Subgroups — C₆₇×#33675 #30712926

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2234	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2224	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2207	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2186	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2177	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2175	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
13	#532×C₆₇	—	2	1	no	—
14	C₄×#41×C₆₇	—	2	1	no	—
15	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
17	#500×C₆₇	C₂²	1	1	yes	characteristic
18	#493×C₆₇	C₂²	1	1	yes	characteristic
19	#486×C₆₇	—	2	1	no	—
20	#466×C₆₇	—	2	1	no	—
21	#466×C₆₇	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×SD₁₆×C₆₇	—	2	1	no	—
25	C₄×SD₁₆×C₆₇	—	2	1	no	—
26	C₄×D₈×C₆₇	—	2	1	no	—
27	C₄×D₈×C₆₇	C₂²	1	2	yes	—
28	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
30	#427×C₆₇	C₂²	1	1	yes	characteristic
31	#416×C₆₇	—	2	1	no	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#367×C₆₇	C₂²	1	2	yes	—
34	#366×C₆₇	C₂²	1	2	yes	—
35	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
37	#358×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	2	yes	—
39	#357×C₆₇	C₂²	1	2	yes	—
40	#357×C₆₇	C₂²	1	2	yes	—
41	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
42	#346×C₆₇	C₂²	1	2	yes	—
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	4	1	no	—
45	#135×C₆₇	—	2	2	no	—
46	#135×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×SD₁₆×C₆₇	—	2	1	no	—
49	C₂×D₈×C₆₇	—	2	1	no	—
50	C₂×D₈×C₆₇	C₂³	1	2	yes	—
51	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₆₇	—	2	1	no	—
55	#127×C₆₇	—	2	1	no	—
56	#124×C₆₇	—	4	1	no	—
57	#121×C₆₇	—	2	2	no	—
58	#121×C₆₇	—	4	1	no	—
59	C₄×Q₈×C₆₇	—	2	1	no	—
60	C₄×Q₈×C₆₇	—	2	1	no	—
61	C₄×D₄×C₆₇	—	2	1	no	—
62	C₄×D₄×C₆₇	—	2	1	no	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₄×D₄×C₆₇	—	4	1	no	—
65	C₄×D₄×C₆₇	—	4	1	no	—
66	C₄×D₄×C₆₇	—	4	1	no	—
67	#117×C₆₇	—	2	1	no	—
68	#117×C₆₇	—	4	1	no	—
69	C₂×C₄²×C₆₇	—	4	1	no	—
70	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
71	SD₃₂×C₆₇	—	2	4	no	—
72	D₁₆×C₆₇	—	2	4	no	—
73	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
74	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
75	#107×C₆₇	—	2	1	no	—
76	#107×C₆₇	C₂³	1	2	yes	—
77	#107×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	C₂³	1	1	yes	characteristic
79	#106×C₆₇	—	2	1	no	—
80	#103×C₆₇	—	2	1	no	—
81	#103×C₆₇	—	2	1	no	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	1	no	—
85	#102×C₆₇	—	2	1	no	—
86	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
87	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
88	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	#41×C₆₇	—	2	4	no	—
91	#41×C₆₇	—	4	2	no	—
92	#41×C₆₇	—	8	4	no	—
93	C₂×Q₈×C₆₇	—	2	1	no	—
94	C₂×Q₈×C₆₇	—	4	1	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	4	1	no	—
98	C₂×D₄×C₆₇	—	4	2	no	—
99	C₂×D₄×C₆₇	—	4	1	no	—
100	C₂×D₄×C₆₇	—	4	1	no	—
101	C₂²×C₄×C₆₇	—	4	1	no	—
102	C₂²×C₄×C₆₇	—	4	1	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₆₇	—	2	1	no	—
109	Q₁₆×C₆₇	#41	1	2	yes	complemented
110	SD₁₆×C₆₇	—	2	4	no	—
111	D₈×C₆₇	—	2	1	no	—
112	D₈×C₆₇	—	2	2	no	—
113	D₈×C₆₇	—	2	2	no	—
114	D₈×C₆₇	—	4	2	no	—
115	D₈×C₆₇	#41	1	2	yes	complemented
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	4	1	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	C₂⁴	1	1	yes	characteristic
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	4	1	no	—
127	C₄⋊C₄×C₆₇	—	4	1	no	—
128	C₄⋊C₄×C₆₇	—	4	1	no	—
129	C₂²⋊C₄×C₆₇	—	4	1	no	—
130	C₂²⋊C₄×C₆₇	—	4	1	no	—
131	C₂²⋊C₄×C₆₇	—	4	1	no	—
132	C₂²⋊C₄×C₆₇	—	4	2	no	—
133	C₂²⋊C₄×C₆₇	—	4	2	no	—
134	C₄²×C₆₇	—	4	1	no	—
135	C₄²×C₆₇	—	4	1	no	—
136	C₄²×C₆₇	—	4	1	no	—
137	C₄²×C₆₇	D₈	1	1	yes	characteristic
138	C₄²×C₆₇	D₈	1	1	yes	characteristic
139	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₁₆×C₆₇	—	2	2	no	—
142	C₁₆×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	4	1	no	—
