Subgroups — C₆₇×#33123 #30712374

Order 17152.

808 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3094	C₂	1	2	yes	maximal, complemented
2	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1978	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1952	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1931	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1900	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#121×C₆₇	C₂²	1	2	yes	—
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	#482×C₆₇	—	2	4	no	—
15	#481×C₆₇	—	2	2	no	—
16	#480×C₆₇	—	2	2	no	—
17	#466×C₆₇	—	2	1	no	—
18	#466×C₆₇	—	2	1	no	—
19	#466×C₆₇	—	2	1	no	—
20	#466×C₆₇	—	2	1	no	—
21	C₂×#108×C₆₇	C₂²	1	2	yes	—
22	#428×C₆₇	C₂²	1	4	yes	—
23	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
24	#418×C₆₇	C₂²	1	4	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#100×C₆₇	C₂²	1	4	yes	—
27	#407×C₆₇	C₂²	1	2	yes	—
28	#405×C₆₇	—	2	1	no	—
29	#405×C₆₇	—	2	1	no	—
30	#405×C₆₇	C₂²	1	2	yes	—
31	#405×C₆₇	C₂²	1	2	yes	—
32	C₂×#97×C₆₇	—	2	1	no	—
33	C₂×C₄×C₈×C₆₇	—	2	1	no	—
34	#343×C₆₇	C₂²	1	4	yes	—
35	#341×C₆₇	C₂²	1	2	yes	—
36	C₂²×D₄×C₆₇	—	2	2	no	—
37	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×D₈×C₆₇	—	2	1	no	—
40	C₂×D₈×C₆₇	—	2	1	no	—
41	C₂×D₈×C₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	—	2	1	no	—
43	C₂×M₁₆×C₆₇	—	2	4	no	—
44	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
45	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	#121×C₆₇	—	2	2	no	—
55	#121×C₆₇	—	2	2	no	—
56	#121×C₆₇	—	2	2	no	—
57	#121×C₆₇	—	2	2	no	—
58	#117×C₆₇	—	2	1	no	—
59	#117×C₆₇	—	2	1	no	—
60	#117×C₆₇	D₄	1	2	yes	—
61	#117×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	—	2	1	no	—
68	#108×C₆₇	—	2	4	no	—
69	#107×C₆₇	—	2	1	no	—
70	#107×C₆₇	—	2	1	no	—
71	#107×C₆₇	—	2	4	no	—
72	#106×C₆₇	—	2	4	no	—
73	#102×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	4	no	—
76	#102×C₆₇	—	2	4	no	—
77	#100×C₆₇	—	2	8	no	—
78	C₂²⋊C₈×C₆₇	—	2	2	no	—
79	C₂²⋊C₈×C₆₇	—	2	2	no	—
80	#97×C₆₇	—	2	2	no	—
81	#97×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	—	2	2	no	—
83	C₄×C₈×C₆₇	—	2	2	no	—
84	C₂⁴×C₆₇	—	2	2	no	—
85	C₂⁴×C₆₇	#41	1	1	yes	characteristic
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	—	2	4	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	8	4	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
104	D₈×C₆₇	—	4	2	no	—
105	D₈×C₆₇	—	4	2	no	—
106	M₁₆×C₆₇	—	2	4	no	—
107	M₁₆×C₆₇	—	2	2	no	—
108	M₁₆×C₆₇	—	2	4	no	—
109	M₁₆×C₆₇	—	2	4	no	—
110	M₁₆×C₆₇	#41	1	4	yes	—
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	4	no	—
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	#41	1	4	yes	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
129	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
130	C₂²⋊C₄×C₆₇	—	2	2	no	—
131	C₂²⋊C₄×C₆₇	—	2	2	no	—
132	C₂²⋊C₄×C₆₇	—	4	2	no	—
133	C₂²⋊C₄×C₆₇	—	4	2	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	4	1	no	—
139	C₂³×C₆₇	—	4	2	no	—
140	C₂³×C₆₇	—	4	4	no	—
141	C₂³×C₆₇	—	8	2	no	—
142	C₂³×C₆₇	—	8	2	no	—
143	C₂³×C₆₇	#121	1	1	yes	characteristic
144	C₂³×C₆₇	#121	1	1	yes	characteristic
145	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
146	D₄×C₆₇	—	4	2	no	—
147	D₄×C₆₇	—	4	2	no	—
148	D₄×C₆₇	—	4	4	no	—
149	D₄×C₆₇	—	8	4	no	—
