Subgroups — C₆₇×#33850 #30713101

Order 17152.

686 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3287	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2217	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2189	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2186	C₂	1	2	yes	maximal, complemented
10	C₄×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	2	yes	—
13	C₂×#135×C₆₇	C₂²	1	2	yes	—
14	#550×C₆₇	—	2	1	no	—
15	#547×C₆₇	—	2	1	no	—
16	#508×C₆₇	—	2	8	no	—
17	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
19	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
20	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
23	#494×C₆₇	C₂²	1	1	yes	characteristic
24	#493×C₆₇	C₂²	1	1	yes	characteristic
25	#467×C₆₇	C₂²	1	2	yes	—
26	#466×C₆₇	C₂²	1	2	yes	—
27	#461×C₆₇	—	2	1	no	—
28	#460×C₆₇	—	2	1	no	—
29	#452×C₆₇	—	2	2	no	—
30	#449×C₆₇	—	2	2	no	—
31	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	#367×C₆₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	#358×C₆₇	C₂²	1	1	yes	characteristic
37	#358×C₆₇	C₂²	1	2	yes	—
38	#357×C₆₇	C₂²	1	1	yes	characteristic
39	#357×C₆₇	C₂²	1	2	yes	—
40	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
41	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	#135×C₆₇	—	2	4	no	—
45	#135×C₆₇	—	2	4	no	—
46	#135×C₆₇	—	4	2	no	—
47	#135×C₆₇	—	4	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×SD₁₆×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	—	2	1	no	—
54	C₂×D₈×C₆₇	C₂³	1	2	yes	—
55	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₆₇	—	2	1	no	—
58	#127×C₆₇	—	2	1	no	—
59	#127×C₆₇	—	4	1	no	—
60	#123×C₆₇	—	4	2	no	—
61	C₂²⋊Q₈×C₆₇	—	2	2	no	—
62	#121×C₆₇	—	2	2	no	—
63	#121×C₆₇	—	4	1	no	—
64	#121×C₆₇	—	4	2	no	—
65	C₄×Q₈×C₆₇	—	2	1	no	—
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	—	4	1	no	—
68	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
70	Q₃₂×C₆₇	—	2	4	no	—
71	SD₃₂×C₆₇	—	2	4	no	—
72	SD₃₂×C₆₇	—	4	2	no	—
73	SD₃₂×C₆₇	D₄	1	4	yes	complemented
74	D₁₆×C₆₇	—	2	4	no	—
75	C₂×C₁₆×C₆₇	—	2	4	no	—
76	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
77	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	C₂³	1	1	yes	characteristic
80	#107×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
82	#103×C₆₇	—	2	1	no	—
83	#103×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	1	no	—
86	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
87	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	#41×C₆₇	—	4	4	no	—
89	#41×C₆₇	—	4	4	no	—
90	#41×C₆₇	—	8	2	no	—
91	#41×C₆₇	—	8	2	no	—
92	C₂×Q₈×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	4	2	no	—
97	C₂×D₄×C₆₇	—	4	2	no	—
98	C₂×D₄×C₆₇	—	4	2	no	—
99	C₂×D₄×C₆₇	—	4	1	no	—
100	C₂×D₄×C₆₇	—	4	1	no	—
101	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	—	4	1	no	—
103	C₂²×C₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	Q₁₆×C₆₇	—	2	2	no	—
107	Q₁₆×C₆₇	—	2	1	no	—
108	Q₁₆×C₆₇	—	2	2	no	—
109	Q₁₆×C₆₇	—	8	1	no	—
110	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
111	SD₁₆×C₆₇	—	4	4	no	—
112	SD₁₆×C₆₇	—	4	4	no	—
113	D₈×C₆₇	—	2	2	no	—
114	D₈×C₆₇	—	2	1	no	—
115	D₈×C₆₇	—	2	2	no	—
116	D₈×C₆₇	—	8	1	no	—
117	D₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	4	1	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₄⋊C₄×C₆₇	—	4	1	no	—
128	C₄⋊C₄×C₆₇	—	4	1	no	—
129	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
130	C₂²⋊C₄×C₆₇	—	4	2	no	—
131	C₂²⋊C₄×C₆₇	—	4	1	no	—
132	C₂²⋊C₄×C₆₇	—	4	2	no	—
133	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
134	C₄²×C₆₇	—	4	1	no	—
135	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₁₆×C₆₇	—	2	2	no	—
137	C₁₆×C₆₇	—	2	1	no	—
138	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
139	C₂³×C₆₇	—	4	1	no	—
140	C₂³×C₆₇	—	4	2	no	—
141	C₂³×C₆₇	C₂×D₈	1	2	yes	—
142	Q₈×C₆₇	—	2	2	no	—
143	Q₈×C₆₇	—	4	1	no	—
144	Q₈×C₆₇	—	4	2	no	—
145	Q₈×C₆₇	—	4	2	no	—
146	D₄×C₆₇	—	2	2	no	—
147	D₄×C₆₇	—	2	2	no	—
148	D₄×C₆₇	—	4	2	no	—
149	D₄×C₆₇	—	4	1	no	—
