Subgroups — C₆₇×#25921 #30705172

Order 17152.

972 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3140	C₂	1	2	yes	maximal
3	C₆₇×#2545	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2028	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2027	C₂	1	2	yes	maximal
6	C₆₇×#1986	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1888	C₂	1	2	yes	maximal
8	C₆₇×#1886	C₂	1	2	yes	maximal, complemented
9	C₄×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	Q₈²×C₆₇	—	2	2	no	—
11	#557×C₆₇	—	2	2	no	—
12	#557×C₆₇	—	2	4	no	—
13	#544×C₆₇	—	2	2	no	—
14	#544×C₆₇	C₂²	1	2	yes	—
15	#533×C₆₇	—	2	2	no	—
16	#533×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	#519×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	#489×C₆₇	—	2	2	no	—
23	#489×C₆₇	—	2	2	no	—
24	#488×C₆₇	—	2	4	no	—
25	#419×C₆₇	C₂²	1	4	yes	—
26	#418×C₆₇	C₂²	1	4	yes	—
27	C₂×#101×C₆₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
29	#396×C₆₇	—	2	2	no	—
30	#391×C₆₇	C₂²	1	1	yes	characteristic
31	#390×C₆₇	C₂²	1	1	yes	characteristic
32	#388×C₆₇	—	2	2	no	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	#343×C₆₇	C₂²	1	2	yes	—
35	#339×C₆₇	C₂²	1	4	yes	—
36	#339×C₆₇	C₂²	1	4	yes	—
37	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
41	C₄⋊Q₈×C₆₇	—	2	2	no	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	C₄⋊Q₈×C₆₇	—	2	8	no	—
44	C₄⋊Q₈×C₆₇	—	2	4	no	—
45	C₄⋊Q₈×C₆₇	—	2	4	no	—
46	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
47	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
48	#125×C₆₇	—	2	4	no	—
49	#125×C₆₇	—	2	8	no	—
50	#125×C₆₇	—	2	4	no	—
51	#125×C₆₇	D₄	1	4	yes	—
52	#124×C₆₇	—	2	4	no	—
53	#124×C₆₇	—	2	2	no	—
54	#124×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	—	2	2	no	—
56	C₂²⋊Q₈×C₆₇	—	2	4	no	—
57	C₂²⋊Q₈×C₆₇	—	2	2	no	—
58	C₄×Q₈×C₆₇	—	2	4	no	—
59	C₄×Q₈×C₆₇	—	2	4	no	—
60	C₄×Q₈×C₆₇	—	2	4	no	—
61	C₄×Q₈×C₆₇	—	2	4	no	—
62	C₄×Q₈×C₆₇	—	2	2	no	—
63	C₄×Q₈×C₆₇	—	2	2	no	—
64	C₄×Q₈×C₆₇	—	2	2	no	—
65	C₄×Q₈×C₆₇	—	2	2	no	—
66	#117×C₆₇	—	2	4	no	—
67	#117×C₆₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	D₄	1	2	yes	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	#103×C₆₇	—	2	4	no	—
81	#103×C₆₇	—	2	4	no	—
82	#102×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	4	no	—
84	#101×C₆₇	—	2	4	no	—
85	#97×C₆₇	—	2	2	no	—
86	C₄×C₈×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂×Q₈×C₆₇	—	2	4	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	#41	1	4	yes	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	M₁₆×C₆₇	—	2	4	no	—
111	M₁₆×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	—	4	4	no	—
125	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
126	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
127	C₂²⋊C₄×C₆₇	—	2	4	no	—
128	C₂²⋊C₄×C₆₇	—	4	4	no	—
129	C₂²⋊C₄×C₆₇	—	4	1	no	—
130	C₂²⋊C₄×C₆₇	—	4	1	no	—
131	C₂²⋊C₄×C₆₇	—	4	2	no	—
132	C₂²⋊C₄×C₆₇	—	4	4	no	—
133	C₄²×C₆₇	—	2	8	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	2	4	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	C₂×D₄	1	2	yes	—
140	C₄²×C₆₇	C₂×D₄	1	2	yes	—
141	C₂³×C₆₇	—	4	2	no	—
142	C₂³×C₆₇	—	4	2	no	—
143	C₂³×C₆₇	—	4	1	no	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	—	4	1	no	—
146	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₆₇	—	2	8	no	—
148	Q₈×C₆₇	—	4	4	no	—
149	Q₈×C₆₇	—	4	4	no	—