150	D₄×C₆₇	—	8	4	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	8	2	no	—
164	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
165	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	2	4	no	—
173	C₈×C₆₇	—	4	4	no	—
174	C₈×C₆₇	#123	1	2	yes	—
175	C₈×C₆₇	#123	1	2	yes	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	2	1	no	—
182	C₂²×C₆₇	—	2	1	no	—
183	C₂²×C₆₇	—	4	4	no	—
184	C₂²×C₆₇	—	8	2	no	—
185	C₂²×C₆₇	—	8	2	no	—
186	C₂²×C₆₇	—	8	2	no	—
187	C₂²×C₆₇	—	16	4	no	—
188	C₂²×C₆₇	#472	1	2	yes	—
189	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
190	C₂²×C₆₇	#529	1	2	yes	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	4	2	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	8	2	no	—
196	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
197	C₄×C₆₇	#525	1	1	yes	characteristic
198	C₄×C₆₇	#532	1	2	yes	—
199	#33123	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₆₇	—	2	2	no	—
201	C₂×C₆₇	—	2	2	no	—
202	C₂×C₆₇	—	4	1	no	—
203	C₂×C₆₇	—	16	2	no	—
204	C₂×C₆₇	#2854	1	1	yes	characteristic
205	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
206	C₂×C₆₇	#3067	1	1	yes	characteristic
207	#3094	C₂×C₆₇	1	2	yes	complemented
208	C₂×#466	C₂×C₆₇	1	1	yes	characteristic
209	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#1978	C₂×C₆₇	1	2	yes	complemented
211	#1952	C₂×C₆₇	1	2	yes	complemented
212	#1931	C₂×C₆₇	1	4	yes	complemented
213	#1900	C₂×C₆₇	1	2	yes	complemented
214	#1810	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₆₇	#33123	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
217	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
218	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
219	C₂×#121	C₂²×C₆₇	1	2	yes	—
220	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
221	#482	—	2	4	no	—
222	#481	—	2	2	no	—
223	#480	—	2	2	no	—
224	#466	—	2	1	no	—
225	#466	—	2	1	no	—
226	#466	—	2	1	no	—
227	#466	—	2	1	no	—
228	C₂×#108	C₂²×C₆₇	1	2	yes	—
229	#428	C₂²×C₆₇	1	4	yes	—
230	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
231	#418	C₂²×C₆₇	1	4	yes	—
232	C₂×#102	C₂²×C₆₇	1	2	yes	—
233	C₂×#100	C₂²×C₆₇	1	4	yes	—
234	#407	C₂²×C₆₇	1	2	yes	—
235	#405	—	2	1	no	—
236	#405	—	2	1	no	—
237	#405	C₂²×C₆₇	1	2	yes	—
238	#405	C₂²×C₆₇	1	2	yes	—
239	C₂×#97	—	2	1	no	—
240	C₂×C₄×C₈	—	2	1	no	—
241	#343	C₂²×C₆₇	1	4	yes	—
242	#341	C₂²×C₆₇	1	2	yes	—
243	C₂²×D₄	—	2	2	no	—
244	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
245	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂×D₈	—	2	1	no	—
247	C₂×D₈	—	2	1	no	—
248	C₂×D₈	—	2	4	no	—
249	C₂×M₁₆	—	2	1	no	—
250	C₂×M₁₆	—	2	4	no	—
251	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
252	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
253	C₂²×C₈	—	2	1	no	—
254	C₂²×C₈	—	2	4	no	—
255	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
256	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
257	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
258	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
261	#121	—	2	2	no	—
262	#121	—	2	2	no	—
263	#121	—	2	2	no	—
264	#121	—	2	2	no	—
265	#117	—	2	1	no	—
266	#117	—	2	1	no	—
267	#117	D₄×C₆₇	1	2	yes	—
268	#117	C₂³×C₆₇	1	2	yes	—
269	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
270	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
