₇	—	2	1	no	—
91	#41×C₆₇	—	2	1	no	—
92	#41×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
101	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	4	4	no	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	SD₁₆×C₆₇	—	4	4	no	—
106	D₈×C₆₇	—	2	2	no	—
107	D₈×C₆₇	—	2	2	no	—
108	D₈×C₆₇	—	2	2	no	—
109	D₈×C₆₇	—	2	2	no	—
110	D₈×C₆₇	—	4	4	no	—
111	M₁₆×C₆₇	—	2	1	no	—
112	M₁₆×C₆₇	—	2	2	no	—
113	M₁₆×C₆₇	—	2	1	no	—
114	M₁₆×C₆₇	—	2	1	no	—
115	M₁₆×C₆₇	—	2	2	no	—
116	M₁₆×C₆₇	—	2	1	no	—
117	M₁₆×C₆₇	—	2	1	no	—
118	M₁₆×C₆₇	—	2	2	no	—
119	M₁₆×C₆₇	—	2	1	no	—
120	M₁₆×C₆₇	—	2	2	no	—
121	M₁₆×C₆₇	—	4	1	no	—
122	M₁₆×C₆₇	—	4	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	4	1	no	—
129	C₂×C₈×C₆₇	—	4	1	no	—
130	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
133	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
134	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
135	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₂²⋊C₄×C₆₇	—	2	2	no	—
140	C₂²⋊C₄×C₆₇	—	2	1	no	—
141	C₂²⋊C₄×C₆₇	—	4	4	no	—
142	C₂²⋊C₄×C₆₇	#41	1	2	yes	—
143	C₄²×C₆₇	—	2	1	no	—
144	C₄²×C₆₇	—	2	1	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	1	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	16	2	no	—
151	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	1	no	—
153	Q₈×C₆₇	—	2	1	no	—
154	D₄×C₆₇	—	2	4	no	—
155	D₄×C₆₇	—	2	1	no	—
156	D₄×C₆₇	—	2	4	no	—
157	D₄×C₆₇	—	2	1	no	—
158	D₄×C₆₇	—	4	1	no	—
159	D₄×C₆₇	—	4	1	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	16	4	no	—
169	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
175	C₈×C₆₇	—	4	1	no	—
176	C₈×C₆₇	—	4	2	no	—
177	C₈×C₆₇	—	4	1	no	—
178	C₈×C₆₇	—	4	2	no	—
179	C₈×C₆₇	—	4	1	no	—
180	C₈×C₆₇	#123	1	1	yes	characteristic
181	C₈×C₆₇	#123	1	1	yes	characteristic
182	C₈×C₆₇	#123	1	1	yes	characteristic
183	C₈×C₆₇	#123	1	1	yes	characteristic
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	—	4	4	no	—
187	C₂²×C₆₇	—	8	4	no	—
188	C₂²×C₆₇	—	8	1	no	—
189	C₂²×C₆₇	—	16	2	no	—
190	C₂²×C₆₇	—	16	2	no	—
191	C₂²×C₆₇	#529	1	1	yes	characteristic
192	C₂²×C₆₇	#546	1	1	yes	characteristic
193	C₂²×C₆₇	#555	1	1	yes	characteristic
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	4	1	no	—
196	C₄×C₆₇	—	8	1	no	—
197	C₄×C₆₇	—	8	2	no	—
198	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
199	C₄×C₆₇	#540	1	1	yes	characteristic
200	C₄×C₆₇	#546	1	1	yes	characteristic
201	C₄×C₆₇	#553	1	1	yes	characteristic
202	#24988	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	2	1	no	—
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	1	no	—
206	C₂×C₆₇	—	8	1	no	—
207	C₂×C₆₇	—	8	4	no	—
208	C₂×C₆₇	#2420	1	1	yes	characteristic
209	C₂×#472	C₂×C₆₇	1	1	yes	characteristic
210	#2948	C₂×C₆₇	1	1	yes	characteristic, complemented
211	#2081	C₂×C₆₇	1	2	yes	complemented
212	#2048	C₂×C₆₇	1	2	yes	complemented
213	#1978	C₂×C₆₇	1	1	yes	characteristic, complemented
214	#1953	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#1904	C₂×C₆₇	1	2	yes	complemented
216	#1833	C₂×C₆₇	1	1	yes	characteristic, complemented
217	#1829	C₂×C₆₇	1	2	yes	complemented
218	#1814	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1757	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₆₇	#24988	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
222	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
224	#518	C₂²×C₆₇	1	2	yes	—
225	#472	—	2	2	no	—
226	#472	—	2	2	no	—
227	#442	—	2	2	no	—
228	C₄×D₈	—	2	2	no	—
229	#430	C₂²×C₆₇	1	2	yes	—
230	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
232	#427	C₂²×C₆₇	1	1	yes	characteristic
233	#421	C₂²×C₆₇	1	2	yes	—
234	#418	C₂²×C₆₇	1	2	yes	—
235	#418	C₂²×C₆₇	1	2	yes	—
236	#413	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#100	C₂²×C₆₇	1	2	yes	—
238	C₂×#100	C₂²×C₆₇	1	2	yes	—
239	#410	C₂²×C₆₇	1	1	yes	characteristic
240	#405	C₂²×C₆₇	1	1	yes	characteristic
241	#405	C₂²×C₆₇	1	1	yes	characteristic
242	#344	C₂²×C₆₇	1	2	yes	—
243	#344	C₂²×C₆₇	1	2	yes	—
244	#344	C₂²×C₆₇	1	1	yes	characteristic
245	#344	C₂²×C₆₇	1	1	yes	characteristic
246	#344	C₂²×C₆₇	1	2	yes	—
247	H(3,ℤ/4)	C₂²×C₆₇	1	2	yes	—
248	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
249	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
250	AGL(1,ℤ/8)	—	2	4	no	—
251	AGL(1,ℤ/8)	—	4	2	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×D₈	—	2	2	no	—
254	C₂×D₈	—	2	1	no	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	2	1	no	—
258	C₂×D₈	D₄×C₆₇	1	2	yes	—
259	C₂×D₈	D₄×C₆₇	1	2	yes	—