144	C₂³×C₆₇	—	4	2	no	—
145	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
146	Q₈×C₆₇	—	2	2	no	—
147	Q₈×C₆₇	—	4	1	no	—
148	Q₈×C₆₇	—	8	2	no	—
149	D₄×C₆₇	—	2	2	no	—
150	D₄×C₆₇	—	4	4	no	—
151	D₄×C₆₇	—	4	2	no	—
152	D₄×C₆₇	—	4	1	no	—
153	D₄×C₆₇	—	4	2	no	—
154	D₄×C₆₇	—	8	2	no	—
155	D₄×C₆₇	—	8	4	no	—
156	D₄×C₆₇	—	8	2	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	4	4	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	1	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	8	2	no	—
167	C₂×C₄×C₆₇	—	8	4	no	—
168	C₂×C₄×C₆₇	—	8	2	no	—
169	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₈×C₆₇	—	2	1	no	—
176	C₈×C₆₇	—	2	1	no	—
177	C₈×C₆₇	—	2	1	no	—
178	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	2	1	no	—
182	C₂²×C₆₇	—	2	1	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	8	1	no	—
185	C₂²×C₆₇	—	8	1	no	—
186	C₂²×C₆₇	—	8	2	no	—
187	C₂²×C₆₇	—	8	2	no	—
188	C₂²×C₆₇	—	8	2	no	—
189	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	8	1	no	—
196	C₄×C₆₇	—	8	2	no	—
197	C₄×C₆₇	—	8	1	no	—
198	C₄×C₆₇	#509	1	2	yes	—
199	C₄×C₆₇	#525	1	1	yes	characteristic
200	C₄×C₆₇	#575	1	1	yes	characteristic
201	#33675	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₆₇	—	4	1	no	—
203	C₂×C₆₇	—	8	2	no	—
204	C₂×C₆₇	—	16	2	no	—
205	C₂×C₆₇	#3100	1	1	yes	characteristic
206	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
207	C₂×C₆₇	#3414	1	1	yes	characteristic
208	#3169	C₂×C₆₇	1	1	yes	characteristic, complemented
209	C₄×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#2234	C₂×C₆₇	1	2	yes	complemented
211	#2225	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#2224	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#2207	C₂×C₆₇	1	1	yes	characteristic, complemented
214	#2206	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#2186	C₂×C₆₇	1	2	yes	complemented
216	#2177	C₂×C₆₇	1	2	yes	complemented
217	#2175	C₂×C₆₇	1	2	yes	complemented
218	#2163	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₆₇	#33675	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
221	#532	—	2	1	no	—
222	C₄×#41	—	2	1	no	—
223	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
224	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
225	#500	C₂²×C₆₇	1	1	yes	characteristic
226	#493	C₂²×C₆₇	1	1	yes	characteristic
227	#486	—	2	1	no	—
228	#466	—	2	1	no	—
229	#466	C₂²×C₆₇	1	2	yes	—
230	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
231	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
232	C₄×SD₁₆	—	2	1	no	—
233	C₄×SD₁₆	—	2	1	no	—
234	C₄×D₈	—	2	1	no	—
235	C₄×D₈	C₂²×C₆₇	1	2	yes	—
236	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
237	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
238	#427	C₂²×C₆₇	1	1	yes	characteristic
239	#416	—	2	1	no	—
240	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
241	#367	C₂²×C₆₇	1	2	yes	—
242	#366	C₂²×C₆₇	1	2	yes	—
243	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
244	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
245	#358	C₂²×C₆₇	1	2	yes	—
246	#357	C₂²×C₆₇	1	2	yes	—
247	#357	C₂²×C₆₇	1	2	yes	—
248	#357	C₂²×C₆₇	1	2	yes	—
249	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
250	#346	C₂²×C₆₇	1	2	yes	—
251	C₂×#41	—	2	1	no	—
252	C₂×#41	—	4	1	no	—
253	#135	—	2	2	no	—
254	#135	—	2	2	no	—
255	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
256	C₂×SD₁₆	—	2	1	no	—
257	C₂×D₈	—	2	1	no	—
258	C₂×D₈	C₂³×C₆₇	1	2	yes	—
259	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
260	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₄⋊Q₈	—	2	1	no	—
263	#127	—	2	1	no	—
264	#124	—	4	1	no	—
265	#121	—	2	2	no	—
266	#121	—	4	1	no	—
267	C₄×Q₈	—	2	1	no	—
268	C₄×Q₈	—	2	1	no	—
269	C₄×D₄	—	2	1	no	—
270	C₄×D₄	—	2	1	no	—
271	C₄×D₄	—	2	2	no	—
272	C₄×D₄	—	4	1	no	—
273	C₄×D₄	—	4	1	no	—
274	C₄×D₄	—	4	1	no	—
275	#117	—	2	1	no	—
276	#117	—	4	1	no	—
277	C₂×C₄²	—	4	1	no	—
278	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
279	SD₃₂	—	2	4	no	—
280	D₁₆	—	2	4	no	—