150	D₄×C₆₇	—	4	2	no	—
151	D₄×C₆₇	—	8	4	no	—
152	D₄×C₆₇	—	8	4	no	—
153	D₄×C₆₇	—	8	2	no	—
154	D₄×C₆₇	—	16	1	no	—
155	D₄×C₆₇	—	16	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	4	1	no	—
159	C₂×C₄×C₆₇	—	4	1	no	—
160	C₂×C₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	8	4	no	—
163	C₂×C₄×C₆₇	—	8	4	no	—
164	C₂×C₄×C₆₇	—	8	2	no	—
165	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
167	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	C₈×C₆₇	—	2	2	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂²×C₆₇	—	2	2	no	—
174	C₂²×C₆₇	—	2	2	no	—
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	8	1	no	—
178	C₂²×C₆₇	—	8	1	no	—
179	C₂²×C₆₇	—	8	2	no	—
180	C₂²×C₆₇	—	8	2	no	—
181	C₂²×C₆₇	—	8	2	no	—
182	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
183	C₄×C₆₇	—	2	1	no	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	4	2	no	—
186	C₄×C₆₇	—	8	1	no	—
187	C₄×C₆₇	—	8	1	no	—
188	C₄×C₆₇	—	8	2	no	—
189	C₄×C₆₇	#508	1	2	yes	complemented
190	C₄×C₆₇	D₄²	1	1	yes	characteristic
191	C₄×C₆₇	#576	1	1	yes	characteristic
192	#33850	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₆₇	—	4	2	no	—
194	C₂×C₆₇	—	8	2	no	—
195	C₂×C₆₇	—	16	2	no	—
196	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
198	C₂×C₆₇	#3416	1	1	yes	characteristic
199	C₂×#508	C₂×C₆₇	1	2	yes	complemented
200	#3287	C₂×C₆₇	1	1	yes	characteristic, complemented
201	#3283	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#2248	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#2217	C₂×C₆₇	1	2	yes	complemented
204	#2209	C₂×C₆₇	1	1	yes	characteristic, complemented
205	#2206	C₂×C₆₇	1	1	yes	characteristic, complemented
206	#2189	C₂×C₆₇	1	2	yes	complemented
207	#2186	C₂×C₆₇	1	2	yes	complemented
208	C₄×SD₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
209	D₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₆₇	#33850	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×#135	C₂²×C₆₇	1	2	yes	—
212	C₂×#135	C₂²×C₆₇	1	2	yes	—
213	#550	—	2	1	no	—
214	#547	—	2	1	no	—
215	#508	—	2	8	no	—
216	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
217	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
218	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
219	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic
220	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
221	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
222	#494	C₂²×C₆₇	1	1	yes	characteristic
223	#493	C₂²×C₆₇	1	1	yes	characteristic
224	#467	C₂²×C₆₇	1	2	yes	—
225	#466	C₂²×C₆₇	1	2	yes	—
226	#461	—	2	1	no	—
227	#460	—	2	1	no	—
228	#452	—	2	2	no	—
229	#449	—	2	2	no	—
230	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
231	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
233	#367	C₂²×C₆₇	1	1	yes	characteristic
234	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
235	#358	C₂²×C₆₇	1	1	yes	characteristic
236	#358	C₂²×C₆₇	1	2	yes	—
237	#357	C₂²×C₆₇	1	1	yes	characteristic
238	#357	C₂²×C₆₇	1	2	yes	—
239	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
240	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×#41	—	2	2	no	—
242	C₂×#41	—	2	2	no	—
243	#135	—	2	4	no	—
244	#135	—	2	4	no	—
245	#135	—	4	2	no	—
246	#135	—	4	2	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
249	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
250	C₂×SD₁₆	—	2	2	no	—
251	C₂×SD₁₆	—	2	2	no	—
252	C₂×D₈	—	2	1	no	—
253	C₂×D₈	C₂³×C₆₇	1	2	yes	—
254	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
255	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
256	C₄⋊Q₈	—	2	1	no	—
257	#127	—	2	1	no	—
258	#127	—	4	1	no	—
259	#123	—	4	2	no	—
260	C₂²⋊Q₈	—	2	2	no	—
261	#121	—	2	2	no	—
262	#121	—	4	1	no	—
263	#121	—	4	2	no	—
264	C₄×Q₈	—	2	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	—	4	1	no	—
267	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
268	C₂×C₄⋊C₄	—	4	1	no	—
269	Q₃₂	—	2	4	no	—
270	SD₃₂	—	2	4	no	—
271	SD₃₂	—	4	2	no	—
272	SD₃₂	D₄×C₆₇	1	4	yes	complemented