260	#131	—	2	1	no	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	#131	—	2	1	no	—
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
266	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
267	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
268	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
269	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
272	C₄×D₄	—	2	4	no	—
273	C₄×D₄	—	4	2	no	—
274	#117	C₂³×C₆₇	1	2	yes	—
275	#117	C₂³×C₆₇	1	1	yes	characteristic
276	#117	C₂³×C₆₇	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	—	2	4	no	—
278	#108	—	2	1	no	—
279	#108	—	2	1	no	—
280	#108	—	2	1	no	—
281	#108	—	2	1	no	—
282	#108	—	4	2	no	—
283	#107	—	2	1	no	—
284	#106	—	2	1	no	—
285	#104	—	4	2	no	—
286	#102	—	2	4	no	—
287	#102	—	4	2	no	—
288	#101	—	2	1	no	—
289	#100	—	2	4	no	—
290	#100	—	2	1	no	—
291	#100	—	4	2	no	—
292	#99	—	2	2	no	—
293	#97	—	2	1	no	—
294	#97	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₄×C₈	—	2	1	no	—
297	C₂⁴	—	2	2	no	—
298	#41	—	2	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	—	2	1	no	—
302	C₂×Q₈	#41×C₆₇	1	1	yes	characteristic
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
310	C₂×D₄	#41×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	4	4	no	—
313	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
314	SD₁₆	—	4	4	no	—
315	D₈	—	2	2	no	—
316	D₈	—	2	2	no	—
317	D₈	—	2	2	no	—
318	D₈	—	2	2	no	—
319	D₈	—	4	4	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	2	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	1	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	2	1	no	—
326	M₁₆	—	2	1	no	—
327	M₁₆	—	2	2	no	—
328	M₁₆	—	2	1	no	—
329	M₁₆	—	2	2	no	—
330	M₁₆	—	4	1	no	—
331	M₁₆	—	4	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	4	1	no	—
338	C₂×C₈	—	4	1	no	—
339	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
341	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
342	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
343	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
344	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₂²⋊C₄	#41×C₆₇	1	2	yes	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	16	2	no	—
360	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	1	no	—
365	D₄	—	2	4	no	—
366	D₄	—	2	1	no	—
367	D₄	—	4	1	no	—
368	D₄	—	4	1	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	16	4	no	—
378	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
380	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
382	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
383	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	1	no	—
389	C₈	#123×C₆₇	1	1	yes	characteristic
390	C₈	#123×C₆₇	1	1	yes	characteristic
391	C₈	#123×C₆₇	1	1	yes	characteristic
392	C₈	#123×C₆₇	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	8	4	no	—
397	C₂²	—	8	1	no	—
398	C₂²	—	16	2	no	—
399	C₂²	—	16	2	no	—
400	C₂²	#529×C₆₇	1	1	yes	characteristic
401	C₂²	#546×C₆₇	1	1	yes	characteristic
402	C₂²	#555×C₆₇	1	1	yes	characteristic
403	C₄	—	4	2	no	—
404	C₄	—	4	1	no	—
405	C₄	—	8	2	no	—
406	C₄	—	8	1	no	—
407	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
408	C₄	#540×C₆₇	1	1	yes	characteristic
409	C₄	#546×C₆₇	1	1	yes	characteristic
410	C₄	#553×C₆₇	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	4	no	—
415	C₂	—	8	1	no	—
416	C₂	C₆₇×#2420	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2420
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#472×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2948	C₂	semidirect
3. C₆₇×#2081	C₂	semidirect
4. C₆₇×#2048	C₂	semidirect
5. C₆₇×#1978	C₂	semidirect
6. C₆₇×#1953	C₂	semidirect
7. C₆₇×#1904	C₂	semidirect
8. C₆₇×#1833	C₂	semidirect
9. C₆₇×#1829	C₂	semidirect
10. C₆₇×#1814	C₂	semidirect
11. C₆₇×#1757	C₂	semidirect
202. #24988	C₆₇	direct (N × H)
210. #2948	C₂×C₆₇	semidirect
211. #2081	C₂×C₆₇	semidirect
212. #2048	C₂×C₆₇	semidirect
213. #1978	C₂×C₆₇	semidirect
214. #1953	C₂×C₆₇	semidirect
215. #1904	C₂×C₆₇	semidirect
216. #1833	C₂×C₆₇	semidirect
217. #1829	C₂×C₆₇	semidirect
218. #1814	C₂×C₆₇	semidirect
219. #1757	C₂×C₆₇	semidirect
220. C₆₇	#24988	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#472×C₆₇	2
C₆₇×#2948	2
C₆₇×#2081	2
C₆₇×#2048	2
C₆₇×#1978	2
C₆₇×#1953	2
C₆₇×#1904	2
C₆₇×#1833	2
C₆₇×#1829	2
C₆₇×#1814	2
C₆₇×#1757	2
#24988	67

Subgroups — C67×#24988 #30704239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#24988 #30704239