272	C₂×C₂²⋊C₄	—	2	1	no	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₄²	—	2	1	no	—
275	#108	—	2	4	no	—
276	#107	—	2	1	no	—
277	#107	—	2	1	no	—
278	#107	—	2	4	no	—
279	#106	—	2	4	no	—
280	#102	—	2	2	no	—
281	#102	—	2	2	no	—
282	#102	—	2	4	no	—
283	#102	—	2	4	no	—
284	#100	—	2	8	no	—
285	C₂²⋊C₈	—	2	2	no	—
286	C₂²⋊C₈	—	2	2	no	—
287	#97	—	2	2	no	—
288	#97	—	2	2	no	—
289	C₄×C₈	—	2	2	no	—
290	C₄×C₈	—	2	2	no	—
291	C₂⁴	—	2	2	no	—
292	C₂⁴	#41×C₆₇	1	1	yes	characteristic
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	4	no	—
296	C₂×D₄	—	2	4	no	—
297	C₂×D₄	—	4	2	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	8	4	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
311	D₈	—	4	2	no	—
312	D₈	—	4	2	no	—
313	M₁₆	—	2	4	no	—
314	M₁₆	—	2	2	no	—
315	M₁₆	—	2	4	no	—
316	M₁₆	—	2	4	no	—
317	M₁₆	#41×C₆₇	1	4	yes	—
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	4	no	—
326	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
327	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₂×C₈	#41×C₆₇	1	4	yes	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	4	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	4	1	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	4	no	—
348	C₂³	—	8	2	no	—
349	C₂³	—	8	2	no	—
350	C₂³	#121×C₆₇	1	1	yes	characteristic
351	C₂³	#121×C₆₇	1	1	yes	characteristic
352	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	4	no	—
356	D₄	—	8	4	no	—
357	D₄	—	8	4	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	8	2	no	—
371	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
372	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
373	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
376	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
377	C₈	—	2	1	no	—
378	C₈	—	2	1	no	—
379	C₈	—	2	4	no	—
380	C₈	—	4	4	no	—
381	C₈	#123×C₆₇	1	2	yes	—
382	C₈	#123×C₆₇	1	2	yes	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	4	4	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	16	4	no	—
395	C₂²	#472×C₆₇	1	2	yes	—
396	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
397	C₂²	#529×C₆₇	1	2	yes	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	2	no	—
403	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
404	C₄	#525×C₆₇	1	1	yes	characteristic
405	C₄	#532×C₆₇	1	2	yes	—
406	C₂	—	2	2	no	—
407	C₂	—	2	2	no	—
408	C₂	—	4	1	no	—
409	C₂	—	16	2	no	—
410	C₂	C₆₇×#2854	1	1	yes	characteristic
411	C₂	C₂×#472×C₆₇	1	1	yes	characteristic
412	C₂	C₆₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#466×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3094	C₂	semidirect
3. C₂×#405×C₆₇	C₂	semidirect
4. C₆₇×#1978	C₂	semidirect
5. C₆₇×#1952	C₂	semidirect
6. C₆₇×#1931	C₂	semidirect
7. C₆₇×#1900	C₂	semidirect
8. C₆₇×#1810	C₂	semidirect
199. #33123	C₆₇	direct (N × H)
207. #3094	C₂×C₆₇	semidirect
209. C₂×#405	C₂×C₆₇	semidirect
210. #1978	C₂×C₆₇	semidirect
211. #1952	C₂×C₆₇	semidirect
212. #1931	C₂×C₆₇	semidirect
213. #1900	C₂×C₆₇	semidirect
214. #1810	C₂×C₆₇	semidirect
215. C₆₇	#33123	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×#3094	2
C₂×#466×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#1978	2
C₆₇×#1952	2
C₆₇×#1931	2
C₆₇×#1900	2
C₆₇×#1810	2
#33123	67

Subgroups — C67×#33123 #30712374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33123 #30712374