150	Q₈×C₆₇	—	8	2	no	—
151	Q₈×C₆₇	—	8	4	no	—
152	D₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	4	1	no	—
160	C₂×C₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
171	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#120	1	2	yes	—
173	C₂×C₄×C₆₇	#124	1	4	yes	—
174	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
175	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
176	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
177	C₈×C₆₇	—	4	4	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	8	2	no	—
181	C₂²×C₆₇	—	8	2	no	—
182	C₂²×C₆₇	—	8	1	no	—
183	C₂²×C₆₇	—	8	1	no	—
184	C₂²×C₆₇	—	8	1	no	—
185	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
186	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
187	C₂²×C₆₇	#536	1	1	yes	characteristic
188	C₄×C₆₇	—	2	4	no	—
189	C₄×C₆₇	—	2	2	no	—
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	4	2	no	—
193	C₄×C₆₇	—	4	2	no	—
194	C₄×C₆₇	—	4	4	no	—
195	C₄×C₆₇	—	4	4	no	—
196	C₄×C₆₇	—	8	1	no	—
197	C₄×C₆₇	—	8	1	no	—
198	C₄×C₆₇	#548	1	4	yes	—
199	#25921	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₆₇	—	2	1	no	—
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	—	16	1	no	—
203	C₂×C₆₇	#2436	1	1	yes	characteristic
204	C₂×C₆₇	#3018	1	1	yes	characteristic
205	C₂×C₆₇	#3020	1	1	yes	characteristic
206	#3502	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#3140	C₂×C₆₇	1	2	yes	—
208	#2545	C₂×C₆₇	1	1	yes	characteristic
209	#2028	C₂×C₆₇	1	2	yes	complemented
210	#2027	C₂×C₆₇	1	2	yes	—
211	#1986	C₂×C₆₇	1	2	yes	complemented
212	#1888	C₂×C₆₇	1	2	yes	—
213	#1886	C₂×C₆₇	1	2	yes	complemented
214	C₄×#101	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₆₇	#25921	1	1	yes	characteristic, direct product factor, Sylow
216	Q₈²	—	2	2	no	—
217	#557	—	2	2	no	—
218	#557	—	2	4	no	—
219	#544	—	2	2	no	—
220	#544	C₂²×C₆₇	1	2	yes	—
221	#533	—	2	2	no	—
222	#533	C₂²×C₆₇	1	2	yes	—
223	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
226	#519	C₂²×C₆₇	1	2	yes	—
227	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
228	#489	—	2	2	no	—
229	#489	—	2	2	no	—
230	#488	—	2	4	no	—
231	#419	C₂²×C₆₇	1	4	yes	—
232	#418	C₂²×C₆₇	1	4	yes	—
233	C₂×#101	C₂²×C₆₇	1	2	yes	—
234	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
235	#396	—	2	2	no	—
236	#391	C₂²×C₆₇	1	1	yes	characteristic
237	#390	C₂²×C₆₇	1	1	yes	characteristic
238	#388	—	2	2	no	—
239	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
240	#343	C₂²×C₆₇	1	2	yes	—
241	#339	C₂²×C₆₇	1	4	yes	—
242	#339	C₂²×C₆₇	1	4	yes	—
243	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
244	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
245	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
247	C₄⋊Q₈	—	2	2	no	—
248	C₄⋊Q₈	—	2	2	no	—
249	C₄⋊Q₈	—	2	8	no	—
250	C₄⋊Q₈	—	2	4	no	—
251	C₄⋊Q₈	—	2	4	no	—
252	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
253	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
254	#125	—	2	4	no	—
255	#125	—	2	8	no	—
256	#125	—	2	4	no	—
257	#125	D₄×C₆₇	1	4	yes	—
258	#124	—	2	4	no	—
259	#124	—	2	2	no	—
260	#124	—	2	2	no	—
261	C₂²⋊Q₈	—	2	2	no	—
262	C₂²⋊Q₈	—	2	4	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	C₄×Q₈	—	2	4	no	—
265	C₄×Q₈	—	2	4	no	—
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	2	4	no	—
268	C₄×Q₈	—	2	2	no	—