273	D₁₆	—	2	4	no	—
274	C₂×C₁₆	—	2	4	no	—
275	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
276	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
277	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
278	#107	C₂³×C₆₇	1	1	yes	characteristic
279	#107	C₂³×C₆₇	1	1	yes	characteristic
280	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
281	#103	—	2	1	no	—
282	#103	—	2	2	no	—
283	#102	—	2	1	no	—
284	#102	—	2	2	no	—
285	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
286	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
287	#41	—	4	4	no	—
288	#41	—	4	4	no	—
289	#41	—	8	2	no	—
290	#41	—	8	2	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	4	2	no	—
296	C₂×D₄	—	4	2	no	—
297	C₂×D₄	—	4	2	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
305	Q₁₆	—	2	2	no	—
306	Q₁₆	—	2	1	no	—
307	Q₁₆	—	2	2	no	—
308	Q₁₆	—	8	1	no	—
309	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
310	SD₁₆	—	4	4	no	—
311	SD₁₆	—	4	4	no	—
312	D₈	—	2	2	no	—
313	D₈	—	2	1	no	—
314	D₈	—	2	2	no	—
315	D₈	—	8	1	no	—
316	D₈	C₂×D₄×C₆₇	1	2	yes	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
321	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
329	C₂²⋊C₄	—	4	2	no	—
330	C₂²⋊C₄	—	4	1	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
333	C₄²	—	4	1	no	—
334	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₁₆	—	2	2	no	—
336	C₁₆	—	2	1	no	—
337	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
338	C₂³	—	4	1	no	—
339	C₂³	—	4	2	no	—
340	C₂³	C₂×D₈×C₆₇	1	2	yes	—
341	Q₈	—	2	2	no	—
342	Q₈	—	4	1	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	D₄	—	2	2	no	—
346	D₄	—	2	2	no	—
347	D₄	—	4	2	no	—
348	D₄	—	4	1	no	—
349	D₄	—	4	2	no	—
350	D₄	—	8	4	no	—
351	D₄	—	8	4	no	—
352	D₄	—	8	2	no	—
353	D₄	—	16	1	no	—
354	D₄	—	16	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	8	4	no	—
362	C₂×C₄	—	8	4	no	—
363	C₂×C₄	—	8	2	no	—
364	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
366	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	1	no	—
370	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
371	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
372	C₂²	—	2	2	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	8	1	no	—
377	C₂²	—	8	1	no	—
378	C₂²	—	8	2	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	8	2	no	—
381	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
382	C₄	—	2	2	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	8	1	no	—
387	C₄	—	8	1	no	—
388	C₄	#508×C₆₇	1	2	yes	complemented
389	C₄	D₄²×C₆₇	1	1	yes	characteristic
390	C₄	#576×C₆₇	1	1	yes	characteristic
391	C₂	—	4	2	no	—
392	C₂	—	8	2	no	—
393	C₂	—	16	2	no	—
394	C₂	D₄×D₈×C₆₇	1	1	yes	characteristic
395	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
396	C₂	C₆₇×#3416	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 ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₆₇	C₂	semidirect
2. C₆₇×#3287	C₂	semidirect
3. C₆₇×#3283	C₂	semidirect
4. C₆₇×#2248	C₂	semidirect
5. C₆₇×#2217	C₂	semidirect
6. C₆₇×#2209	C₂	semidirect
7. C₆₇×#2206	C₂	semidirect
8. C₆₇×#2189	C₂	semidirect
9. C₆₇×#2186	C₂	semidirect
10. C₄×SD₃₂×C₆₇	C₂	semidirect
11. D₄×C₁₆×C₆₇	C₂	semidirect
73. SD₃₂×C₆₇	D₄	semidirect
189. C₄×C₆₇	#508	semidirect
192. #33850	C₆₇	direct (N × H)
199. C₂×#508	C₂×C₆₇	semidirect
200. #3287	C₂×C₆₇	semidirect
201. #3283	C₂×C₆₇	semidirect
202. #2248	C₂×C₆₇	semidirect
203. #2217	C₂×C₆₇	semidirect
204. #2209	C₂×C₆₇	semidirect
205. #2206	C₂×C₆₇	semidirect
206. #2189	C₂×C₆₇	semidirect
207. #2186	C₂×C₆₇	semidirect
208. C₄×SD₃₂	C₂×C₆₇	semidirect
209. D₄×C₁₆	C₂×C₆₇	semidirect
210. C₆₇	#33850	direct (N × H)
272. SD₃₂	D₄×C₆₇	semidirect
388. C₄	#508×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₆₇	2
C₆₇×#3287	2
C₆₇×#3283	2
C₆₇×#2248	2
C₆₇×#2217	2
C₆₇×#2209	2
C₆₇×#2206	2
C₆₇×#2189	2
C₆₇×#2186	2
C₄×SD₃₂×C₆₇	2
D₄×C₁₆×C₆₇	2
#33850	67

Subgroups — C67×#33850 #30713101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33850 #30713101