281	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
282	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
283	#107	—	2	1	no	—
284	#107	C₂³×C₆₇	1	1	yes	characteristic
285	#107	C₂³×C₆₇	1	1	yes	characteristic
286	#107	C₂³×C₆₇	1	2	yes	—
287	#106	—	2	1	no	—
288	#103	—	2	1	no	—
289	#103	—	2	1	no	—
290	#102	—	2	1	no	—
291	#102	—	2	1	no	—
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
295	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
296	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
298	#41	—	2	4	no	—
299	#41	—	4	2	no	—
300	#41	—	8	4	no	—
301	C₂×Q₈	—	2	1	no	—
302	C₂×Q₈	—	4	1	no	—
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	4	1	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	—	4	1	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
316	Q₁₆	—	2	1	no	—
317	Q₁₆	#41×C₆₇	1	2	yes	complemented
318	SD₁₆	—	2	4	no	—
319	D₈	—	2	1	no	—
320	D₈	—	2	2	no	—
321	D₈	—	2	2	no	—
322	D₈	—	4	2	no	—
323	D₈	#41×C₆₇	1	2	yes	complemented
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	4	1	no	—
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	C₂⁴×C₆₇	1	1	yes	characteristic
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	4	1	no	—
337	C₂²⋊C₄	—	4	1	no	—
338	C₂²⋊C₄	—	4	1	no	—
339	C₂²⋊C₄	—	4	1	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₄²	—	4	1	no	—
343	C₄²	—	4	1	no	—
344	C₄²	—	4	1	no	—
345	C₄²	D₈×C₆₇	1	1	yes	characteristic
346	C₄²	D₈×C₆₇	1	1	yes	characteristic
347	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₁₆	—	2	2	no	—
350	C₁₆	—	2	2	no	—
351	C₂³	—	4	1	no	—
352	C₂³	—	4	2	no	—
353	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
354	Q₈	—	2	2	no	—
355	Q₈	—	4	1	no	—
356	Q₈	—	8	2	no	—
357	D₄	—	2	2	no	—
358	D₄	—	4	4	no	—
359	D₄	—	4	2	no	—
360	D₄	—	4	1	no	—
361	D₄	—	4	2	no	—
362	D₄	—	8	2	no	—
363	D₄	—	8	4	no	—
364	D₄	—	8	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	—	8	2	no	—
376	C₂×C₄	—	8	2	no	—
377	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
378	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
380	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
382	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
383	C₈	—	2	1	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	1	no	—
386	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
387	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	8	1	no	—
393	C₂²	—	8	1	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	—	8	2	no	—
397	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	4	2	no	—
403	C₄	—	8	2	no	—
404	C₄	—	8	1	no	—
405	C₄	—	8	1	no	—
406	C₄	#509×C₆₇	1	2	yes	—
407	C₄	#525×C₆₇	1	1	yes	characteristic
408	C₄	#575×C₆₇	1	1	yes	characteristic
409	C₂	—	4	1	no	—
410	C₂	—	8	2	no	—
411	C₂	—	16	2	no	—
412	C₂	C₆₇×#3100	1	1	yes	characteristic
413	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
414	C₂	C₆₇×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3169	C₂	semidirect
2. C₄×#135×C₆₇	C₂	semidirect
3. C₆₇×#2234	C₂	semidirect
4. C₆₇×#2225	C₂	semidirect
5. C₆₇×#2224	C₂	semidirect
6. C₆₇×#2207	C₂	semidirect
7. C₆₇×#2206	C₂	semidirect
8. C₆₇×#2186	C₂	semidirect
9. C₆₇×#2177	C₂	semidirect
10. C₆₇×#2175	C₂	semidirect
11. C₆₇×#2163	C₂	semidirect
109. Q₁₆×C₆₇	#41	semidirect
115. D₈×C₆₇	#41	semidirect
201. #33675	C₆₇	direct (N × H)
208. #3169	C₂×C₆₇	semidirect
209. C₄×#135	C₂×C₆₇	semidirect
210. #2234	C₂×C₆₇	semidirect
211. #2225	C₂×C₆₇	semidirect
212. #2224	C₂×C₆₇	semidirect
213. #2207	C₂×C₆₇	semidirect
214. #2206	C₂×C₆₇	semidirect
215. #2186	C₂×C₆₇	semidirect
216. #2177	C₂×C₆₇	semidirect
217. #2175	C₂×C₆₇	semidirect
218. #2163	C₂×C₆₇	semidirect
219. C₆₇	#33675	direct (N × H)
317. Q₁₆	#41×C₆₇	semidirect
323. D₈	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3169	2
C₄×#135×C₆₇	2
C₆₇×#2234	2
C₆₇×#2225	2
C₆₇×#2224	2
C₆₇×#2207	2
C₆₇×#2206	2
C₆₇×#2186	2
C₆₇×#2177	2
C₆₇×#2175	2
C₆₇×#2163	2
#33675	67

Subgroups — C67×#33675 #30712926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33675 #30712926