269	C₄×Q₈	—	2	2	no	—
270	C₄×Q₈	—	2	2	no	—
271	C₄×Q₈	—	2	2	no	—
272	#117	—	2	4	no	—
273	#117	C₂³×C₆₇	1	4	yes	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
277	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
278	C₂×C₂²⋊C₄	—	2	1	no	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	1	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₄²	D₄×C₆₇	1	2	yes	—
283	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
286	#103	—	2	4	no	—
287	#103	—	2	4	no	—
288	#102	—	2	4	no	—
289	#102	—	2	4	no	—
290	#101	—	2	4	no	—
291	#97	—	2	2	no	—
292	C₄×C₈	—	2	2	no	—
293	#95	—	2	1	no	—
294	#95	—	2	2	no	—
295	C₂⁴	—	2	1	no	—
296	C₂×Q₈	—	2	4	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	#41×C₆₇	1	4	yes	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
316	M₁₆	—	2	4	no	—
317	M₁₆	—	2	4	no	—
318	C₂×C₈	—	2	4	no	—
319	C₄⋊C₄	—	2	4	no	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	8	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	4	2	no	—
330	C₄⋊C₄	—	4	4	no	—
331	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
332	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
333	C₂²⋊C₄	—	2	4	no	—
334	C₂²⋊C₄	—	4	4	no	—
335	C₂²⋊C₄	—	4	1	no	—
336	C₂²⋊C₄	—	4	1	no	—
337	C₂²⋊C₄	—	4	2	no	—
338	C₂²⋊C₄	—	4	4	no	—
339	C₄²	—	2	8	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	4	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	C₂×D₄×C₆₇	1	2	yes	—
346	C₄²	C₂×D₄×C₆₇	1	2	yes	—
347	C₂³	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	1	no	—
350	C₂³	—	4	1	no	—
351	C₂³	—	4	1	no	—
352	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
353	Q₈	—	2	8	no	—
354	Q₈	—	4	4	no	—
355	Q₈	—	4	4	no	—
356	Q₈	—	8	2	no	—
357	Q₈	—	8	4	no	—
358	D₄	—	4	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
377	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
378	C₂×C₄	#120×C₆₇	1	2	yes	—
379	C₂×C₄	#124×C₆₇	1	4	yes	—
380	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
381	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
382	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
383	C₈	—	4	4	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
392	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
393	C₂²	#536×C₆₇	1	1	yes	characteristic
394	C₄	—	2	2	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	4	no	—
398	C₄	—	4	2	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	4	no	—
401	C₄	—	4	4	no	—
402	C₄	—	8	1	no	—
403	C₄	—	8	1	no	—
404	C₄	#548×C₆₇	1	4	yes	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	—	16	1	no	—
408	C₂	C₆₇×#2436	1	1	yes	characteristic
409	C₂	C₆₇×#3018	1	1	yes	characteristic
410	C₂	C₆₇×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3502	C₂	semidirect
4. C₆₇×#2028	C₂	semidirect
6. C₆₇×#1986	C₂	semidirect
8. C₆₇×#1886	C₂	semidirect
9. C₄×#101×C₆₇	C₂	semidirect
199. #25921	C₆₇	direct (N × H)
206. #3502	C₂×C₆₇	semidirect
209. #2028	C₂×C₆₇	semidirect
211. #1986	C₂×C₆₇	semidirect
213. #1886	C₂×C₆₇	semidirect
214. C₄×#101	C₂×C₆₇	semidirect
215. C₆₇	#25921	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×#3502	2
C₆₇×#3140	2
C₆₇×#2545	2
C₆₇×#2028	2
C₆₇×#2027	2
C₆₇×#1986	2
C₆₇×#1888	2
C₆₇×#1886	2
C₄×#101×C₆₇	2
#25921	67

Subgroups — C67×#25921 #30705172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25921 #30705172